Integrand size = 24, antiderivative size = 726 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=c \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{2} a c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {7 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+c \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:
c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)-1/2*a*c*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x )^2+3*I*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+ 1)^(1/2))/(a^2*c*x^2+c)^(1/2)+c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3+1/3*(a^2 *c*x^2+c)^(3/2)*arctan(a*x)^3-2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^3*arctan h((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-c^(3/2)*arctanh(a*c^(1/ 2)*x/(a^2*c*x^2+c)^(1/2))+7*I*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2, I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-7*I*c^2*(a^2*x^2+1)^(1/ 2)*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/ 2)-3*I*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1) ^(1/2))/(a^2*c*x^2+c)^(1/2)+7*I*c^2*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^ 2*x^2+1)^(1/2))*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2)-6*c^2*(a^2*x^2+1)^(1/2)* arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+7* c^2*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2 +c)^(1/2)-7*c^2*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)) /(a^2*c*x^2+c)^(1/2)+6*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(3,(1+I*a* x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*I*c^2*(a^2*x^2+1)^(1/2)*polylo g(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-6*I*c^2*(a^2*x^2+1)^( 1/2)*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)
Time = 1.02 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (-3 i \pi ^4-24 \coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+\frac {24 \arctan (a x)}{\sqrt {1+a^2 x^2}}+\frac {24 a^2 x^2 \arctan (a x)}{\sqrt {1+a^2 x^2}}-\frac {12 a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}-\frac {12 a^3 x^3 \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+24 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+\frac {32 \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+\frac {40 a^2 x^2 \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+\frac {8 a^4 x^4 \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+6 i \arctan (a x)^4+24 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-72 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+72 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-24 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+72 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+72 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-168 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+168 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+144 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-144 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+168 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-168 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-144 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-144 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )\right )}{24 \sqrt {1+a^2 x^2}} \] Input:
Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x,x]
Output:
(c*Sqrt[c + a^2*c*x^2]*((-3*I)*Pi^4 - 24*ArcCoth[(a*x)/Sqrt[1 + a^2*x^2]] + (24*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + (24*a^2*x^2*ArcTan[a*x])/Sqrt[1 + a ^2*x^2] - (12*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - (12*a^3*x^3*ArcTan[a* x]^2)/Sqrt[1 + a^2*x^2] + (24*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + (32*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (40*a^2*x^2*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (8*a^4*x^4*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (6*I)*ArcTan[a* x]^4 + 24*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 72*ArcTan[a*x]^2*L og[1 - I*E^(I*ArcTan[a*x])] + 72*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x]) ] - 24*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (72*I)*ArcTan[a*x]^2*Pol yLog[2, E^((-I)*ArcTan[a*x])] + (72*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcT an[a*x])] - (168*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (168* I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 144*ArcTan[a*x]*PolyLog[3 , E^((-I)*ArcTan[a*x])] - 144*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 168*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 168*PolyLog[3, I*E^(I*ArcTan[a*x ])] - (144*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (144*I)*PolyLog[4, -E^(I* ArcTan[a*x])]))/(24*Sqrt[1 + a^2*x^2])
Time = 6.27 (sec) , antiderivative size = 629, normalized size of antiderivative = 0.87, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5485, 5465, 5415, 224, 219, 5425, 5423, 3042, 4669, 3011, 2720, 5485, 5465, 5425, 5423, 3042, 4669, 3011, 2720, 5493, 5491, 3042, 4671, 3011, 7143, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 5415 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 5425 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 5423 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle c \left (a^2 c \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 5425 |
\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 5423 |
\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 5491 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^3}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
Input:
Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x,x]
Output:
a^2*c*(((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*a^2*c) - (-((Sqrt[c + a^2* c*x^2]*ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + (Sqrt[c ]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (c*Sqrt[1 + a^2*x^2]*((- 2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcT an[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]) ))/(2*a*Sqrt[c + a^2*c*x^2]))/a) + c*(a^2*c*((Sqrt[c + a^2*c*x^2]*ArcTan[a *x]^3)/(a^2*c) - (3*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*Ar cTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLo g[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[ a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])])))/(a^2*Sqrt[c + a^2*c*x^2])) + ( c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])] + 3*(I*Ar cTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyL og[3, -E^(I*ArcTan[a*x])] + PolyLog[4, -E^(I*ArcTan[a*x])])) - 3*(I*ArcTan [a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])] + PolyLog[4, E^(I*ArcTan[a*x])]))))/Sqrt[c + a^2*c*x^2 ])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt Q[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan[c*x])^ p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & & IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2] ), x_Symbol] :> Simp[1/Sqrt[d] Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTa n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 5.58 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 \arctan \left (a x \right )^{2} x^{2} a^{2}-3 \arctan \left (a x \right ) a x +8 \arctan \left (a x \right )^{2}+6\right )}{6}-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+7 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+14 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-14 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+12 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-12 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-14 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+14 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\) | \(511\) |
Input:
int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x,x,method=_RETURNVERBOSE)
Output:
1/6*c*(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)*(2*arctan(a*x)^2*x^2*a^2-3*arc tan(a*x)*a*x+8*arctan(a*x)^2+6)-1/2*c*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan( a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a ^2*x^2+1)^(1/2))-6*I*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2)) +6*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*arctan(a*x)^2* ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+7*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2* x^2+1)^(1/2))+14*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1 4*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*arctan(a*x)*po lylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*arctan(a*x)*polylog(3,(1+I*a*x)/( a^2*x^2+1)^(1/2))+12*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*I*polylo g(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))-1 4*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+14*polylog(3,I*(1+I*a*x)/(a^2* x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x,x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \] Input:
integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**3/x,x)
Output:
Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)**3/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x, x)
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x} \,d x \] Input:
int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x,x)
Output:
int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx=\sqrt {c}\, c \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}{x}d x +\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x d x \right ) a^{2}\right ) \] Input:
int((a^2*c*x^2+c)^(3/2)*atan(a*x)^3/x,x)
Output:
sqrt(c)*c*(int((sqrt(a**2*x**2 + 1)*atan(a*x)**3)/x,x) + int(sqrt(a**2*x** 2 + 1)*atan(a*x)**3*x,x)*a**2)