Integrand size = 22, antiderivative size = 477 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=-\frac {c x \sqrt {c+a^2 c x^2}}{20 a}+\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)}{20 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{10 a^2}-\frac {9 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{40 a}-\frac {3 x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{20 a}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{5 a^2 c}-\frac {c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^2}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 a^2 \sqrt {c+a^2 c x^2}} \]
1/10*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a^2-3/20*x*(a^2*c*x^2+c)^(3/2)*arctan (a*x)^2/a+1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/a^2/c-1/2*c^(3/2)*arctanh( a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^2+9/20*I*c^2*arctan((1+I*a*x)/(a^2*x^2+ 1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^2/(a^2*c*x^2+c)^(1/2)-9/20*I*c ^2*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2) /a^2/(a^2*c*x^2+c)^(1/2)+9/20*I*c^2*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2 *x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^2/(a^2*c*x^2+c)^(1/2)+9/20*c^2*polylog( 3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^2/(a^2*c*x^2+c)^(1/2 )-9/20*c^2*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^2/ (a^2*c*x^2+c)^(1/2)-1/20*c*x*(a^2*c*x^2+c)^(1/2)/a+9/20*c*arctan(a*x)*(a^2 *c*x^2+c)^(1/2)/a^2-9/40*c*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a
Time = 2.70 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.92 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (960 \left (i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-\text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+48 \left (-11 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+10 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+11 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-11 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-11 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+11 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+80 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \left (6+4 \arctan (a x)^2+6 \cos (2 \arctan (a x))-3 \arctan (a x) \sin (2 \arctan (a x))\right )-\left (1+a^2 x^2\right )^{5/2} \left (\frac {48 a x}{\left (1+a^2 x^2\right )^2}+32 \arctan (a x)^3 (-1+5 \cos (2 \arctan (a x)))+6 \arctan (a x) (25+36 \cos (2 \arctan (a x))+11 \cos (4 \arctan (a x)))+\arctan (a x)^2 (6 \sin (2 \arctan (a x))-33 \sin (4 \arctan (a x)))\right )\right )}{960 a^2 \sqrt {1+a^2 x^2}} \]
(c*Sqrt[c + a^2*c*x^2]*(960*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - A rcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTa n[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I) *E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]) + 48*((-11*I)*ArcTa n[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x ]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 80*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] *(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[ a*x]]) - (1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3* (-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*Ar cTan[a*x]]))))/(960*a^2*Sqrt[1 + a^2*x^2])
Time = 1.44 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.80, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5465, 5415, 211, 224, 219, 5415, 224, 219, 5425, 5423, 3042, 4669, 3011, 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.
| \(\displaystyle \int x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx}{5 a}\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\) |
((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/(5*a^2*c) - (3*(-1/6*((c + a^2*c*x^2 )^(3/2)*ArcTan[a*x])/a + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 + (c*(( x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x ^2]])/(2*a)))/6 + (3*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])]*Arc Tan[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*ArcTan[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])))/4)) /(5*a)
3.5.22.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((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[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]
Time = 3.63 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (8 a^{4} \arctan \left (a x \right )^{3} x^{4}-6 a^{3} \arctan \left (a x \right )^{2} x^{3}+16 \arctan \left (a x \right )^{3} x^{2} a^{2}+4 a^{2} \arctan \left (a x \right ) x^{2}-15 a \arctan \left (a x \right )^{2} x +8 \arctan \left (a x \right )^{3}-2 a x +22 \arctan \left (a x \right )\right )}{40 a^{2}}+\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(421\) |
1/40*c/a^2*(c*(a*x-I)*(I+a*x))^(1/2)*(8*a^4*arctan(a*x)^3*x^4-6*a^3*arctan (a*x)^2*x^3+16*arctan(a*x)^3*x^2*a^2+4*a^2*arctan(a*x)*x^2-15*a*arctan(a*x )^2*x+8*arctan(a*x)^3-2*a*x+22*arctan(a*x))+3/40*c*(c*(a*x-I)*(I+a*x))^(1/ 2)*(-I*arctan(a*x)^3+3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6 *I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(3,-I*(1 +I*a*x)/(a^2*x^2+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+3/40*c*(c*(a*x-I)*(I+a*x ))^(1/2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/ 2))+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I *(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+I*c/a^2*(c*(a*x-I)*(I +a*x))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)
\[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
\[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x \arctan \left (a x\right )^{3} \,d x } \]
Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]