Integrand size = 21, antiderivative size = 438 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \]
-1/6*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a+1/4*x*(a^2*c*x^2+c)^(3/2)*arctan(a* x)^2+5/6*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a-3/4*I*c^2*arct an((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a/(a^2*c*x ^2+c)^(1/2)+3/4*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/(a^2*c*x^2+c)^(1/2)-3/4*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/(a^2*c*x^2+c)^(1/2)-3/4 *c^2*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c* x^2+c)^(1/2)+3/4*c^2*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^ (1/2)/a/(a^2*c*x^2+c)^(1/2)+1/12*c*x*(a^2*c*x^2+c)^(1/2)-3/4*c*arctan(a*x) *(a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)
Time = 0.83 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (2 a x \sqrt {1+a^2 x^2}+2 a^3 x^3 \sqrt {1+a^2 x^2}-94 \sqrt {1+a^2 x^2} \arctan (a x)+2 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+69 a x \sqrt {1+a^2 x^2} \arctan (a x)^2+21 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)^2-72 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+80 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+6 \arctan (a x) \cos (3 \arctan (a x))+12 a^2 x^2 \arctan (a x) \cos (3 \arctan (a x))+6 a^4 x^4 \arctan (a x) \cos (3 \arctan (a x))+72 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-72 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-72 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+72 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+2 \sin (3 \arctan (a x))+4 a^2 x^2 \sin (3 \arctan (a x))+2 a^4 x^4 \sin (3 \arctan (a x))-3 \arctan (a x)^2 \sin (3 \arctan (a x))-6 a^2 x^2 \arctan (a x)^2 \sin (3 \arctan (a x))-3 a^4 x^4 \arctan (a x)^2 \sin (3 \arctan (a x))\right )}{96 a \sqrt {1+a^2 x^2}} \]
(c*Sqrt[c + a^2*c*x^2]*(2*a*x*Sqrt[1 + a^2*x^2] + 2*a^3*x^3*Sqrt[1 + a^2*x ^2] - 94*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 2*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTa n[a*x] + 69*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 21*a^3*x^3*Sqrt[1 + a^2* x^2]*ArcTan[a*x]^2 - (72*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 80*A rcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 12*a^ 2*x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 6*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[ a*x]] + (72*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (72*I)*Arc Tan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 72*PolyLog[3, (-I)*E^(I*ArcTan[ a*x])] + 72*PolyLog[3, I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]] + 4*a^2 *x^2*Sin[3*ArcTan[a*x]] + 2*a^4*x^4*Sin[3*ArcTan[a*x]] - 3*ArcTan[a*x]^2*S in[3*ArcTan[a*x]] - 6*a^2*x^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3*a^4*x^4 *ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(96*a*Sqrt[1 + a^2*x^2])
Time = 1.24 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.79, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {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.
| \(\displaystyle \int \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\) |
-1/6*((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/a + (x*(c + a^2*c*x^2)^(3/2)*ArcT an[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*S qrt[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*Arc Tan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLo g[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
3.4.18.3.1 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[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 = 1.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{3} \arctan \left (a x \right )^{2} x^{3}-4 a^{2} \arctan \left (a x \right ) x^{2}+15 a \arctan \left (a x \right )^{2} x +2 a x -22 \arctan \left (a x \right )\right )}{24 a}+\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (9 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-40 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a \sqrt {a^{2} x^{2}+1}}\) | \(304\) |
1/24*c/a*(c*(a*x-I)*(I+a*x))^(1/2)*(6*a^3*arctan(a*x)^2*x^3-4*a^2*arctan(a *x)*x^2+15*a*arctan(a*x)^2*x+2*a*x-22*arctan(a*x))+1/24*I*c*(c*(a*x-I)*(I+ a*x))^(1/2)*(9*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-9*I*arc tan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+18*arctan(a*x)*polylog(2,-I *(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^ 2+1)^(1/2))+18*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*I*polylog(3, I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-40*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2)))/a/( a^2*x^2+1)^(1/2)
\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2} \,d x } \]
\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2} \,d x } \]
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]