Optimal. Leaf size=334 \[ -\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{10 a^2 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{30 a^2}+\frac {3 c \sqrt {a^2 c x^2+c}}{20 a^2}+\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a}-\frac {3 c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{20 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4930, 4878, 4890, 4886} \[ -\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{10 a^2 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{30 a^2}+\frac {3 c \sqrt {a^2 c x^2+c}}{20 a^2}+\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a}-\frac {3 c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{20 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4878
Rule 4886
Rule 4890
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {2 \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx}{5 a}\\ &=\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {(3 c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx}{10 a}\\ &=\frac {3 c \sqrt {c+a^2 c x^2}}{20 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}\\ &=\frac {3 c \sqrt {c+a^2 c x^2}}{20 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 a \sqrt {c+a^2 c x^2}}\\ &=\frac {3 c \sqrt {c+a^2 c x^2}}{20 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{10 a^2 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{20 a^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.31, size = 601, normalized size = 1.80 \[ \frac {c \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \left (80 \left (-\frac {4 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac {4 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac {3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+\frac {3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+2 \cos \left (2 \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+2\right )-\left (a^2 x^2+1\right ) \left (-\frac {176 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+\frac {176 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}-\frac {110 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+\frac {110 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-32 \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-22 \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )+160 \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )+72 \cos \left (2 \tan ^{-1}(a x)\right )+22 \cos \left (4 \tan ^{-1}(a x)\right )-55 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )-11 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (5 \tan ^{-1}(a x)\right )+55 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+11 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (5 \tan ^{-1}(a x)\right )+50\right )\right )}{960 a^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{3} + c x\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.06, size = 237, normalized size = 0.71 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 \arctan \left (a x \right )^{2} x^{4} a^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+24 \arctan \left (a x \right )^{2} x^{2} a^{2}+2 a^{2} x^{2}-15 \arctan \left (a x \right ) x a +12 \arctan \left (a x \right )^{2}+11\right )}{60 a^{2}}+\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{20 a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x \arctan \left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________