Optimal. Leaf size=305 \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac {2 i \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {4 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2 x \tan ^{-1}(a x)}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.40, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4964, 4930, 4890, 4886, 4894} \[ -\frac {2 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac {2}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {4 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2 x \tan ^{-1}(a x)}{a^3 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4886
Rule 4890
Rule 4894
Rule 4930
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=\frac {\tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac {2 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c}\\ &=-\frac {2}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {2 x \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {2 x \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^4 c^2}+\frac {4 i \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 )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 209, normalized size = 0.69 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {4 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+\frac {4 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {4 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+\frac {4 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+3 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-2 \cos \left (2 \tan ^{-1}(a x)\right )-2\right )}{2 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 2.78, size = 294, normalized size = 0.96 \[ \frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}+\frac {\arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{4} c^{2}}-\frac {2 i \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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