Optimal. Leaf size=315 \[ \frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {5 i \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}-\frac {5 i \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c}}{3 a^4 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^4 c}-\frac {10 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{3 a^4 \sqrt {a^2 c x^2+c}}-\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^3 c} \]
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Rubi [A] time = 0.43, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4952, 261, 4890, 4886, 4930} \[ \frac {5 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}-\frac {5 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c}}{3 a^4 c}+\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^4 c}-\frac {10 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{3 a^4 \sqrt {a^2 c x^2+c}}-\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^3 c} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4886
Rule 4890
Rule 4930
Rule 4952
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {2 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {2 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {\int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {4 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {10 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 )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 279, normalized size = 0.89 \[ \frac {\left (a^2 x^2+1\right ) \sqrt {c \left (a^2 x^2+1\right )} \left (\frac {20 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac {20 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac {15 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {15 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-2 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-6 \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )+2 \cos \left (2 \tan ^{-1}(a x)\right )+5 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )-5 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+2\right )}{12 a^4 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
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 = 3.08, size = 206, normalized size = 0.65 \[ \frac {\left (\arctan \left (a x \right )^{2} x^{2} a^{2}-\arctan \left (a x \right ) x a -2 \arctan \left (a x \right )^{2}+1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 c \,a^{4}}+\frac {5 i \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, a^{4} c} \]
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 \[ \int \frac {x^{3} \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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