Optimal. Leaf size=315 \[ \frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{3 a^2 c}-\frac {10 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.32, 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 = {5072, 267,
5010, 5006, 5050} \begin {gather*} \frac {x^2 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {2 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{3 a^4 c}-\frac {10 i \sqrt {a^2 x^2+1} \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \text {ArcTan}(a x)}{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 {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 {x \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{3 a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5006
Rule 5010
Rule 5050
Rule 5072
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 = 0.47, size = 279, normalized size = 0.89 \begin {gather*} \frac {\left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2-2 \text {ArcTan}(a x)^2+2 \cos (2 \text {ArcTan}(a x))-6 \text {ArcTan}(a x)^2 \cos (2 \text {ArcTan}(a x))+\frac {15 \text {ArcTan}(a x) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {1+a^2 x^2}}+5 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\frac {15 \text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {1+a^2 x^2}}-5 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+\frac {20 i \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-\frac {20 i \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \text {ArcTan}(a x) \sin (2 \text {ArcTan}(a x))\right )}{12 a^4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.00, size = 206, normalized size = 0.65
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right )^{2} a^{2} x^{2}-\arctan \left (a x \right ) a x -2 \arctan \left (a x \right )^{2}+1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 c \,a^{4}}+\frac {5 i \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, a^{4} c}\) | \(206\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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