Optimal. Leaf size=385 \[ -\frac {11 \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4 c}+\frac {x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2-\frac {11 i c \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 i c \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.99, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5070, 5072,
267, 5010, 5006, 5050, 272, 45} \begin {gather*} \frac {x^2 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{15 a^2}+\frac {1}{5} x^4 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}-\frac {x^3 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{10 a}-\frac {2 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{15 a^4}-\frac {11 i c \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {a^2 c x^2+c}}+\frac {11 i c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {a^2 c x^2+c}}-\frac {11 i c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{30 a^4 c}-\frac {11 \sqrt {a^2 c x^2+c}}{60 a^4}+\frac {x \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{12 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 267
Rule 272
Rule 5006
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rubi steps
\begin {align*} \int x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx &=c \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x^5 \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}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{5} (4 c) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {(2 c) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {(2 c) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}-\frac {1}{5} (2 a c) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{10} c \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {(4 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+\frac {(8 c) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {(3 c) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {(8 c) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^4}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{20} c \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {(3 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {(4 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {(16 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {(3 c) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {(4 c) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (c \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 {\left (4 c \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}}{12 a^4}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {10 i c \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 c \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 c \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 {1}{20} c \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {11 \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {11 i c \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 )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 i c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 360, normalized size = 0.94 \begin {gather*} -\frac {\left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (50-32 \text {ArcTan}(a x)^2+72 \cos (2 \text {ArcTan}(a x))+160 \text {ArcTan}(a x)^2 \cos (2 \text {ArcTan}(a x))+22 \cos (4 \text {ArcTan}(a x))-\frac {110 \text {ArcTan}(a x) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-11 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+\frac {110 \text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+11 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-\frac {176 i \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \text {ArcTan}(a x) \sin (2 \text {ArcTan}(a x))-22 \text {ArcTan}(a x) \sin (4 \text {ArcTan}(a x))\right )}{960 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.71, size = 235, normalized size = 0.61
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 \arctan \left (a x \right )^{2} a^{4} x^{4}-6 \arctan \left (a x \right ) a^{3} x^{3}+4 \arctan \left (a x \right )^{2} a^{2} x^{2}+2 a^{2} x^{2}+5 \arctan \left (a x \right ) a x -8 \arctan \left (a x \right )^{2}-9\right )}{60 a^{4}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(235\) |
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 x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \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 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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