Optimal. Leaf size=120 \[ \frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^4 c}+\frac {5 \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^4 \sqrt {c}}-\frac {x \sqrt {a^2 c x^2+c}}{6 a^3 c} \]
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Rubi [A] time = 0.15, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4952, 321, 217, 206, 4930} \[ -\frac {x \sqrt {a^2 c x^2+c}}{6 a^3 c}+\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^4 c}+\frac {5 \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^4 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 4930
Rule 4952
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^2 c}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{6 a^3}+\frac {2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^2 c}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{3 a^3}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^2 c}+\frac {5 \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^4 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 91, normalized size = 0.76 \[ \frac {-a x \sqrt {a^2 c x^2+c}+5 \sqrt {c} \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )+2 \left (a^2 x^2-2\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{6 a^4 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 80, normalized size = 0.67 \[ -\frac {2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - 2 \, {\left (a^{2} x^{2} - 2\right )} \arctan \left (a x\right )\right )} - 5 \, \sqrt {c} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{12 \, a^{4} c} \]
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 [C] time = 3.16, size = 165, normalized size = 1.38 \[ \frac {\left (2 \arctan \left (a x \right ) x^{2} a^{2}-a x -4 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,a^{4}}+\frac {5 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}-\frac {5 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 89, normalized size = 0.74 \[ -\frac {a {\left (\frac {\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}}{a^{2}} - \frac {4 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} - 2 \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \arctan \left (a x\right )}{6 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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