Optimal. Leaf size=107 \[ -\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^4 c^{3/2}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2}+\frac {\tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {x}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.20, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4964, 4930, 217, 206, 191} \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^4 c^{3/2}}-\frac {x}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 206
Rule 217
Rule 4930
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=\frac {\tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^4 c^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c}\\ &=-\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^4 c^2}-\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 )}{a^3 c}\\ &=-\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^4 c^2}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 107, normalized size = 1.00 \[ \frac {-a x \sqrt {a^2 c x^2+c}-\sqrt {c} \left (a^2 x^2+1\right ) \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )+\left (a^2 x^2+2\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 102, normalized size = 0.95 \[ \frac {{\left (a^{2} x^{2} + 1\right )} \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - {\left (a^{2} x^{2} + 2\right )} \arctan \left (a x\right )\right )}}{2 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \]
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 [C] time = 3.02, size = 242, normalized size = 2.26 \[ \frac {\left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}+\frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{4} c^{2}}+\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c^{2}}-\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c^{2}} \]
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 \[ \int \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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