Optimal. Leaf size=191 \[ -\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^{3/2}}+\frac {16}{3 a^2 c^3 x^5 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}+\frac {32}{3 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}+\frac {80 \text {Int}\left (\frac {1}{x^6 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right )}{3 a^2}+80 \text {Int}\left (\frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right )+\frac {224}{3} a^2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right ) \]
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Rubi [A]
time = 0.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (16 a) \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^3 x^4 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16}{3 a^2 c^3 x^5 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {32}{3 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+32 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+48 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {80 \int \frac {1}{x^6 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (224 a^2\right ) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A]
time = 9.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.60, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{6} x^{10} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{8} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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