Optimal. Leaf size=86 \[ \frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {x}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {x}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.52, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4964, 4968, 4970, 3312, 3302, 4904, 4406} \[ \frac {\text {CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {\text {CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {x}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {x}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 4406
Rule 4904
Rule 4964
Rule 4968
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a^3}+\frac {3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^3 c}-\frac {\int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a c}\\ &=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {\operatorname {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {\operatorname {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {1}{8 x}-\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^3}\\ &=\frac {x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 83, normalized size = 0.97 \[ \frac {-2 a^3 x^3+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text {Ci}\left (2 \tan ^{-1}(a x)\right )-\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text {Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.47, size = 292, normalized size = 3.40 \[ -\frac {4 \, a^{3} x^{3} + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 58, normalized size = 0.67 \[ \frac {4 \Ci \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \Ci \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-2 \sin \left (2 \arctan \left (a x \right )\right )+\sin \left (4 \arctan \left (a x \right )\right )}{8 a^{4} c^{3} \arctan \left (a x \right )} \]
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 \[ -\frac {{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \mathit {sage}_{0} x \arctan \left (a x\right ) + x^{3}}{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )} \]
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 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x^{3}}{a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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