Optimal. Leaf size=177 \[ -\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {2}{a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac {x}{2 a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac {x}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.64, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4964, 4932, 4970, 4406, 12, 3299, 4968, 4902} \[ -\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {x}{2 a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac {x}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {2}{a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 4902
Rule 4932
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)^3} \, dx &=-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac {1-a^2 x^2}{2 a^4 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)^2} \, dx}{2 a^3}+\frac {3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^2 c}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {1}{2 a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a^3}+\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a^2}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 a^3 c}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {6 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a^2}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^2 c}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {6 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{4 a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^3}\\ &=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{a^4 c^3}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 72, normalized size = 0.41 \[ \frac {\frac {a^2 x^2 \left (\left (a^2 x^2-3\right ) \tan ^{-1}(a x)-a x\right )}{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\text {Si}\left (2 \tan ^{-1}(a x)\right )+2 \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.42, size = 328, normalized size = 1.85 \[ -\frac {2 \, a^{3} x^{3} - {\left (2 i \, a^{4} x^{4} + 4 i \, a^{2} x^{2} + 2 i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (-2 i \, a^{4} x^{4} - 4 i \, a^{2} x^{2} - 2 i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 3 \, a^{2} x^{2}\right )} \arctan \left (a x\right )}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )^{2}} \]
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 = 0.23, size = 90, normalized size = 0.51 \[ -\frac {8 \Si \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-16 \Si \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{16 a^{4} c^{3} \arctan \left (a x \right )^{2}} \]
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 \[ -\frac {2 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )} \mathit {sage}_{0} x \arctan \left (a x\right )^{2} + a x^{3} - {\left (a^{2} x^{4} - 3 \, x^{2}\right )} \arctan \left (a x\right )}{2 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )} \arctan \left (a x\right )^{2}} \]
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 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x^{3}}{a^{6} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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