Optimal. Leaf size=44 \[ \frac {1}{6} \sin ^3(5 x)+\frac {1}{2} \sin (5 x)+\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{2} \tanh ^{-1}(\sin (5 x)) \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2592, 288, 302, 206} \[ \frac {1}{6} \sin ^3(5 x)+\frac {1}{2} \sin (5 x)+\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{2} \tanh ^{-1}(\sin (5 x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 302
Rule 2592
Rubi steps
\begin {align*} \int \sin ^3(5 x) \tan ^3(5 x) \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (5 x)\right )\\ &=\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (5 x)\right )\\ &=\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{2} \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (5 x)\right )\\ &=\frac {1}{2} \sin (5 x)+\frac {1}{6} \sin ^3(5 x)+\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (5 x)\right )\\ &=-\frac {1}{2} \tanh ^{-1}(\sin (5 x))+\frac {1}{2} \sin (5 x)+\frac {1}{6} \sin ^3(5 x)+\frac {1}{10} \sin ^3(5 x) \tan ^2(5 x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.18 \[ -\frac {1}{15} \sin ^3(5 x) \tan ^2(5 x)-\frac {1}{3} \sin (5 x) \tan ^2(5 x)-\frac {1}{2} \tanh ^{-1}(\sin (5 x))+\frac {1}{2} \tan (5 x) \sec (5 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 65, normalized size = 1.48 \[ -\frac {15 \, \cos \left (5 \, x\right )^{2} \log \left (\sin \left (5 \, x\right ) + 1\right ) - 15 \, \cos \left (5 \, x\right )^{2} \log \left (-\sin \left (5 \, x\right ) + 1\right ) + 2 \, {\left (2 \, \cos \left (5 \, x\right )^{4} - 14 \, \cos \left (5 \, x\right )^{2} - 3\right )} \sin \left (5 \, x\right )}{60 \, \cos \left (5 \, x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 51, normalized size = 1.16 \[ \frac {1}{15} \, \sin \left (5 \, x\right )^{3} - \frac {\sin \left (5 \, x\right )}{10 \, {\left (\sin \left (5 \, x\right )^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \left (5 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \left (5 \, x\right ) + 1\right ) + \frac {2}{5} \, \sin \left (5 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 50, normalized size = 1.14 \[ \frac {\sin ^{7}\left (5 x \right )}{10 \cos \left (5 x \right )^{2}}+\frac {\left (\sin ^{5}\left (5 x \right )\right )}{10}+\frac {\left (\sin ^{3}\left (5 x \right )\right )}{6}+\frac {\sin \left (5 x \right )}{2}-\frac {\ln \left (\sec \left (5 x \right )+\tan \left (5 x \right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 1.11 \[ \frac {1}{15} \, \sin \left (5 \, x\right )^{3} - \frac {\sin \left (5 \, x\right )}{10 \, {\left (\sin \left (5 \, x\right )^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \left (5 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\sin \left (5 \, x\right ) - 1\right ) + \frac {2}{5} \, \sin \left (5 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 69, normalized size = 1.57 \[ \frac {5\,{\mathrm {tan}\left (\frac {5\,x}{2}\right )}^9+\frac {20\,{\mathrm {tan}\left (\frac {5\,x}{2}\right )}^7}{3}-\frac {22\,{\mathrm {tan}\left (\frac {5\,x}{2}\right )}^5}{3}+\frac {20\,{\mathrm {tan}\left (\frac {5\,x}{2}\right )}^3}{3}+5\,\mathrm {tan}\left (\frac {5\,x}{2}\right )}{5\,{\left ({\mathrm {tan}\left (\frac {5\,x}{2}\right )}^2-1\right )}^2\,{\left ({\mathrm {tan}\left (\frac {5\,x}{2}\right )}^2+1\right )}^3}-\mathrm {atanh}\left (\mathrm {tan}\left (\frac {5\,x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 51, normalized size = 1.16 \[ \frac {\log {\left (\sin {\left (5 x \right )} - 1 \right )}}{4} - \frac {\log {\left (\sin {\left (5 x \right )} + 1 \right )}}{4} + \frac {\sin ^{3}{\left (5 x \right )}}{15} + \frac {2 \sin {\left (5 x \right )}}{5} - \frac {\sin {\left (5 x \right )}}{5 \left (2 \sin ^{2}{\left (5 x \right )} - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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