Optimal. Leaf size=66 \[ \frac {5 \sin ^3(a+b x)}{6 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\sin (a+b x))}{2 b} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2592, 288, 302, 206} \[ \frac {5 \sin ^3(a+b x)}{6 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 302
Rule 2592
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=\frac {5 \sin (a+b x)}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=-\frac {5 \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 52, normalized size = 0.79 \[ \frac {(24 \cos (2 (a+b x))-\cos (4 (a+b x))+37) \tan (a+b x) \sec (a+b x)-60 \tanh ^{-1}(\sin (a+b x))}{24 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 84, normalized size = 1.27 \[ -\frac {15 \, \cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) - 15 \, \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (2 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} - 3\right )} \sin \left (b x + a\right )}{12 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 68, normalized size = 1.03 \[ \frac {4 \, \sin \left (b x + a\right )^{3} - \frac {6 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - 15 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 15 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right ) + 24 \, \sin \left (b x + a\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 1.20 \[ \frac {\sin ^{7}\left (b x +a \right )}{2 b \cos \left (b x +a \right )^{2}}+\frac {\sin ^{5}\left (b x +a \right )}{2 b}+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{6 b}+\frac {5 \sin \left (b x +a \right )}{2 b}-\frac {5 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 66, normalized size = 1.00 \[ \frac {4 \, \sin \left (b x + a\right )^{3} - \frac {6 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\sin \left (b x + a\right ) - 1\right ) + 24 \, \sin \left (b x + a\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.24, size = 147, normalized size = 2.23 \[ \frac {5\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^9+\frac {20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^7}{3}-\frac {22\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^5}{3}+\frac {20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3}{3}+5\,\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6-2\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}-\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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