Optimal. Leaf size=28 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4288, 2592, 321, 206} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2592
Rule 4288
Rubi steps
\begin {align*} \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx &=\frac {1}{2} \int \sin (a+b x) \tan (a+b x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=-\frac {\sin (a+b x)}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.96 \[ \frac {1}{2} \left (\frac {\tanh ^{-1}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 36, normalized size = 1.29 \[ \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 618, normalized size = 22.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 32, normalized size = 1.14 \[ -\frac {\sin \left (b x +a \right )}{2 b}+\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 124, normalized size = 4.43 \[ -\frac {\log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} - 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} + 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 2 \, \sin \left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 23, normalized size = 0.82 \[ -\frac {\frac {\sin \left (a+b\,x\right )}{2}-\frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2}}{b} \]
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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