Optimal. Leaf size=13 \[ \frac{\sec (a+b x)}{4 b} \]
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Rubi [A] time = 0.0353868, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2606, 8} \[ \frac{\sec (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx &=\frac{1}{4} \int \sec (a+b x) \tan (a+b x) \, dx\\ &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\sec (a+b x))}{4 b}\\ &=\frac{\sec (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0094908, size = 13, normalized size = 1. \[ \frac{\sec (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 14, normalized size = 1.1 \begin{align*}{\frac{1}{4\,b\cos \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14944, size = 112, normalized size = 8.62 \begin{align*} \frac{\cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) + \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{2 \,{\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.465013, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{4 \, b \cos \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.61941, size = 431, normalized size = 33.15 \begin{align*} -\frac{6 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{11} - \tan \left (\frac{1}{2} \, a\right )^{12} - 2 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{9} + 12 \, \tan \left (\frac{1}{2} \, a\right )^{10} - 36 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{7} + 27 \, \tan \left (\frac{1}{2} \, a\right )^{8} - 36 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{5} - 2 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{3} - 27 \, \tan \left (\frac{1}{2} \, a\right )^{4} + 6 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right ) - 12 \, \tan \left (\frac{1}{2} \, a\right )^{2} + 1}{2 \,{\left (\tan \left (\frac{1}{2} \, b x + 2 \, a\right )^{2} \tan \left (\frac{1}{2} \, a\right )^{6} - 15 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right )^{2} \tan \left (\frac{1}{2} \, a\right )^{4} + 12 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{5} - \tan \left (\frac{1}{2} \, a\right )^{6} + 15 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right )^{2} \tan \left (\frac{1}{2} \, a\right )^{2} - 40 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right )^{3} + 15 \, \tan \left (\frac{1}{2} \, a\right )^{4} - \tan \left (\frac{1}{2} \, b x + 2 \, a\right )^{2} + 12 \, \tan \left (\frac{1}{2} \, b x + 2 \, a\right ) \tan \left (\frac{1}{2} \, a\right ) - 15 \, \tan \left (\frac{1}{2} \, a\right )^{2} + 1\right )}{\left (\tan \left (\frac{1}{2} \, a\right )^{6} - 15 \, \tan \left (\frac{1}{2} \, a\right )^{4} + 15 \, \tan \left (\frac{1}{2} \, a\right )^{2} - 1\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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