Optimal. Leaf size=60 \[ -\frac{8 \sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{2 \sin (a+b x) \cos ^3(a+b x)}{3 b}+\frac{\sin (a+b x) \cos (a+b x)}{b}+x \]
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Rubi [A] time = 0.0743127, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4288, 2568, 2635, 8} \[ -\frac{8 \sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{2 \sin (a+b x) \cos ^3(a+b x)}{3 b}+\frac{\sin (a+b x) \cos (a+b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac{8}{3} \int \cos ^4(a+b x) \, dx\\ &=\frac{2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac{8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+2 \int \cos ^2(a+b x) \, dx\\ &=\frac{\cos (a+b x) \sin (a+b x)}{b}+\frac{2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac{8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+\int 1 \, dx\\ &=x+\frac{\cos (a+b x) \sin (a+b x)}{b}+\frac{2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac{8 \cos ^5(a+b x) \sin (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.099401, size = 40, normalized size = 0.67 \[ -\frac{-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{12 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 55, normalized size = 0.9 \begin{align*} 16\,{\frac{-1/6\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+3/2\,\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) +1/16\,bx+a/16}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19902, size = 58, normalized size = 0.97 \begin{align*} \frac{12 \, b x - \sin \left (6 \, b x + 6 \, a\right ) - 3 \, \sin \left (4 \, b x + 4 \, a\right ) + 3 \, \sin \left (2 \, b x + 2 \, a\right )}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.485046, size = 115, normalized size = 1.92 \begin{align*} \frac{3 \, b x -{\left (8 \, \cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{3 \, b} \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 [A] time = 1.38985, size = 74, normalized size = 1.23 \begin{align*} \frac{3 \, b x + 3 \, a + \frac{3 \, \tan \left (b x + a\right )^{5} + 8 \, \tan \left (b x + a\right )^{3} - 3 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{3}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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