Optimal. Leaf size=111 \[ -\frac{32 \sin ^3(a+b x) \cos ^7(a+b x)}{5 b}-\frac{12 \sin (a+b x) \cos ^7(a+b x)}{5 b}+\frac{2 \sin (a+b x) \cos ^5(a+b x)}{5 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{2 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{4 b}+\frac{3 x}{4} \]
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Rubi [A] time = 0.120024, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4288, 2568, 2635, 8} \[ -\frac{32 \sin ^3(a+b x) \cos ^7(a+b x)}{5 b}-\frac{12 \sin (a+b x) \cos ^7(a+b x)}{5 b}+\frac{2 \sin (a+b x) \cos ^5(a+b x)}{5 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{2 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{4 b}+\frac{3 x}{4} \]
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 ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx\\ &=-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}+\frac{96}{5} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}+\frac{12}{5} \int \cos ^6(a+b x) \, dx\\ &=\frac{2 \cos ^5(a+b x) \sin (a+b x)}{5 b}-\frac{12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}+2 \int \cos ^4(a+b x) \, dx\\ &=\frac{\cos ^3(a+b x) \sin (a+b x)}{2 b}+\frac{2 \cos ^5(a+b x) \sin (a+b x)}{5 b}-\frac{12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}+\frac{3}{2} \int \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos (a+b x) \sin (a+b x)}{4 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{2 b}+\frac{2 \cos ^5(a+b x) \sin (a+b x)}{5 b}-\frac{12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}+\frac{3 \int 1 \, dx}{4}\\ &=\frac{3 x}{4}+\frac{3 \cos (a+b x) \sin (a+b x)}{4 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{2 b}+\frac{2 \cos ^5(a+b x) \sin (a+b x)}{5 b}-\frac{12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac{32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.197873, size = 62, normalized size = 0.56 \[ \frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{160 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 83, normalized size = 0.8 \begin{align*} 64\,{\frac{1}{b} \left ( -1/10\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{7}-{\frac{3\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( bx+a \right ) }{160} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) }+{\frac{3\,bx}{256}}+{\frac{3\,a}{256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19874, size = 88, normalized size = 0.79 \begin{align*} \frac{120 \, b x + 2 \, \sin \left (10 \, b x + 10 \, a\right ) + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 10 \, \sin \left (6 \, b x + 6 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right ) + 20 \, \sin \left (2 \, b x + 2 \, a\right )}{160 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.512515, size = 177, normalized size = 1.59 \begin{align*} \frac{15 \, b x +{\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{20 \, 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 = 1.55702, size = 101, normalized size = 0.91 \begin{align*} \frac{15 \, b x + 15 \, a + \frac{15 \, \tan \left (b x + a\right )^{9} + 70 \, \tan \left (b x + a\right )^{7} + 128 \, \tan \left (b x + a\right )^{5} - 70 \, \tan \left (b x + a\right )^{3} - 15 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{5}}}{20 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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