Optimal. Leaf size=46 \[ \frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4288, 2564, 270} \[ \frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^4(a+b x) \, dx\\ &=\frac {32 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \sin ^5(a+b x)}{5 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^9(a+b x)}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 38, normalized size = 0.83 \[ \frac {32 \left (35 \sin ^9(a+b x)-90 \sin ^7(a+b x)+63 \sin ^5(a+b x)\right )}{315 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 53, normalized size = 1.15 \[ \frac {32 \, {\left (35 \, \cos \left (b x + a\right )^{8} - 50 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 36, normalized size = 0.78 \[ \frac {32 \, {\left (35 \, \sin \left (b x + a\right )^{9} - 90 \, \sin \left (b x + a\right )^{7} + 63 \, \sin \left (b x + a\right )^{5}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.97, size = 69, normalized size = 1.50 \[ \frac {-\frac {32 \left (\sin ^{3}\left (b x +a \right )\right ) \left (\cos ^{6}\left (b x +a \right )\right )}{9}-\frac {32 \sin \left (b x +a \right ) \left (\cos ^{6}\left (b x +a \right )\right )}{21}+\frac {32 \left (\frac {8}{3}+\cos ^{4}\left (b x +a \right )+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{3}\right ) \sin \left (b x +a \right )}{105}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 58, normalized size = 1.26 \[ \frac {35 \, \sin \left (9 \, b x + 9 \, a\right ) + 45 \, \sin \left (7 \, b x + 7 \, a\right ) - 252 \, \sin \left (5 \, b x + 5 \, a\right ) - 420 \, \sin \left (3 \, b x + 3 \, a\right ) + 1890 \, \sin \left (b x + a\right )}{2520 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 36, normalized size = 0.78 \[ \frac {32\,\left (35\,{\sin \left (a+b\,x\right )}^9-90\,{\sin \left (a+b\,x\right )}^7+63\,{\sin \left (a+b\,x\right )}^5\right )}{315\,b} \]
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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