Optimal. Leaf size=46 \[ \frac {32 \sin ^7(a+b x)}{7 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4288, 2564, 270} \[ \frac {32 \sin ^7(a+b x)}{7 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^2(a+b x) \, dx\\ &=\frac {32 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \sin ^3(a+b x)}{3 b}-\frac {64 \sin ^5(a+b x)}{5 b}+\frac {32 \sin ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 37, normalized size = 0.80 \[ \frac {4 \sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{105 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 43, normalized size = 0.93 \[ -\frac {32 \, {\left (15 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 36, normalized size = 0.78 \[ \frac {32 \, {\left (15 \, \sin \left (b x + a\right )^{7} - 42 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3}\right )}}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.96, size = 51, normalized size = 1.11 \[ \frac {-\frac {32 \sin \left (b x +a \right ) \left (\cos ^{6}\left (b x +a \right )\right )}{7}+\frac {32 \left (\frac {8}{3}+\cos ^{4}\left (b x +a \right )+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{3}\right ) \sin \left (b x +a \right )}{35}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 47, normalized size = 1.02 \[ -\frac {15 \, \sin \left (7 \, b x + 7 \, a\right ) + 63 \, \sin \left (5 \, b x + 5 \, a\right ) + 35 \, \sin \left (3 \, b x + 3 \, a\right ) - 525 \, \sin \left (b x + a\right )}{210 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 36, normalized size = 0.78 \[ \frac {32\,\left (15\,{\sin \left (a+b\,x\right )}^7-42\,{\sin \left (a+b\,x\right )}^5+35\,{\sin \left (a+b\,x\right )}^3\right )}{105\,b} \]
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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