Optimal. Leaf size=29 \[ \frac {4 \cos ^8(a+b x)}{b}-\frac {16 \cos ^6(a+b x)}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 29, 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, 2565, 14} \[ \frac {4 \cos ^8(a+b x)}{b}-\frac {16 \cos ^6(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac {32 \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {32 \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \cos ^6(a+b x)}{3 b}+\frac {4 \cos ^8(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 48, normalized size = 1.66 \[ \frac {-72 \cos (2 (a+b x))-12 \cos (4 (a+b x))+8 \cos (6 (a+b x))+3 \cos (8 (a+b x))}{96 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.90 \[ \frac {4 \, {\left (3 \, \cos \left (b x + a\right )^{8} - 4 \, \cos \left (b x + a\right )^{6}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 139, normalized size = 4.79 \[ \frac {128 \, {\left (\frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}}\right )}}{3 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 35, normalized size = 1.21 \[ \frac {-4 \left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{6}\left (b x +a \right )\right )-\frac {4 \left (\cos ^{6}\left (b x +a \right )\right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 50, normalized size = 1.72 \[ \frac {3 \, \cos \left (8 \, b x + 8 \, a\right ) + 8 \, \cos \left (6 \, b x + 6 \, a\right ) - 12 \, \cos \left (4 \, b x + 4 \, a\right ) - 72 \, \cos \left (2 \, b x + 2 \, a\right )}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 25, normalized size = 0.86 \[ \frac {4\,{\cos \left (a+b\,x\right )}^6\,\left (3\,{\cos \left (a+b\,x\right )}^2-4\right )}{3\,b} \]
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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