Optimal. Leaf size=44 \[ -\frac {32 \cos ^{12}(a+b x)}{3 b}+\frac {128 \cos ^{10}(a+b x)}{5 b}-\frac {16 \cos ^8(a+b x)}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4288, 2565, 266, 43} \[ -\frac {32 \cos ^{12}(a+b x)}{3 b}+\frac {128 \cos ^{10}(a+b x)}{5 b}-\frac {16 \cos ^8(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac {128 \operatorname {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \operatorname {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(a+b x)\right )}{b}\\ &=-\frac {64 \operatorname {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(a+b x)\right )}{b}\\ &=-\frac {16 \cos ^8(a+b x)}{b}+\frac {128 \cos ^{10}(a+b x)}{5 b}-\frac {32 \cos ^{12}(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 48, normalized size = 1.09 \[ \frac {16 \left (-10 \sin ^{12}(a+b x)+36 \sin ^{10}(a+b x)-45 \sin ^8(a+b x)+20 \sin ^6(a+b x)\right )}{15 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 36, normalized size = 0.82 \[ -\frac {16 \, {\left (10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 183, normalized size = 4.16 \[ -\frac {4096 \, {\left (\frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {39 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {42 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {39 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}}\right )}}{15 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 53, normalized size = 1.20 \[ \frac {-\frac {32 \left (\sin ^{4}\left (b x +a \right )\right ) \left (\cos ^{8}\left (b x +a \right )\right )}{3}-\frac {64 \left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{8}\left (b x +a \right )\right )}{15}-\frac {16 \left (\cos ^{8}\left (b x +a \right )\right )}{15}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 72, normalized size = 1.64 \[ -\frac {5 \, \cos \left (12 \, b x + 12 \, a\right ) + 12 \, \cos \left (10 \, b x + 10 \, a\right ) - 30 \, \cos \left (8 \, b x + 8 \, a\right ) - 100 \, \cos \left (6 \, b x + 6 \, a\right ) + 75 \, \cos \left (4 \, b x + 4 \, a\right ) + 600 \, \cos \left (2 \, b x + 2 \, a\right )}{960 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 35, normalized size = 0.80 \[ -\frac {16\,{\cos \left (a+b\,x\right )}^8\,\left (10\,{\cos \left (a+b\,x\right )}^4-24\,{\cos \left (a+b\,x\right )}^2+15\right )}{15\,b} \]
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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