Optimal. Leaf size=60 \[ -\frac {8 \sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac {2 \sin (a+b x) \cos ^3(a+b x)}{3 b}+\frac {\sin (a+b x) \cos (a+b x)}{b}+x \]
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Rubi [A] time = 0.07, antiderivative size = 60, 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, 2568, 2635, 8} \[ -\frac {8 \sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac {2 \sin (a+b x) \cos ^3(a+b x)}{3 b}+\frac {\sin (a+b x) \cos (a+b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac {8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {8}{3} \int \cos ^4(a+b x) \, dx\\ &=\frac {2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac {8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+2 \int \cos ^2(a+b x) \, dx\\ &=\frac {\cos (a+b x) \sin (a+b x)}{b}+\frac {2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac {8 \cos ^5(a+b x) \sin (a+b x)}{3 b}+\int 1 \, dx\\ &=x+\frac {\cos (a+b x) \sin (a+b x)}{b}+\frac {2 \cos ^3(a+b x) \sin (a+b x)}{3 b}-\frac {8 \cos ^5(a+b x) \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 40, normalized size = 0.67 \[ -\frac {-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{12 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 47, normalized size = 0.78 \[ \frac {3 \, b x - {\left (8 \, \cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 55, normalized size = 0.92 \[ \frac {3 \, b x + 3 \, a + \frac {3 \, \tan \left (b x + a\right )^{5} + 8 \, \tan \left (b x + a\right )^{3} - 3 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 55, normalized size = 0.92 \[ \frac {-\frac {8 \sin \left (b x +a \right ) \left (\cos ^{5}\left (b x +a \right )\right )}{3}+\frac {2 \left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{3}+b x +a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 43, normalized size = 0.72 \[ \frac {12 \, b x - \sin \left (6 \, b x + 6 \, a\right ) - 3 \, \sin \left (4 \, b x + 4 \, a\right ) + 3 \, \sin \left (2 \, b x + 2 \, a\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 65, normalized size = 1.08 \[ x+\frac {{\mathrm {tan}\left (a+b\,x\right )}^5+\frac {8\,{\mathrm {tan}\left (a+b\,x\right )}^3}{3}-\mathrm {tan}\left (a+b\,x\right )}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^6+3\,{\mathrm {tan}\left (a+b\,x\right )}^4+3\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]
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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