Optimal. Leaf size=21 \[ \frac {2 \sin (a+b x) \cos (a+b x)}{b}+2 x \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4288, 2635, 8} \[ \frac {2 \sin (a+b x) \cos (a+b x)}{b}+2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^2(a+b x) \, dx\\ &=\frac {2 \cos (a+b x) \sin (a+b x)}{b}+2 \int 1 \, dx\\ &=2 x+\frac {2 \cos (a+b x) \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.95 \[ \frac {2 (a+b x)+\sin (2 (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 22, normalized size = 1.05 \[ \frac {2 \, {\left (b x + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 29, normalized size = 1.38 \[ \frac {2 \, {\left (b x + a + \frac {\tan \left (b x + a\right )}{\tan \left (b x + a\right )^{2} + 1}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 28, normalized size = 1.33 \[ \frac {2 \cos \left (b x +a \right ) \sin \left (b x +a \right )+2 b x +2 a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 18, normalized size = 0.86 \[ \frac {2 \, b x + \sin \left (2 \, b x + 2 \, a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 17, normalized size = 0.81 \[ 2\,x+\frac {\sin \left (2\,a+2\,b\,x\right )}{b} \]
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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