Optimal. Leaf size=21 \[ \frac{2 \sin (a+b x) \cos (a+b x)}{b}+2 x \]
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Rubi [A] time = 0.0343239, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 4288
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0243541, 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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Maple [A] time = 0.025, size = 28, normalized size = 1.3 \begin{align*} 4\,{\frac{1/2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) +1/2\,bx+a/2}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12614, size = 24, normalized size = 1.14 \begin{align*} \frac{2 \, b x + \sin \left (2 \, b x + 2 \, a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.476171, size = 53, normalized size = 2.52 \begin{align*} \frac{2 \,{\left (b x + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29708, size = 39, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (b x + a + \frac{\tan \left (b x + a\right )}{\tan \left (b x + a\right )^{2} + 1}\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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