Optimal. Leaf size=16 \[ b x-\frac{a \cot (c+d x)}{d} \]
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Rubi [A] time = 0.0232825, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 8} \[ b x-\frac{a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 8
Rubi steps
\begin{align*} \int \csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{a \cot (c+d x)}{d}+b \int 1 \, dx\\ &=b x-\frac{a \cot (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0182045, size = 16, normalized size = 1. \[ b x-\frac{a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 22, normalized size = 1.4 \begin{align*}{\frac{-\cot \left ( dx+c \right ) a+ \left ( dx+c \right ) b}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42677, size = 31, normalized size = 1.94 \begin{align*} \frac{{\left (d x + c\right )} b - \frac{a}{\tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5839, size = 76, normalized size = 4.75 \begin{align*} \frac{b d x \sin \left (d x + c\right ) - a \cos \left (d x + c\right )}{d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15831, size = 53, normalized size = 3.31 \begin{align*} \frac{2 \,{\left (d x + c\right )} b + a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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