Optimal. Leaf size=43 \[ -\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.0357116, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3012, 3767, 8} \[ -\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc ^4(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}+\frac{1}{3} (2 a+3 b) \int \csc ^2(c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{(2 a+3 b) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=-\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0264045, size = 49, normalized size = 1.14 \[ -\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 35, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) -b\cot \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950237, size = 38, normalized size = 0.88 \begin{align*} -\frac{3 \,{\left (a + b\right )} \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58397, size = 132, normalized size = 3.07 \begin{align*} -\frac{{\left (2 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (a + b\right )} \cos \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \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 = 1.15288, size = 50, normalized size = 1.16 \begin{align*} -\frac{3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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