Optimal. Leaf size=101 \[ -\frac{\sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
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Rubi [A] time = 0.102305, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3196, 511, 510} \[ -\frac{\sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx &=\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac{b \sin ^2(c+d x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2} \left (1+\frac{b x^2}{a}\right )^p}{x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{F_1\left (-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right ) \sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac{b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.237309, size = 102, normalized size = 1.01 \[ -\frac{\sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{a+b \sin ^2(c+d x)}{a}\right )^{-p} F_1\left (-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.506, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{4} \left ( a+ \left ( \sin \left ( dx+c \right ) \right ) ^{2}b \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \left (d x + c\right )^{4}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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