Optimal. Leaf size=83 \[ -\frac{\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p} F_1\left (\frac{1}{2};2,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right )}{f} \]
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Rubi [A] time = 0.0835968, antiderivative size = 83, 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 = {3186, 430, 429} \[ -\frac{\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p} F_1\left (\frac{1}{2};2,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\left (\left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{b x^2}{a+b}\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{F_1\left (\frac{1}{2};2,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p}}{f}\\ \end{align*}
Mathematica [F] time = 6.18479, size = 0, normalized size = 0. \[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.895, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{3} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \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 (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{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 (f x + e\right )^{2} + a + b\right )}^{p} \csc \left (f x + e\right )^{3}, 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 (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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