Optimal. Leaf size=83 \[ -\frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\cos ^2(e+f x),\frac {b \cos ^2(e+f x)}{a+b}\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p}}{f} \]
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Rubi [A]
time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 = {3265, 441, 440}
\begin {gather*} -\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};1,-p;\frac {3}{2};\cos ^2(e+f x),\frac {b \cos ^2(e+f x)}{a+b}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 3265
Rubi steps
\begin {align*} \int \csc (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^p}{1-x^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 ) \text {Subst}\left (\int \frac {\left (1-\frac {b x^2}{a+b}\right )^p}{1-x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {F_1\left (\frac {1}{2};1,-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*}
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Mathematica [F]
time = 15.14, size = 0, normalized size = 0.00 \begin {gather*} \int \csc (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \csc \left (f x +e \right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.41, size = 25, normalized size = 0.30 \begin {gather*} {\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \csc \left (f x + e\right ), x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{p} \csc {\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p}{\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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