Optimal. Leaf size=128 \[ -\frac {(3 a+2 b (p+1)) \cot (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )}{3 f (a+b)}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{3 f (a+b)} \]
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Rubi [A] time = 0.11, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4132, 453, 365, 364} \[ -\frac {(3 a+2 b (p+1)) \cot (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )}{3 f (a+b)}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{3 f (a+b)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 453
Rule 4132
Rubi steps
\begin {align*} \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b+b x^2\right )^p}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{3 (a+b) f}+\frac {(3 a+2 b (1+p)) \operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{3 (a+b) f}+\frac {\left ((3 a+2 b (1+p)) \left (a+b+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a+b}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a+b}\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{3 (a+b) f}-\frac {(3 a+2 b (1+p)) \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right ) \left (a+b+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a+b}\right )^{-p}}{3 (a+b) f}\\ \end {align*}
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Mathematica [A] time = 2.25, size = 132, normalized size = 1.03 \[ -\frac {\cot (e+f x) \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \left (a+b \sec ^2(e+f x)\right )^p \left ((3 a+2 b (p+1)) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )+\cot ^2(e+f x) \left (a+b \tan ^2(e+f x)+b\right ) \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^p\right )}{3 f (a+b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.68, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{4}\left (f x +e \right )\right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p}{{\sin \left (e+f\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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