Optimal. Leaf size=120 \[ -\frac {(3 a-b (1-2 p)) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )}{3 a f}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{3 a f} \]
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Rubi [A] time = 0.12, antiderivative size = 120, 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 = {3663, 453, 365, 364} \[ -\frac {(3 a-b (1-2 p)) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )}{3 a f}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{3 a f} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 453
Rule 3663
Rubi steps
\begin {align*} \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b x^2\right )^p}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{3 a f}-\frac {(-3 a-b (-3+2 (1+p))) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{3 a f}-\frac {\left ((-3 a-b (-3+2 (1+p))) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{3 a f}-\frac {(3 a-b (1-2 p)) \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{3 a f}\\ \end {align*}
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Mathematica [A] time = 1.58, size = 111, normalized size = 0.92 \[ \frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (-\left ((3 a+b (2 p-1)) \tan ^2(e+f x) \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )\right )-a-b \tan ^2(e+f x)\right )}{3 a f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{4}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^4} \,d x \]
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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