Optimal. Leaf size=92 \[ \frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac {b \sec ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right )}{3 f} \]
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Rubi [A] time = 0.12, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3664, 511, 510} \[ \frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac {b \sec ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 3664
Rubi steps
\begin {align*} \int \csc ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a-b+b x^2\right )^p}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\left (\left (a-b+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a-b}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a-b}\right )^p}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right ) \sec ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a-b}\right )^{-p}}{3 f}\\ \end {align*}
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Mathematica [B] time = 20.74, size = 252, normalized size = 2.74 \[ \frac {b (2 p-3) \cot (e+f x) \csc (e+f x) \left (a+b \tan ^2(e+f x)\right )^p F_1\left (\frac {1}{2}-p;-\frac {1}{2},-p;\frac {3}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right )}{f (2 p-1) \left (b (2 p-3) F_1\left (\frac {1}{2}-p;-\frac {1}{2},-p;\frac {3}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right )-\cot ^2(e+f x) \left (2 a p F_1\left (\frac {3}{2}-p;-\frac {1}{2},1-p;\frac {5}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right )+b F_1\left (\frac {3}{2}-p;\frac {1}{2},-p;\frac {5}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}, x\right ) \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{3}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^3} \,d x \]
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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