Optimal. Leaf size=81 \[ \frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (\frac {b \sec ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right )}{3 f} \]
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Rubi [A] time = 0.09, antiderivative size = 81, 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 = {4134, 511, 510} \[ \frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (\frac {b \sec ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 4134
Rubi steps
\begin {align*} \int \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^p}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\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}\right ) \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{3 f}\\ \end {align*}
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Mathematica [B] time = 4.41, size = 266, normalized size = 3.28 \[ \frac {\csc ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^p F_1\left (\frac {1}{2}-p;-\frac {1}{2},-p;\frac {3}{2}-p;-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right )}{f (2 p-1) \left (\sec (e+f x) F_1\left (\frac {1}{2}-p;-\frac {1}{2},-p;\frac {3}{2}-p;-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right )-\frac {\cot (e+f x) \csc (e+f x) \left (2 p (a+b) F_1\left (\frac {3}{2}-p;-\frac {1}{2},1-p;\frac {5}{2}-p;-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right )+b F_1\left (\frac {3}{2}-p;\frac {1}{2},-p;\frac {5}{2}-p;-\cot ^2(e+f x),-\frac {(a+b) \cot ^2(e+f x)}{b}\right )\right )}{b (2 p-3)}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}, x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.56, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{3}\left (f x +e \right )\right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p}{{\sin \left (e+f\,x\right )}^3} \,d x \]
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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