Optimal. Leaf size=92 \[ \frac {F_1\left (\frac {3}{2};2,-p;\frac {5}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right ) \sec ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a-b}\right )^{-p}}{3 f} \]
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Rubi [A]
time = 0.07, 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 = {3745, 525, 524}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3745
Rubi steps
\begin {align*} \int \csc ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(252\) vs. \(2(92)=184\).
time = 15.05, size = 252, normalized size = 2.74 \begin {gather*} \frac {b (-3+2 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 ) \cot (e+f x) \csc (e+f x) \left (a+b \tan ^2(e+f x)\right )^p}{f (-1+2 p) \left (b (-3+2 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 )-\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 ) \cot ^2(e+f x)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.22, 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 \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 = 0.00, 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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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