Optimal. Leaf size=88 \[ -\frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right ) \sec (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a-b}\right )^{-p}}{f} \]
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Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3745, 441, 440}
\begin {gather*} -\frac {\sec (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 {1}{2};1,-p;\frac {3}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 3745
Rubi steps
\begin {align*} \int \csc (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^p}{-1+x^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 {\left (1+\frac {b x^2}{a-b}\right )^p}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a-b}\right ) \sec (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}}{f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1215\) vs. \(2(88)=176\).
time = 14.07, size = 1215, normalized size = 13.81 \begin {gather*} \frac {\csc (e+f x) \left (a+b \tan ^2(e+f x)\right )^{2 p} \left (\frac {2 F_1\left (-\frac {1}{2}-p;-\frac {1}{2},-p;\frac {1}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}-F_1\left (1;\frac {1}{2},-p;2;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^2(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right )}{2 f \left (b p \sec ^2(e+f x) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^{-1+p} \left (\frac {2 F_1\left (-\frac {1}{2}-p;-\frac {1}{2},-p;\frac {1}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}-F_1\left (1;\frac {1}{2},-p;2;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^2(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right )+\frac {1}{2} \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {2 F_1\left (-\frac {1}{2}-p;-\frac {1}{2},-p;\frac {1}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {4 a p F_1\left (-\frac {1}{2}-p;-\frac {1}{2},-p;\frac {1}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-1-p} \sqrt {\csc ^2(e+f x)} \sqrt {\sec ^2(e+f x)}}{b (1+2 p)}+\frac {2 \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-p} \left (-\frac {2 a \left (-\frac {1}{2}-p\right ) p F_1\left (\frac {1}{2}-p;-\frac {1}{2},1-p;\frac {3}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \cot (e+f x) \csc ^2(e+f x)}{b \left (\frac {1}{2}-p\right )}-\frac {\left (-\frac {1}{2}-p\right ) 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 ^2(e+f x)}{\frac {1}{2}-p}\right ) \sqrt {\sec ^2(e+f x)}}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {2 F_1\left (-\frac {1}{2}-p;-\frac {1}{2},-p;\frac {1}{2}-p;-\cot ^2(e+f x),-\frac {a \cot ^2(e+f x)}{b}\right ) \left (1+\frac {a \cot ^2(e+f x)}{b}\right )^{-p} \sqrt {\sec ^2(e+f x)} \tan (e+f x)}{(1+2 p) \sqrt {\csc ^2(e+f x)}}+\frac {2 b p F_1\left (1;\frac {1}{2},-p;2;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x) \tan ^3(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-1-p}}{a}-2 F_1\left (1;\frac {1}{2},-p;2;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x) \tan (e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}-\tan ^2(e+f x) \left (\frac {b p F_1\left (2;\frac {1}{2},1-p;3;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x) \tan (e+f x)}{a}-\frac {1}{2} F_1\left (2;\frac {3}{2},-p;3;-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x) \tan (e+f x)\right ) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \csc \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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