Optimal. Leaf size=88 \[ -\frac{\sec (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac{b \sec ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right )}{f} \]
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Rubi [A] time = 0.0805196, antiderivative size = 88, normalized size of antiderivative = 1., 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 = {3664, 430, 429} \[ -\frac{\sec (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac{b \sec ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \csc (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac{\operatorname{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 ) \operatorname{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*}
Mathematica [B] time = 15.0711, size = 1215, normalized size = 13.81 \[ \frac{\csc (e+f x) \left (b \tan ^2(e+f x)+a\right )^{2 p} \left (\frac{2 F_1\left (-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right ) \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \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 (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p}\right )}{2 f \left (b p \sec ^2(e+f x) \tan (e+f x) \left (\frac{2 F_1\left (-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right ) \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \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 (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p}\right ) \left (b \tan ^2(e+f x)+a\right )^{p-1}+\frac{1}{2} \left (\frac{4 a p F_1\left (-p-\frac{1}{2};-\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) \sqrt{\csc ^2(e+f x)} \sqrt{\sec ^2(e+f x)} \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p-1}}{b (2 p+1)}+\frac{2 F_1\left (-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right ) \sqrt{\sec ^2(e+f x)} \tan (e+f x) \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 \left (-\frac{2 a \left (-p-\frac{1}{2}\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 (-p-\frac{1}{2}\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)} \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 F_1\left (-p-\frac{1}{2};-\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) \sqrt{\sec ^2(e+f x)} \left (\frac{a \cot ^2(e+f x)}{b}+1\right )^{-p}}{(2 p+1) \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 (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p-1}}{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 (\frac{b \tan ^2(e+f x)}{a}+1\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 (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p}\right ) \left (b \tan ^2(e+f x)+a\right )^p\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.214, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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