Optimal. Leaf size=127 \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]
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Rubi [A] time = 0.182889, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3681, 3667, 511, 510} \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]
Antiderivative was successfully verified.
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Rule 3681
Rule 3667
Rule 511
Rule 510
Rubi steps
\begin{align*} \int (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\left ((d \csc (e+f x))^m \left (\frac{\sin (e+f x)}{d}\right )^m\right ) \int \left (\frac{\sin (e+f x)}{d}\right )^{-m} \left (a+b \tan ^2(e+f x)\right )^p \, dx\\ &=\frac{\left ((d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan ^m(e+f x)\right ) \operatorname{Subst}\left (\int x^{-m} \left (1+x^2\right )^{-1+\frac{m}{2}} \left (a+b x^2\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left ((d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan ^m(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac{b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int x^{-m} \left (1+x^2\right )^{-1+\frac{m}{2}} \left (1+\frac{b x^2}{a}\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{F_1\left (\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right ) (d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac{b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)}\\ \end{align*}
Mathematica [B] time = 3.50409, size = 292, normalized size = 2.3 \[ -\frac{a (m-3) \cos ^2(e+f x) \cot (e+f x) (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p F_1\left (\frac{1}{2}-\frac{m}{2};1-\frac{m}{2},-p;\frac{3}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )}{f (m-1) \left (-2 b p F_1\left (\frac{3}{2}-\frac{m}{2};1-\frac{m}{2},1-p;\frac{5}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )-a (m-2) F_1\left (\frac{3}{2}-\frac{m}{2};2-\frac{m}{2},-p;\frac{5}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )+a (m-3) \cot ^2(e+f x) F_1\left (\frac{1}{2}-\frac{m}{2};1-\frac{m}{2},-p;\frac{3}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.773, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \right ) ^{m} \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} \left (d \csc \left (f x + e\right )\right )^{m}\,{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} \left (d \csc \left (f x + e\right )\right )^{m}, 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} \left (d \csc \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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