Optimal. Leaf size=220 \[ \frac{\sqrt{2} a \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt{\csc (e+f x)+1}}-\frac{\sqrt{2} (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt{\csc (e+f x)+1}} \]
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Rubi [A] time = 0.225604, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3838, 3834, 139, 138} \[ \frac{\sqrt{2} a \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt{\csc (e+f x)+1}}-\frac{\sqrt{2} (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt{\csc (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 3838
Rule 3834
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx &=\frac{\int \csc (e+f x) (a+b \csc (e+f x))^{1+m} \, dx}{b}-\frac{a \int \csc (e+f x) (a+b \csc (e+f x))^m \, dx}{b}\\ &=\frac{\cot (e+f x) \operatorname{Subst}\left (\int \frac{(a+b x)^{1+m}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\csc (e+f x)\right )}{b f \sqrt{1-\csc (e+f x)} \sqrt{1+\csc (e+f x)}}-\frac{(a \cot (e+f x)) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\csc (e+f x)\right )}{b f \sqrt{1-\csc (e+f x)} \sqrt{1+\csc (e+f x)}}\\ &=-\frac{\left (a \cot (e+f x) (a+b \csc (e+f x))^m \left (-\frac{a+b \csc (e+f x)}{-a-b}\right )^{-m}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^m}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\csc (e+f x)\right )}{b f \sqrt{1-\csc (e+f x)} \sqrt{1+\csc (e+f x)}}-\frac{\left ((-a-b) \cot (e+f x) (a+b \csc (e+f x))^m \left (-\frac{a+b \csc (e+f x)}{-a-b}\right )^{-m}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{1+m}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\csc (e+f x)\right )}{b f \sqrt{1-\csc (e+f x)} \sqrt{1+\csc (e+f x)}}\\ &=-\frac{\sqrt{2} (a+b) F_1\left (\frac{1}{2};\frac{1}{2},-1-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m}}{b f \sqrt{1+\csc (e+f x)}}+\frac{\sqrt{2} a F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m}}{b f \sqrt{1+\csc (e+f x)}}\\ \end{align*}
Mathematica [F] time = 2.71815, size = 0, normalized size = 0. \[ \int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.247, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{2} \left ( a+b\csc \left ( fx+e \right ) \right ) ^{m}\, 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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}\,{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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \csc{\left (e + f x \right )}\right )^{m} \csc ^{2}{\left (e + f x \right )}\, dx \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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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