Optimal. Leaf size=84 \[ -\frac {\sqrt {2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac {1}{2};\frac {1}{2},3;m+\frac {3}{2};\frac {1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt {1-\csc (e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 84, 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 = {3828, 3827, 136} \[ -\frac {\sqrt {2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac {1}{2};\frac {1}{2},3;m+\frac {3}{2};\frac {1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt {1-\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 136
Rule 3827
Rule 3828
Rubi steps
\begin {align*} \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx &=\left ((1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int (1+\csc (e+f x))^m \sin ^2(e+f x) \, dx\\ &=\frac {\left (\cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x} x^3} \, dx,x,\csc (e+f x)\right )}{f \sqrt {1-\csc (e+f x)}}\\ &=-\frac {\sqrt {2} F_1\left (\frac {1}{2}+m;\frac {1}{2},3;\frac {3}{2}+m;\frac {1}{2} (1+\csc (e+f x)),1+\csc (e+f x)\right ) \cot (e+f x) (a+a \csc (e+f x))^m}{f (1+2 m) \sqrt {1-\csc (e+f x)}}\\ \end {align*}
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Mathematica [F] time = 7.71, size = 0, normalized size = 0.00 \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} {\left (a \csc \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.69, size = 0, normalized size = 0.00 \[ \int \left (a +a \csc \left (f x +e \right )\right )^{m} \left (\sin ^{2}\left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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