Optimal. Leaf size=119 \[ \frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b (e \sin (c+d x))^{m+1} \text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)} \]
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Rubi [A] time = 0.157583, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3872, 2838, 2564, 364, 2643} \[ \frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b (e \sin (c+d x))^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right )}{d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2838
Rule 2564
Rule 364
Rule 2643
Rubi steps
\begin{align*} \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx &=-\int (-b-a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^m \, dx\\ &=a \int (e \sin (c+d x))^m \, dx+b \int \sec (c+d x) (e \sin (c+d x))^m \, dx\\ &=\frac{a \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m) \sqrt{\cos ^2(c+d x)}}+\frac{b \operatorname{Subst}\left (\int \frac{x^m}{1-\frac{x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}\\ &=\frac{a \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m) \sqrt{\cos ^2(c+d x)}}+\frac{b \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.106581, size = 98, normalized size = 0.82 \[ \frac{\tan (c+d x) (e \sin (c+d x))^m \left (a \sqrt{\cos ^2(c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},\sin ^2(c+d x)\right )+b \cos (c+d x) \text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},\sin ^2(c+d x)\right )\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.619, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) \left ( e\sin \left ( dx+c \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 \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\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 \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m}, 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 (e \sin{\left (c + d x \right )}\right )^{m} \left (a + b \sec{\left (c + d x \right )}\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 \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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