Optimal. Leaf size=72 \[ \frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right )}{4 a f (m+1)}-\frac{(a \sin (e+f x)+a)^m}{2 f m} \]
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Rubi [A] time = 0.0504618, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2707, 79, 68} \[ \frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right )}{4 a f (m+1)}-\frac{(a \sin (e+f x)+a)^m}{2 f m} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 79
Rule 68
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)^{-1+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac{(a+a \sin (e+f x))^m}{2 f m}+\frac{\operatorname{Subst}\left (\int \frac{(a+x)^m}{a-x} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac{(a+a \sin (e+f x))^m}{2 f m}+\frac{\, _2F_1\left (1,1+m;2+m;\frac{1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{4 a f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0706524, size = 63, normalized size = 0.88 \[ \frac{(a (\sin (e+f x)+1))^m \left (m (\sin (e+f x)+1) \, _2F_1\left (1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right )-2 (m+1)\right )}{4 f m (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.76, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\tan \left ( fx+e \right ) \, 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 (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \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 (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \tan{\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 (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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