Optimal. Leaf size=130 \[ -\frac{e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac{1}{2} (-m-2 n+1)} F_1\left (1-n;\frac{1-m}{2},\frac{1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n)} \]
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Rubi [A] time = 0.276536, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3876, 2886, 135, 133} \[ -\frac{e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac{1}{2} (-m-2 n+1)} F_1\left (1-n;\frac{1-m}{2},\frac{1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n)} \]
Antiderivative was successfully verified.
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Rule 3876
Rule 2886
Rule 135
Rule 133
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^n (e \sin (c+d x))^m \, dx &=\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int (-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n (e \sin (c+d x))^m \, dx\\ &=-\frac{\left (e (-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac{1-m}{2}-n} (-a+a \cos (c+d x))^{\frac{1-m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \operatorname{Subst}\left (\int (-x)^{-n} (-a-a x)^{\frac{1}{2} (-1+m)+n} (-a+a x)^{\frac{1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\left (e (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac{1}{2}-\frac{m}{2}-n} (-a-a \cos (c+d x))^{-\frac{1}{2}+\frac{1-m}{2}+\frac{m}{2}} (-a+a \cos (c+d x))^{\frac{1-m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \operatorname{Subst}\left (\int (-x)^{-n} (1+x)^{\frac{1}{2} (-1+m)+n} (-a+a x)^{\frac{1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\left (e (1-\cos (c+d x))^{\frac{1}{2}-\frac{m}{2}} (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac{1}{2}-\frac{m}{2}-n} (-a-a \cos (c+d x))^{-\frac{1}{2}+\frac{1-m}{2}+\frac{m}{2}} (-a+a \cos (c+d x))^{-\frac{1}{2}+\frac{1-m}{2}+\frac{m}{2}} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-1+m)} (-x)^{-n} (1+x)^{\frac{1}{2} (-1+m)+n} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{e F_1\left (1-n;\frac{1-m}{2},\frac{1}{2} (1-m-2 n);2-n;\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{\frac{1-m}{2}} \cos (c+d x) (1+\cos (c+d x))^{\frac{1}{2} (1-m-2 n)} (a+a \sec (c+d x))^n (e \sin (c+d x))^{-1+m}}{d (1-n)}\\ \end{align*}
Mathematica [B] time = 1.84603, size = 276, normalized size = 2.12 \[ \frac{4 (m+3) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) (a (\sec (c+d x)+1))^n (e \sin (c+d x))^m F_1\left (\frac{m+1}{2};n,m+1;\frac{m+3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}{d (m+1) \left ((m+3) (\cos (c+d x)+1) F_1\left (\frac{m+1}{2};n,m+1;\frac{m+3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-4 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \left ((m+1) F_1\left (\frac{m+3}{2};n,m+2;\frac{m+5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-n F_1\left (\frac{m+3}{2};n+1,m+1;\frac{m+5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.728, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n} \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 (a \sec \left (d x + c\right ) + a\right )}^{n} \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 (a \sec \left (d x + c\right ) + a\right )}^{n} \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(-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 (a \sec \left (d x + c\right ) + a\right )}^{n} \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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