Optimal. Leaf size=73 \[ \frac{2^{m-\frac{1}{2}} \sec (c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d} \]
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Rubi [A] time = 0.083032, antiderivative size = 73, normalized size of antiderivative = 1., 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 = {2689, 70, 69} \[ \frac{2^{m-\frac{1}{2}} \sec (c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\left (a^2 \sec (c+d x) \sqrt{a-a \sin (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{3}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\left (2^{-\frac{3}{2}+m} a \sec (c+d x) \sqrt{a-a \sin (c+d x)} (a+a \sin (c+d x))^m \left (\frac{a+a \sin (c+d x)}{a}\right )^{\frac{1}{2}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{3}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{2^{-\frac{1}{2}+m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right ) \sec (c+d x) (1+\sin (c+d x))^{\frac{1}{2}-m} (a+a \sin (c+d x))^m}{d}\\ \end{align*}
Mathematica [C] time = 25.916, size = 5807, normalized size = 79.55 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{2} \left ( a+a\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 \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\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 (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{2}, 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 \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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