Optimal. Leaf size=86 \[ -\frac{a 2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0884103, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2689, 70, 69} \[ -\frac{a 2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx &=\frac{\left (a^2 \sqrt{e \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{3}{4}+m}}{(a-a x)^{3/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt [4]{a-a \sin (c+d x)} \sqrt [4]{a+a \sin (c+d x)}}\\ &=\frac{\left (2^{-\frac{3}{4}+m} a^2 \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{-1+m} \left (\frac{a+a \sin (c+d x)}{a}\right )^{\frac{3}{4}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{3}{4}+m}}{(a-a x)^{3/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt [4]{a-a \sin (c+d x)}}\\ &=-\frac{2^{\frac{5}{4}+m} a \sqrt{e \cos (c+d x)} \, _2F_1\left (\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{3}{4}-m} (a+a \sin (c+d x))^{-1+m}}{d e}\\ \end{align*}
Mathematica [A] time = 0.0770356, size = 83, normalized size = 0.97 \[ -\frac{2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{-m-\frac{1}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left (\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}{\frac{1}{\sqrt{e\cos \left ( dx+c \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{e \cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \left (\sin{\left (c + d x \right )} + 1\right )\right )^{m}}{\sqrt{e \cos{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]