Optimal. Leaf size=86 \[ -\frac{a^3 c^2 2^{m+\frac{1}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.139956, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2736, 2689, 70, 69} \[ -\frac{a^3 c^2 2^{m+\frac{1}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2736
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (a+a \sin (e+f x))^{-2+m} \, dx\\ &=\frac{\left (a^4 c^2 \cos ^5(e+f x)\right ) \operatorname{Subst}\left (\int (a-a x)^{3/2} (a+a x)^{-\frac{1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}\\ &=\frac{\left (2^{-\frac{1}{2}+m} a^4 c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-3+m} \left (\frac{a+a \sin (e+f x)}{a}\right )^{\frac{1}{2}-m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{1}{2}+m} (a-a x)^{3/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2}}\\ &=-\frac{2^{\frac{1}{2}+m} a^3 c^2 \cos ^5(e+f x) \, _2F_1\left (\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac{1}{2}-m} (a+a \sin (e+f x))^{-3+m}}{5 f}\\ \end{align*}
Mathematica [C] time = 155.36, size = 88512, normalized size = 1029.21 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.758, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{2}\, 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{\left (c \sin \left (f x + e\right ) - c\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{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 (-{\left (c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}, 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*} c^{2} \left (\int - 2 \left (a \sin{\left (e + f x \right )} + a\right )^{m} \sin{\left (e + f x \right )}\, dx + \int \left (a \sin{\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (a \sin{\left (e + f x \right )} + a\right )^{m}\, dx\right ) \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{\left (c \sin \left (f x + e\right ) - c\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]