Optimal. Leaf size=410 \[ -\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left (c d (A m+A+B m+C m+C)-d^2 (A m+B (m+1)-C (m+1))+c^2 (-(2 C m+C))\right ) \left (\frac{(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right )}{d f (m+1) (c-d) (c+d)^2}+\frac{\cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left (c^2-d^2\right )}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (\frac{c+d \sin (e+f x)}{c-d}\right )^m (c+d \sin (e+f x))^{-m} F_1\left (m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.05988, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3043, 2987, 2788, 132, 140, 139, 138} \[ -\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left (c d (A m+A+B m+C m+C)-d^2 (A m+B (m+1)-C (m+1))+c^2 (-(2 C m+C))\right ) \left (\frac{(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right )}{d f (m+1) (c-d) (c+d)^2}+\frac{\cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left (c^2-d^2\right )}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (\frac{c+d \sin (e+f x)}{c-d}\right )^m (c+d \sin (e+f x))^{-m} F_1\left (m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3043
Rule 2987
Rule 2788
Rule 132
Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac{\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \left (-a (A d (c+c m-d m)+(c C-B d) (d-c m+d m))-a C \left (c^2-d^2\right ) (1+m) \sin (e+f x)\right ) \, dx}{a d \left (c^2-d^2\right ) (1+m)}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}+\frac{C \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-1-m} \, dx}{a d}+\frac{\left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx}{d \left (c^2-d^2\right ) (1+m)}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}+\frac{(a C \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(a+a x)^{\frac{1}{2}+m} (c+d x)^{-1-m}}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^2 \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m} (c+d x)^{-1-m}}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f (1+m) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac{2^{\frac{1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac{(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac{\left (a C \cos (e+f x) \sqrt{\frac{a-a \sin (e+f x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{\frac{1}{2}+m} (c+d x)^{-1-m}}{\sqrt{\frac{1}{2}-\frac{x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt{2} d f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac{2^{\frac{1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac{(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac{\left (a^2 C \cos (e+f x) \sqrt{\frac{a-a \sin (e+f x)}{a}} (c+d \sin (e+f x))^{-m} \left (\frac{a (c+d \sin (e+f x))}{a c-a d}\right )^m\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{\frac{1}{2}+m} \left (\frac{a c}{a c-a d}+\frac{a d x}{a c-a d}\right )^{-1-m}}{\sqrt{\frac{1}{2}-\frac{x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt{2} d (a c-a d) f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac{2^{\frac{1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac{(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac{\sqrt{2} C F_1\left (\frac{3}{2}+m;\frac{1}{2},1+m;\frac{5}{2}+m;\frac{1}{2} (1+\sin (e+f x)),-\frac{d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt{1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-m} \left (\frac{c+d \sin (e+f x)}{c-d}\right )^m}{(c-d) d f (3+2 m) (a-a \sin (e+f x))}\\ \end{align*}
Mathematica [B] time = 40.0668, size = 5193, normalized size = 12.67 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.364, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-2-m} \left ( A+B\sin \left ( fx+e \right ) +C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (C \cos \left (f x + e\right )^{2} - B \sin \left (f x + e\right ) - A - C\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]