Optimal. Leaf size=413 \[ \frac{\sqrt{2} \cos (e+f x) \left (c d (A+C)-d^2 (4 A m+A-C)-2 c^2 (2 C m+C)\right ) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{1}{2};\frac{1}{2},\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) \left (c^2-d^2\right ) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left (d^2 (2 A m+A-C)+2 c^2 C (m+1)\right ) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) \left (c^2-d^2\right ) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (A d^2+c^2 C\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{d f \left (c^2-d^2\right ) \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.957646, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3044, 2987, 2788, 140, 139, 138} \[ \frac{\sqrt{2} \cos (e+f x) \left (c d (A+C)-d^2 (4 A m+A-C)-2 c^2 (2 C m+C)\right ) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{1}{2};\frac{1}{2},\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) \left (c^2-d^2\right ) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left (d^2 (2 A m+A-C)+2 c^2 C (m+1)\right ) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) \left (c^2-d^2\right ) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (A d^2+c^2 C\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{d f \left (c^2-d^2\right ) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2987
Rule 2788
Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{(a+a \sin (e+f x))^m \left (-\frac{1}{2} a \left (2 c C \left (\frac{d}{2}-c m\right )+2 A d \left (\frac{c}{2}-d m\right )\right )-\frac{1}{2} a \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) \sin (e+f x)\right )}{\sqrt{c+d \sin (e+f x)}} \, dx}{a d \left (c^2-d^2\right )}\\ &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) \int \frac{(a+a \sin (e+f x))^{1+m}}{\sqrt{c+d \sin (e+f x)}} \, dx}{a d \left (c^2-d^2\right )}-\frac{\left (2 \left (\frac{1}{2} a^2 \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right )-\frac{1}{2} a^2 \left (2 c C \left (\frac{d}{2}-c m\right )+2 A d \left (\frac{c}{2}-d m\right )\right )\right )\right ) \int \frac{(a+a \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx}{a^2 d \left (c^2-d^2\right )}\\ &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (a \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{\frac{1}{2}+m}}{\sqrt{a-a x} \sqrt{c+d x}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}-\frac{\left (2 \left (\frac{1}{2} a^2 \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right )-\frac{1}{2} a^2 \left (2 c C \left (\frac{d}{2}-c m\right )+2 A d \left (\frac{c}{2}-d m\right )\right )\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{\sqrt{a-a x} \sqrt{c+d x}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (a \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) \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}}{\sqrt{\frac{1}{2}-\frac{x}{2}} \sqrt{c+d x}} \, dx,x,\sin (e+f x)\right )}{\sqrt{2} d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}-\frac{\left (\sqrt{2} \left (\frac{1}{2} a^2 \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right )-\frac{1}{2} a^2 \left (2 c C \left (\frac{d}{2}-c m\right )+2 A d \left (\frac{c}{2}-d m\right )\right )\right ) \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}}{\sqrt{\frac{1}{2}-\frac{x}{2}} \sqrt{c+d x}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}\\ &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (a \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) \cos (e+f x) \sqrt{\frac{a-a \sin (e+f x)}{a}} \sqrt{\frac{a (c+d \sin (e+f x))}{a c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{\frac{1}{2}+m}}{\sqrt{\frac{1}{2}-\frac{x}{2}} \sqrt{\frac{a c}{a c-a d}+\frac{a d x}{a c-a d}}} \, dx,x,\sin (e+f x)\right )}{\sqrt{2} d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{\left (\sqrt{2} \left (\frac{1}{2} a^2 \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right )-\frac{1}{2} a^2 \left (2 c C \left (\frac{d}{2}-c m\right )+2 A d \left (\frac{c}{2}-d m\right )\right )\right ) \cos (e+f x) \sqrt{\frac{a-a \sin (e+f x)}{a}} \sqrt{\frac{a (c+d \sin (e+f x))}{a c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{\sqrt{\frac{1}{2}-\frac{x}{2}} \sqrt{\frac{a c}{a c-a d}+\frac{a d x}{a c-a d}}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \left (c (A+C) d-d^2 (A-C+4 A m)-2 c^2 (C+2 C m)\right ) F_1\left (\frac{1}{2}+m;\frac{1}{2},\frac{1}{2};\frac{3}{2}+m;\frac{1}{2} (1+\sin (e+f x)),-\frac{d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}{d \left (c^2-d^2\right ) f (1+2 m) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \left (2 c^2 C (1+m)+d^2 (A-C+2 A m)\right ) F_1\left (\frac{3}{2}+m;\frac{1}{2},\frac{1}{2};\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} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}{d \left (c^2-d^2\right ) f (3+2 m) (a-a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 32.0562, size = 19675, normalized size = 47.64 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.68, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{{\left (C \sin \left (f x + e\right )^{2} + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{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 (\frac{{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} \sqrt{d \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{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 \frac{{\left (C \sin \left (f x + e\right )^{2} + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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