Optimal. Leaf size=424 \[ \frac {\sqrt {2} \cos (e+f x) \left (3 c d (A+C)+d^2 (-4 A m+A+3 C)-2 c^2 (2 C m+C)\right ) (a \sin (e+f x)+a)^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac {1}{2};\frac {1}{2},\frac {3}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \cos (e+f x) \left (2 c^2 C (m+1)-d^2 (-2 A m+A+3 C)\right ) (a \sin (e+f x)+a)^{m+1} \sqrt {\frac {c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac {3}{2};\frac {1}{2},\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (A d^2+c^2 C\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 1.04, antiderivative size = 424, normalized size of antiderivative = 1.00, 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} \[ \frac {\sqrt {2} \cos (e+f x) \left (3 c d (A+C)+d^2 (-4 A m+A+3 C)-2 c^2 (2 C m+C)\right ) (a \sin (e+f x)+a)^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac {1}{2};\frac {1}{2},\frac {3}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \cos (e+f x) \left (2 c^2 C (m+1)-d^2 (-2 A m+A+3 C)\right ) (a \sin (e+f x)+a)^{m+1} \sqrt {\frac {c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac {3}{2};\frac {1}{2},\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (A d^2+c^2 C\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
Rule 2788
Rule 2987
Rule 3044
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))^{5/2}} \, dx &=\frac {2 \left (c^2 C+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {(a+a \sin (e+f x))^m \left (-\frac {1}{2} a \left (2 c C \left (\frac {3 d}{2}-c m\right )+2 A d \left (\frac {3 c}{2}-d m\right )\right )-\frac {1}{2} a \left (2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 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}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \int \frac {(a+a \sin (e+f x))^{1+m}}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a d \left (c^2-d^2\right )}+\frac {\left (3 c (A+C) d+d^2 (A+3 C-4 A m)-2 c^2 (C+2 C m)\right ) \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 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}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a \left (2 c^2 C (1+m)-d^2 (A+3 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} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 d \left (c^2-d^2\right ) f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 \left (3 c (A+C) d+d^2 (A+3 C-4 A m)-2 c^2 (C+2 C m)\right ) \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 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}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a \left (2 c^2 C (1+m)-d^2 (A+3 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}} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \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 (a^2 \left (3 c (A+C) d+d^2 (A+3 C-4 A m)-2 c^2 (C+2 C 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}} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} 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}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a^2 \left (2 c^2 C (1+m)-d^2 (A+3 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}} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d (a c-a 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 (a^3 \left (3 c (A+C) d+d^2 (A+3 C-4 A m)-2 c^2 (C+2 C 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}} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d (a c-a 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}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\sqrt {2} \left (3 c (A+C) d+d^2 (A+3 C-4 A m)-2 c^2 (C+2 C m)\right ) F_1\left (\frac {1}{2}+m;\frac {1}{2},\frac {3}{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}}}{3 (c-d)^2 d (c+d) 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+3 C-2 A m)\right ) F_1\left (\frac {3}{2}+m;\frac {1}{2},\frac {3}{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}}}{3 (c-d)^2 d (c+d) f (3+2 m) (a-a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 32.46, size = 25065, normalized size = 59.12 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,{\sin \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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