Optimal. Leaf size=406 \[ \frac {\sqrt {2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+7)+2 B m+7 B+2 C m-5 C)) \sqrt {c+d \sin (e+f x)} F_1\left (m+\frac {1}{2};\frac {1}{2},-\frac {3}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}-\frac {\sqrt {2} (c-d) \cos (e+f x) (2 c C (m+1)-d (B (2 m+7)+2 C m)) (a \sin (e+f x)+a)^{m+1} \sqrt {c+d \sin (e+f x)} F_1\left (m+\frac {3}{2};\frac {1}{2},-\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)} \]
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Rubi [A] time = 1.03, antiderivative size = 403, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 6, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3045, 2987, 2788, 140, 139, 138} \[ \frac {\sqrt {2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+7)+2 B m+7 B+2 C m-5 C)) \sqrt {c+d \sin (e+f x)} F_1\left (m+\frac {1}{2};\frac {1}{2},-\frac {3}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {\sqrt {2} (c-d) \cos (e+f x) (B d (2 m+7)-2 c C (m+1)+2 C d m) (a \sin (e+f x)+a)^{m+1} \sqrt {c+d \sin (e+f x)} F_1\left (m+\frac {3}{2};\frac {1}{2},-\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
Rule 2788
Rule 2987
Rule 3045
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} a \left (2 A d \left (\frac {7}{2}+m\right )+2 C \left (\frac {5 d}{2}+c m\right )\right )+\frac {1}{2} a (2 C d m-2 c C (1+m)+B d (7+2 m)) \sin (e+f x)\right ) \, dx}{a d (7+2 m)}\\ &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {(2 C d m-2 c C (1+m)+B d (7+2 m)) \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{3/2} \, dx}{a d (7+2 m)}+\frac {(2 c (C+2 C m)-d (7 B-5 C+2 B m+2 C m-A (7+2 m))) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx}{d (7+2 m)}\\ &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {(a (2 C d m-2 c C (1+m)+B d (7+2 m)) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (2 c (C+2 C m)-d (7 B-5 C+2 B m+2 C m-A (7+2 m))) \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\left (a (2 C d m-2 c C (1+m)+B d (7+2 m)) \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)^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (2 c (C+2 C m)-d (7 B-5 C+2 B m+2 C m-A (7+2 m))) \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)^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\left ((a c-a d) (2 C d m-2 c C (1+m)+B d (7+2 m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {c+d \sin (e+f x)}\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 )^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}}+\frac {\left (a (a c-a d) (2 c (C+2 C m)-d (7 B-5 C+2 B m+2 C m-A (7+2 m))) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {c+d \sin (e+f x)}\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 )^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}}\\ &=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\sqrt {2} (c-d) (2 c (C+2 C m)-d (7 B-5 C+2 B m+2 C m-A (7+2 m))) 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 {c+d \sin (e+f x)}}{d f (1+2 m) (7+2 m) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {\sqrt {2} (c-d) (2 C d m-2 c C (1+m)+B d (7+2 m)) 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 {c+d \sin (e+f x)}}{d f (3+2 m) (7+2 m) (a-a \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}\\ \end {align*}
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Mathematica [B] time = 9.63, size = 6591, normalized size = 16.23 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left ({\left (C c + B d\right )} \cos \left (f x + e\right )^{2} - {\left (A + C\right )} c - B d + {\left (C d \cos \left (f x + e\right )^{2} - B c - {\left (A + C\right )} d\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 4.47, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\left (C\,{\sin \left (e+f\,x\right )}^2+B\,\sin \left (e+f\,x\right )+A\right ) \,d x \]
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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