Optimal. Leaf size=269 \[ \frac {d (A-B) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},2;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}-\frac {B \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},1;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d) (1-\sin (e+f x)) \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.47, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2987, 2788, 137, 136} \[ \frac {d (A-B) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},2;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}-\frac {B \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},1;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d) (1-\sin (e+f x)) \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 2788
Rule 2987
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx &=(A-B) \int \frac {(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx+\frac {B \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac {\left (a^2 (A-B) \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x} (a+a x)^2} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {(a B \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x} (a+a x)} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (a^2 (A-B) \cos (e+f x) \sqrt {\frac {d (a-a \sin (e+f x))}{a c+a d}}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{(a+a x)^2 \sqrt {\frac {a d}{a c+a d}-\frac {a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a B \cos (e+f x) \sqrt {\frac {d (a-a \sin (e+f x))}{a c+a d}}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{(a+a x) \sqrt {\frac {a d}{a c+a d}-\frac {a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {B F_1\left (1+n;\frac {1}{2},1;2+n;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d) f (1+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) d F_1\left (1+n;\frac {1}{2},2;2+n;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d)^2 f (1+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 17.23, size = 603, normalized size = 2.24 \[ \frac {\sec (e+f x) (c+d \sin (e+f x))^n \left (a A (\sin (e+f x)+1) \left (a \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1) \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\frac {d (\sin (e+f x)+1)}{d-c}\right )-\frac {4 \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac {c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (2 a (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )+a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )\right )}{4 n^2-1}\right )+a B (\sin (e+f x)+1) \left (a \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1) \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\frac {d (\sin (e+f x)+1)}{d-c}\right )-\frac {4 \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac {c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )-2 a (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )\right )}{4 n^2-1}\right )\right )}{8 a^3 f \sqrt {a (\sin (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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