Optimal. Leaf size=220 \[ -\frac {(A-B) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},1;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.40, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {2987, 2788, 137, 136, 2776, 70, 69} \[ -\frac {(A-B) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},1;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 136
Rule 137
Rule 2776
Rule 2788
Rule 2987
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx &=(A-B) \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx+\frac {B \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, 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)} \, 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}} \, 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) \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) (c+d \sin (e+f x))^n \left (-\frac {a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {c}{c+d}+\frac {d x}{c+d}\right )^n}{\sqrt {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 {(A-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) (1-\sin (e+f x)) \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 5.57, size = 244, normalized size = 1.11 \[ \frac {\cos (e+f x) \sqrt {a (\sin (e+f x)+1)} (c+d \sin (e+f x))^n \left (\frac {4 (A-B) \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac {c-d}{d \sin (e+f x)+d}+1\right )^{-n} F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )}{2 n+1}-(A+B) \sqrt {2-2 \sin (e+f x)} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\frac {d (\sin (e+f x)+1)}{d-c}\right )\right )}{4 a f (\sin (e+f x)-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.97, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sin \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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