Optimal. Leaf size=223 \[ -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {1}{2};\frac {5}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{2 \sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}}-\frac {B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{\sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}} \]
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Rubi [A] time = 0.34, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2987, 2784, 139, 138} \[ -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {1}{2};\frac {5}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{2 \sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}}-\frac {B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{\sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2784
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))^2} \, dx &=(A-B) \int \frac {(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx+\frac {B \int \frac {(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac {((A-B) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {1-x} (1+x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {(B \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=\frac {\left ((A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n}{\sqrt {1-x} (1+x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {\left (B \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {B F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{\sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}}-\frac {(A-B) F_1\left (\frac {1}{2};\frac {5}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{2 \sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}}\\ \end {align*}
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Mathematica [F] time = 28.14, size = 0, normalized size = 0.00 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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