Optimal. Leaf size=36 \[ \text {Int}\left (\frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2},x\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx &=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx\\ \end {align*}
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Mathematica [A] time = 35.60, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 6.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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