Optimal. Leaf size=26 \[ \text {Int}\left (\sec ^2(e+f x) \left (a+b \sin ^n(e+f x)\right )^p,x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int \sec ^2(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \sec ^2(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\int \sec ^2(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx\\ \end {align*}
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Mathematica [A] time = 4.38, size = 0, normalized size = 0.00 \[ \int \sec ^2(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right )^{2}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p}{{\cos \left (e+f\,x\right )}^2} \,d x \]
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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