Optimal. Leaf size=90 \[ \frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right )}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3185, 430, 429} \[ \frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 3185
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(e+f x)\right )^p \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1}{2};\frac {1}{2},-p;\frac {3}{2};\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \sqrt {\cos ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 145, normalized size = 1.61 \[ \frac {2^{-p-1} \csc (2 (e+f x)) \sqrt {-\frac {b \sin ^2(e+f x)}{a}} \sqrt {\frac {b \cos ^2(e+f x)}{a+b}} (2 a-b \cos (2 (e+f x))+b)^{p+1} F_1\left (p+1;\frac {1}{2},\frac {1}{2};p+2;\frac {2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac {2 a+b-b \cos (2 (e+f x))}{2 a}\right )}{b f (p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.36, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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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