Optimal. Leaf size=99 \[ \frac {\tan ^3(e+f x) \sec ^2(e+f x)^p \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {(a+b) \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {3}{2};p+2,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {(a+b) \tan ^2(e+f x)}{a}\right )}{3 f} \]
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Rubi [A] time = 0.17, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3174, 511, 510} \[ \frac {\tan ^3(e+f x) \sec ^2(e+f x)^p \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {(a+b) \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {3}{2};p+2,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {(a+b) \tan ^2(e+f x)}{a}\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 3174
Rubi steps
\begin {align*} \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=\frac {\left (\sec ^2(e+f x)^p \left (a+b \sin ^2(e+f x)\right )^p \left (a+(a+b) \tan ^2(e+f x)\right )^{-p}\right ) \operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^{-2-p} \left (a+(a+b) x^2\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left (\sec ^2(e+f x)^p \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {(a+b) \tan ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^{-2-p} \left (1+\frac {(a+b) x^2}{a}\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {3}{2};2+p,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {(a+b) \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x)^p \left (a+b \sin ^2(e+f x)\right )^p \tan ^3(e+f x) \left (1+\frac {(a+b) \tan ^2(e+f x)}{a}\right )^{-p}}{3 f}\\ \end {align*}
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Mathematica [B] time = 0.69, size = 240, normalized size = 2.42 \[ -\frac {2^{-p-2} \csc (2 (e+f x)) \sqrt {-\frac {b \sin ^2(e+f x)}{a}} \sqrt {\frac {b \cos ^2(e+f x)}{a+b}} (2 a-b \cos (2 (e+f x))+b)^{p+1} \left (2 a (p+2) F_1\left (p+1;\frac {1}{2},\frac {1}{2};p+2;\frac {2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac {2 a+b-b \cos (2 (e+f x))}{2 a}\right )-(p+1) (2 a-b \cos (2 (e+f x))+b) F_1\left (p+2;\frac {1}{2},\frac {1}{2};p+3;\frac {2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac {2 a+b-b \cos (2 (e+f x))}{2 a}\right )\right )}{b^2 f (p+1) (p+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} {\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.34, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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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