Optimal. Leaf size=138 \[ \frac {2 (g \tan (e+f x))^{p+3}}{a^2 f g^3 (p+3)}-\frac {2 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac {p+5}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left (\frac {p+2}{2},\frac {p+5}{2};\frac {p+4}{2};\sin ^2(e+f x)\right )}{a^2 f g^2 (p+2)}+\frac {(g \tan (e+f x))^{p+1}}{a^2 f g (p+1)} \]
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Rubi [A] time = 0.27, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2711, 2607, 14, 16, 2617, 32} \[ -\frac {2 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac {p+5}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left (\frac {p+2}{2},\frac {p+5}{2};\frac {p+4}{2};\sin ^2(e+f x)\right )}{a^2 f g^2 (p+2)}+\frac {2 (g \tan (e+f x))^{p+3}}{a^2 f g^3 (p+3)}+\frac {(g \tan (e+f x))^{p+1}}{a^2 f g (p+1)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 16
Rule 32
Rule 2607
Rule 2617
Rule 2711
Rubi steps
\begin {align*} \int \frac {(g \tan (e+f x))^p}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int \left (a^2 \sec ^4(e+f x) (g \tan (e+f x))^p-2 a^2 \sec ^3(e+f x) \tan (e+f x) (g \tan (e+f x))^p+a^2 \sec ^2(e+f x) \tan ^2(e+f x) (g \tan (e+f x))^p\right ) \, dx}{a^4}\\ &=\frac {\int \sec ^4(e+f x) (g \tan (e+f x))^p \, dx}{a^2}+\frac {\int \sec ^2(e+f x) \tan ^2(e+f x) (g \tan (e+f x))^p \, dx}{a^2}-\frac {2 \int \sec ^3(e+f x) \tan (e+f x) (g \tan (e+f x))^p \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int (g x)^p \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\int \sec ^2(e+f x) (g \tan (e+f x))^{2+p} \, dx}{a^2 g^2}-\frac {2 \int \sec ^3(e+f x) (g \tan (e+f x))^{1+p} \, dx}{a^2 g}\\ &=-\frac {2 \cos ^2(e+f x)^{\frac {5+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {5+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{2+p}}{a^2 f g^2 (2+p)}+\frac {\operatorname {Subst}\left (\int \left ((g x)^p+\frac {(g x)^{2+p}}{g^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\operatorname {Subst}\left (\int (g x)^{2+p} \, dx,x,\tan (e+f x)\right )}{a^2 f g^2}\\ &=\frac {(g \tan (e+f x))^{1+p}}{a^2 f g (1+p)}-\frac {2 \cos ^2(e+f x)^{\frac {5+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {5+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{2+p}}{a^2 f g^2 (2+p)}+\frac {2 (g \tan (e+f x))^{3+p}}{a^2 f g^3 (3+p)}\\ \end {align*}
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Mathematica [B] time = 13.96, size = 667, normalized size = 4.83 \[ \frac {2^{p+1} \tan \left (\frac {1}{2} (e+f x)\right ) \left (1-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^p \left (-\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\tan ^2\left (\frac {1}{2} (e+f x)\right )-1}\right )^p \tan ^{-p}(e+f x) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4 (g \tan (e+f x))^p \left (\frac {\tan ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (p+2,\frac {p+3}{2};\frac {p+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+3}-\frac {6 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+3}{2},p+3;\frac {p+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+3}+\frac {12 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+3}{2},p+4;\frac {p+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+3}-\frac {2 \tan \left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+2}{2},p+2;\frac {p+4}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+2}+\frac {6 \tan \left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+2}{2},p+3;\frac {p+4}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+2}-\frac {8 \tan \left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+2}{2},p+4;\frac {p+4}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+2}+\frac {\, _2F_1\left (\frac {p+1}{2},p+2;\frac {p+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+1}-\frac {2 \, _2F_1\left (\frac {p+1}{2},p+3;\frac {p+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+1}+\frac {2 \, _2F_1\left (\frac {p+1}{2},p+4;\frac {p+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+1}+\frac {2 \tan ^4\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (p+4,\frac {p+5}{2};\frac {p+7}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+5}+\frac {2 \tan ^3\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (p+3,\frac {p+4}{2};\frac {p+6}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+4}-\frac {8 \tan ^3\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (\frac {p+4}{2},p+4;\frac {p+6}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{p+4}\right )}{f (a \sin (e+f x)+a)^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (g \tan \left (f x + e\right )\right )^{p}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (g \tan \left (f x + e\right )\right )^{p}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \tan \left (f x +e \right )\right )^{p}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {\left (g \tan \left (f x + e\right )\right )^{p}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (g \tan {\left (e + f x \right )}\right )^{p}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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