Optimal. Leaf size=72 \[ -\frac {(a+a \sin (e+f x))^m}{2 f m}+\frac {\, _2F_1\left (1,1+m;2+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{4 a f (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2786, 80, 70}
\begin {gather*} \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (e+f x)+1)\right )}{4 a f (m+1)}-\frac {(a \sin (e+f x)+a)^m}{2 f m} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 80
Rule 2786
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {x (a+x)^{-1+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {(a+a \sin (e+f x))^m}{2 f m}+\frac {\text {Subst}\left (\int \frac {(a+x)^m}{a-x} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac {(a+a \sin (e+f x))^m}{2 f m}+\frac {\, _2F_1\left (1,1+m;2+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{4 a f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 63, normalized size = 0.88 \begin {gather*} \frac {(a (1+\sin (e+f x)))^m \left (-2 (1+m)+m \, _2F_1\left (1,1+m;2+m;\frac {1}{2} (1+\sin (e+f x))\right ) (1+\sin (e+f x))\right )}{4 f m (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \tan {\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {tan}\left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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