Optimal. Leaf size=70 \[ \frac {(e \cos (c+d x))^{-2 (1+m)} \, _2F_1\left (2,-1-m;-m;\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{1+m}}{4 a d e (1+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 7, 70}
\begin {gather*} \frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left (2,-m-1;-m;\frac {1}{2} (1-\sin (c+d x))\right )}{4 a d e (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 7
Rule 70
Rule 2768
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{-2-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (2+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (2+2 m)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (-4-2 m)} (a+a x)^{\frac {1}{2} (-4-2 m)+m} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {\left (a^2 (e \cos (c+d x))^{-2-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (2+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (2+2 m)}\right ) \text {Subst}\left (\int \frac {(a-a x)^{\frac {1}{2} (-4-2 m)}}{(a+a x)^2} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {(e \cos (c+d x))^{-2 (1+m)} \, _2F_1\left (2,-1-m;-m;\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{1+m}}{4 a d e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 76, normalized size = 1.09 \begin {gather*} \frac {(e \cos (c+d x))^{-2 m} \, _2F_1\left (2,-1-m;-m;\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^2(c+d x) (a (1+\sin (c+d x)))^{1+m}}{4 a d e^3 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{-3-2 m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{2\,m+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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