Optimal. Leaf size=70 \[ \frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left (2,-m-1;-m;\frac {1}{2} (1-\sin (c+d x))\right )}{4 a d e (m+1)} \]
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Rubi [A] time = 0.08, 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 = {2689, 7, 68} \[ \frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left (2,-m-1;-m;\frac {1}{2} (1-\sin (c+d x))\right )}{4 a d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 7
Rule 68
Rule 2689
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 ) \operatorname {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 ) \operatorname {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 = 0.13, size = 76, normalized size = 1.09 \[ \frac {\sec ^2(c+d x) (a (\sin (c+d x)+1))^{m+1} (e \cos (c+d x))^{-2 m} \, _2F_1\left (2,-m-1;-m;\frac {1}{2} (1-\sin (c+d x))\right )}{4 a d e^3 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.37, 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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{2\,m+3}} \,d x \]
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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