Optimal. Leaf size=40 \[ \frac {(a \sin (c+d x)+a)^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (\sin (c+d x)+1)\right )}{2 d m} \]
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Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 68} \[ \frac {(a \sin (c+d x)+a)^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (\sin (c+d x)+1)\right )}{2 d m} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2667
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\left (1,m;1+m;\frac {1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^m}{2 d m}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 63, normalized size = 1.58 \[ \frac {(a (\sin (c+d x)+1))^m \left (m (\sin (c+d x)+1) \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (c+d x)+1)\right )+2 (m+1)\right )}{4 d m (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int \sec \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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