Optimal. Leaf size=40 \[ \frac {\, _2F_1\left (1,m;1+m;\frac {1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^m}{2 d m} \]
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Rubi [A]
time = 0.03, 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 = {2746, 70}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 2746
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {a \text {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 \begin {gather*} \frac {(a (1+\sin (c+d x)))^m \left (2 (1+m)+m \, _2F_1\left (1,1+m;2+m;\frac {1}{2} (1+\sin (c+d x))\right ) (1+\sin (c+d x))\right )}{4 d m (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, 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 \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 (c + d x \right )} + 1\right )\right )^{m} \sec {\left (c + d 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 = -0.02 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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