Optimal. Leaf size=51 \[ -\frac {a^2 (a \sin (c+d x)+a)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac {1}{2} (\sin (c+d x)+1)\right )}{8 d (2-m)} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 68} \[ -\frac {a^2 (a \sin (c+d x)+a)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac {1}{2} (\sin (c+d x)+1)\right )}{8 d (2-m)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2667
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {(a+x)^{-3+m}}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2 \, _2F_1\left (3,-2+m;-1+m;\frac {1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^{-2+m}}{8 d (2-m)}\\ \end {align*}
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Mathematica [B] time = 0.74, size = 163, normalized size = 3.20 \[ \frac {(a (\sin (c+d x)+1))^m \left (\frac {6 (\sin (c+d x)+1) \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (c+d x)+1)\right )}{m+1}+\frac {3 (\sin (c+d x)+1) \, _2F_1\left (2,m+1;m+2;\frac {1}{2} (\sin (c+d x)+1)\right )}{m+1}+\frac {(\sin (c+d x)+1) \, _2F_1\left (3,m+1;m+2;\frac {1}{2} (\sin (c+d x)+1)\right )}{m+1}+\frac {12}{(m-1) (\sin (c+d x)+1)}+\frac {8}{(m-2) (\sin (c+d x)+1)^2}+\frac {12}{m}\right )}{64 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5}, x\right ) \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{5}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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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