Optimal. Leaf size=51 \[ -\frac {a^2 \, _2F_1\left (3,-2+m;-1+m;\frac {1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^{-2+m}}{8 d (2-m)} \]
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Rubi [A]
time = 0.04, 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 = {2746, 70}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 2746
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {a^5 \text {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 [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 47.34, size = 26029, normalized size = 510.37 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.14, 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 \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 = -0.02 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\cos \left (c+d\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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