Optimal. Leaf size=47 \[ -\frac {a \, _2F_1\left (2,-1+m;m;\frac {1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^{-1+m}}{4 d (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 47, 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 (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (2,m-1;m;\frac {1}{2} (\sin (c+d x)+1)\right )}{4 d (1-m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 2746
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {a^3 \text {Subst}\left (\int \frac {(a+x)^{-2+m}}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \, _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 d (1-m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 37.80, size = 10814, normalized size = 230.09 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\sec ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 ^{3}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________