Optimal. Leaf size=47 \[ -\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right )}{4 d (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0545228, antiderivative size = 47, normalized size of antiderivative = 1., 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} \[ -\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right )}{4 d (1-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{a^3 \operatorname{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 [B] time = 0.337913, size = 111, normalized size = 2.36 \[ \frac{(a (\sin (c+d x)+1))^m \left (\frac{2 (\sin (c+d x)+1) \, _2F_1\left (1,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right )}{m+1}+\frac{(\sin (c+d x)+1) \, _2F_1\left (2,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right )}{m+1}+4 \left (\frac{1}{(m-1) (\sin (c+d x)+1)}+\frac{1}{m}\right )\right )}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]