Optimal. Leaf size=115 \[ -\frac {2^{\frac {3}{2}+\frac {m}{2}} a (e \cos (c+d x))^{3-m} \, _2F_1\left (\frac {1}{2} (-1-m),\frac {3-m}{2};\frac {5-m}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-m)} (a+a \sin (c+d x))^{-1+m}}{d e (3-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 72, 71}
\begin {gather*} -\frac {a 2^{\frac {m}{2}+\frac {3}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{3-m} \, _2F_1\left (\frac {1}{2} (-m-1),\frac {3-m}{2};\frac {5-m}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d e (3-m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{2-m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{3-m} (a-a \sin (c+d x))^{\frac {1}{2} (-3+m)} (a+a \sin (c+d x))^{\frac {1}{2} (-3+m)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1-m}{2}} (a+a x)^{\frac {1-m}{2}+m} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac {\left (2^{\frac {1}{2}+\frac {m}{2}} a^2 (e \cos (c+d x))^{3-m} (a-a \sin (c+d x))^{\frac {1}{2} (-3+m)} (a+a \sin (c+d x))^{\frac {1}{2}+\frac {1}{2} (-3+m)+\frac {m}{2}} \left (\frac {a+a \sin (c+d x)}{a}\right )^{-\frac {1}{2}-\frac {m}{2}}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1-m}{2}+m} (a-a x)^{\frac {1-m}{2}} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=-\frac {2^{\frac {3}{2}+\frac {m}{2}} a (e \cos (c+d x))^{3-m} \, _2F_1\left (\frac {1}{2} (-1-m),\frac {3-m}{2};\frac {5-m}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-m)} (a+a \sin (c+d x))^{-1+m}}{d e (3-m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 113, normalized size = 0.98 \begin {gather*} \frac {2^{\frac {3+m}{2}} e^2 \cos ^3(c+d x) (e \cos (c+d x))^{-m} \, _2F_1\left (\frac {1}{2} (-1-m),\frac {3-m}{2};\frac {5-m}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-3-m)} (a (1+\sin (c+d x)))^m}{d (-3+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{2-m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx\]
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} \left (e \cos {\left (c + d x \right )}\right )^{2 - m}\, 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 = -0.01 \begin {gather*} \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{2-m}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________