Optimal. Leaf size=85 \[ \frac {c (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-1-m+n;-m+n;\frac {1}{2} (1-\sin (e+f x))\right ) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{4 f g^3 (1+m-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2932, 12, 2746,
70} \begin {gather*} \frac {c (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-m+n-1;n-m;\frac {1}{2} (1-\sin (e+f x))\right )}{4 f g^3 (m-n+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 70
Rule 2746
Rule 2932
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{-3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{-m+n}}{g^3} \, dx\\ &=\frac {\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \sec ^3(e+f x) (c-c \sin (e+f x))^{-m+n} \, dx}{g^3}\\ &=-\frac {\left (c^3 (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \text {Subst}\left (\int \frac {(c+x)^{-2-m+n}}{(c-x)^2} \, dx,x,-c \sin (e+f x)\right )}{f g^3}\\ &=\frac {c (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-1-m+n;-m+n;\frac {1}{2} (1-\sin (e+f x))\right ) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{4 f g^3 (1+m-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 36.63, size = 135, normalized size = 1.59 \begin {gather*} \frac {(g \cos (e+f x))^{-2 m} \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) \, _2F_1\left (-2-m+n,-1-m+n;-m+n;-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )^{-m+n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n}{8 f g^3 (1+m-n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (g \cos \left (f x +e \right )\right )^{-3-2 m} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{2\,m+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________