Optimal. Leaf size=115 \[ -\frac {a 2^{\frac {m}{2}+\frac {1}{2}} (\sin (c+d x)+1)^{\frac {1-m}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{1-m} \, _2F_1\left (\frac {1-m}{2},\frac {1-m}{2};\frac {3-m}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d e (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2689, 70, 69} \[ -\frac {a 2^{\frac {m}{2}+\frac {1}{2}} (\sin (c+d x)+1)^{\frac {1-m}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{1-m} \, _2F_1\left (\frac {1-m}{2},\frac {1-m}{2};\frac {3-m}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d e (1-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 2689
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{1-m} (a-a \sin (c+d x))^{\frac {1}{2} (-1+m)} (a+a \sin (c+d x))^{\frac {1}{2} (-1+m)}\right ) \operatorname {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1-m)} (a+a x)^{\frac {1}{2} (-1-m)+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))^{1-m} (a-a \sin (c+d x))^{\frac {1}{2} (-1+m)} (a+a \sin (c+d x))^{-\frac {1}{2}+\frac {1}{2} (-1+m)+\frac {m}{2}} \left (\frac {a+a \sin (c+d x)}{a}\right )^{\frac {1}{2}-\frac {m}{2}}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{2} (-1-m)+m} (a-a x)^{\frac {1}{2} (-1-m)} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=-\frac {2^{\frac {1}{2}+\frac {m}{2}} a (e \cos (c+d x))^{1-m} \, _2F_1\left (\frac {1-m}{2},\frac {1-m}{2};\frac {3-m}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1-m}{2}} (a+a \sin (c+d x))^{-1+m}}{d e (1-m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 108, normalized size = 0.94 \[ \frac {2^{\frac {m+1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m} \, _2F_1\left (\frac {1-m}{2},\frac {1-m}{2};\frac {3-m}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (m-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (d x +c \right )\right )^{m} \left (e \cos \left (d x +c \right )\right )^{-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{- m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________