Optimal. Leaf size=115 \[ -\frac {2 e \cos (c+d x) (1-\cos (c+d x))^{\frac {1-m}{2}} (\cos (c+d x)+1)^{1-\frac {m}{2}} F_1\left (\frac {3}{2};\frac {1-m}{2},\frac {2-m}{2};\frac {5}{2};\cos (c+d x),-\cos (c+d x)\right ) (e \sin (c+d x))^{m-1}}{3 d \sqrt {a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3876, 2886, 135, 133} \[ -\frac {2 e \cos (c+d x) (1-\cos (c+d x))^{\frac {1-m}{2}} (\cos (c+d x)+1)^{1-\frac {m}{2}} F_1\left (\frac {3}{2};\frac {1-m}{2},\frac {2-m}{2};\frac {5}{2};\cos (c+d x),-\cos (c+d x)\right ) (e \sin (c+d x))^{m-1}}{3 d \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 135
Rule 2886
Rule 3876
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^m}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {\sqrt {-a-a \cos (c+d x)} \int \frac {\sqrt {-\cos (c+d x)} (e \sin (c+d x))^m}{\sqrt {-a-a \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {\left (e (-a-a \cos (c+d x))^{\frac {1}{2}+\frac {1-m}{2}} (-a+a \cos (c+d x))^{\frac {1-m}{2}} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int \sqrt {-x} (-a-a x)^{-\frac {1}{2}+\frac {1}{2} (-1+m)} (-a+a x)^{\frac {1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {\left (e (1+\cos (c+d x))^{1-\frac {m}{2}} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{\frac {1-m}{2}} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int \sqrt {-x} (1+x)^{-\frac {1}{2}+\frac {1}{2} (-1+m)} (-a+a x)^{\frac {1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {\left (e (1-\cos (c+d x))^{\frac {1}{2}-\frac {m}{2}} (1+\cos (c+d x))^{1-\frac {m}{2}} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-1+m)} \sqrt {-x} (1+x)^{-\frac {1}{2}+\frac {1}{2} (-1+m)} \, dx,x,\cos (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {2 e F_1\left (\frac {3}{2};\frac {1-m}{2},\frac {2-m}{2};\frac {5}{2};\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{\frac {1-m}{2}} \cos (c+d x) (1+\cos (c+d x))^{1-\frac {m}{2}} (e \sin (c+d x))^{-1+m}}{3 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 2.07, size = 277, normalized size = 2.41 \[ \frac {4 (m+3) \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {m+1}{2};-\frac {1}{2},m+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (e \sin (c+d x))^m}{d (m+1) \sqrt {a (\sec (c+d x)+1)} \left ((\cos (c+d x)-1) \left (2 (m+1) F_1\left (\frac {m+3}{2};-\frac {1}{2},m+2;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+F_1\left (\frac {m+3}{2};\frac {1}{2},m+1;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )+(m+3) (\cos (c+d x)+1) F_1\left (\frac {m+1}{2};-\frac {1}{2},m+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \sin \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x +c \right )\right )^{m}}{\sqrt {a +a \sec \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^m}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin {\left (c + d x \right )}\right )^{m}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________