Optimal. Leaf size=104 \[ \frac {a^4 (A+B) (a \sin (e+f x)+a)^{m-2}}{4 f (a-a \sin (e+f x))^2}-\frac {a^2 (A (4-m)-B m) (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (2,m-2;m-1;\frac {1}{2} (\sin (e+f x)+1)\right )}{16 f (2-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2836, 78, 68} \[ \frac {a^4 (A+B) (a \sin (e+f x)+a)^{m-2}}{4 f (a-a \sin (e+f x))^2}-\frac {a^2 (A (4-m)-B m) (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (2,m-2;m-1;\frac {1}{2} (\sin (e+f x)+1)\right )}{16 f (2-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 78
Rule 2836
Rubi steps
\begin {align*} \int \sec ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {(a+x)^{-3+m} \left (A+\frac {B x}{a}\right )}{(a-x)^3} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {a^4 (A+B) (a+a \sin (e+f x))^{-2+m}}{4 f (a-a \sin (e+f x))^2}+\frac {\left (a^4 (A (4-m)-B m)\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-3+m}}{(a-x)^2} \, dx,x,a \sin (e+f x)\right )}{4 f}\\ &=-\frac {a^2 (A (4-m)-B m) \, _2F_1\left (2,-2+m;-1+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{-2+m}}{16 f (2-m)}+\frac {a^4 (A+B) (a+a \sin (e+f x))^{-2+m}}{4 f (a-a \sin (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 76, normalized size = 0.73 \[ \frac {a^2 (a (\sin (e+f x)+1))^{m-2} \left (\frac {4 (A+B)}{(\sin (e+f x)-1)^2}-\frac {(A (m-4)+B m) \, _2F_1\left (2,m-2;m-1;\frac {1}{2} (\sin (e+f x)+1)\right )}{m-2}\right )}{16 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (f x + e\right )^{5} \sin \left (f x + e\right ) + A \sec \left (f x + e\right )^{5}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.85, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{5}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\cos \left (e+f\,x\right )}^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________