\(\int (a+b \sec ^2(e+f x))^p (d \sin (e+f x))^m \, dx\) [132]

Optimal result

Integrand size = 25, antiderivative size = 123 \[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1}{2}+p,-p,\frac {3+m}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^2(e+f x)^{\frac {1}{2}+p} \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \left (\frac {a+b-a \sin ^2(e+f x)}{a+b}\right )^{-p} \tan (e+f x)}{f (1+m)} \]

[Out]

AppellF1(1/2+1/2*m,1/2+p,-p,3/2+1/2*m,sin(f*x+e)^2,a*sin(f*x+e)^2/(a+b))*(cos(f*x+e)^2)^(1/2+p)*(a+b*sec(f*x+e
)^2)^p*(d*sin(f*x+e))^m*tan(f*x+e)/f/(1+m)/(((a+b-a*sin(f*x+e)^2)/(a+b))^p)

Rubi [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx \]

[In]

Int[(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m,x]

[Out]

Defer[Int][(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(286\) vs. \(2(123)=246\).

Time = 4.20 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.33 \[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {2+m}{2},-p,\frac {3+m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right ) \cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) (d \sin (e+f x))^m}{f (1+m) \left (\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {2+m}{2},-p,\frac {3+m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right )-\frac {\left (-2 b p \operatorname {AppellF1}\left (\frac {3+m}{2},\frac {2+m}{2},1-p,\frac {5+m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right )+(a+b) (2+m) \operatorname {AppellF1}\left (\frac {3+m}{2},\frac {4+m}{2},-p,\frac {5+m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a+b}\right )\right ) \tan ^2(e+f x)}{(a+b) (3+m)}\right )} \]

[In]

Integrate[(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m,x]

[Out]

(AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]*(a
+ b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(d*Sin[e + f*x])^m)/(f*(1 + m)*(AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)
/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - ((-2*b*p*AppellF1[(3 + m)/2, (2 + m)/2, 1 - p, (5 + m)/2
, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(2 + m)*AppellF1[(3 + m)/2, (4 + m)/2, -p, (5 + m)
/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)/((a + b)*(3 + m))))

Maple [F]

\[\int \left (a +b \sec \left (f x +e \right )^{2}\right )^{p} \left (d \sin \left (f x +e \right )\right )^{m}d x\]

[In]

int((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x)

[Out]

int((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x)

Fricas [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)

Sympy [F(-1)]

Timed out. \[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(f*x+e)**2)**p*(d*sin(f*x+e))**m,x)

[Out]

Timed out

Maxima [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)

Giac [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \sec ^2(e+f x)\right )^p (d \sin (e+f x))^m \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^m\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \]

[In]

int((d*sin(e + f*x))^m*(a + b/cos(e + f*x)^2)^p,x)

[Out]

int((d*sin(e + f*x))^m*(a + b/cos(e + f*x)^2)^p, x)