3.15.0 ∫ sec4 (e+fx)(a+b sin(e+fx))5∕2 _____________________________________________

\(\int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1480]

   Optimal result
   Rubi [F]
   Mathematica [B] (warning: unable to verify)
   Maple [B] (warning: unable to verify)
   Fricas [F]
   Sympy [F(-1)]
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 35, antiderivative size = 366 \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {5 a \sec (e+f x) (b+a \sin (e+f x)) \sqrt {a+b \sin (e+f x)}}{6 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}-\frac {5 a (a+b)^{3/2} \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{6 \sqrt {d} f}-\frac {5 a b (a+b) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {b+a \csc (e+f x)}{-a+b}} E\left (\arcsin \left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {-a+b}{a+b}\right ) (1+\sin (e+f x)) \tan (e+f x)}{6 f \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]

[Out]

1/3*sec(f*x+e)^3*(a+b*sin(f*x+e))^(5/2)*(d*sin(f*x+e))^(1/2)/d/f+5/6*a*sec(f*x+e)*(b+a*sin(f*x+e))*(a+b*sin(f*

x+e))^(1/2)/f/(d*sin(f*x+e))^(1/2)-5/6*a*(a+b)^(3/2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*s

in(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x

+e)/f/d^(1/2)-5/6*a*b*(a+b)*EllipticE(((-b-a*csc(f*x+e))/(a-b))^(1/2),((-a+b)/(a+b))^(1/2))*(1+sin(f*x+e))*(-a

*(-1+csc(f*x+e))/(a+b))^(1/2)*((b+a*csc(f*x+e))/(-a+b))^(1/2)*tan(f*x+e)/f/(a*(1+csc(f*x+e))/(a-b))^(1/2)/(d*s

in(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)

Rubi [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx \]

[In]

Int[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2))/(3*d*f) + (5*a*Defer[Int][(Sec[e + f*x]^2*(a

+ b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x])/6

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}+\frac {1}{6} (5 a) \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4665\) vs. \(2(366)=732\).

Time = 28.72 (sec) , antiderivative size = 4665, normalized size of antiderivative = 12.75 \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]^3*(a^2 + b^2 + 2*a*b*Sin[e + f*x]))/3 + (Sec[e + f*x]*(5

*a^2 - 2*b^2 + 5*a*b*Sin[e + f*x]))/6))/(f*Sqrt[d*Sin[e + f*x]]) + (5*a*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*

Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]]*((5*a^2*Sqrt[a + b*Sin[e + f*x]])/(12*Sqrt[Sin[e + f*x]]) - (5*a*b*Sqr

t[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/6)*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f

*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x

)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*Elliptic

F[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b

+ Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[

-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(96*f*Sqrt[d*Sin[

e + f*x]]*((5*a*b*Cos[e + f*x]*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^(7/2)*(-2*b*Tan[(e + f*x)/2]

^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[S

qrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-

a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2

+ b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^

2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-

b + Sqrt[-a^2 + b^2])])))/(192*Sqrt[a + b*Sin[e + f*x]]) + (35*a*Cos[e + f*x]*Csc[(e + f*x)/2]^4*Sec[(e + f*x)

/2]^2*Sin[e + f*x]^(5/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(

e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e

+ f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*Ell

ipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2]

)/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b +

Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/192 - (5*a*C

sc[(e + f*x)/2]^5*Sec[(e + f*x)/2]*Sin[e + f*x]^(7/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Tan[(e + f*x)/2]^2 + (2*S

qrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b +

Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2

])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/

Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt

[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[

-a^2 + b^2])])))/48 + (5*a*Csc[(e + f*x)/2]^3*Sec[(e + f*x)/2]^3*Sin[e + f*x]^(7/2)*Sqrt[a + b*Sin[e + f*x]]*(

-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-

(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2

+ b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e

+ f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])

/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a

*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/96 + (5*a*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^(7

/2)*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] - (a*Sqrt[-a^2 + b^2]*Sec[(e + f*x)/2]^

2*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b

^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f

*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*S

qrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e +

f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(2*(-b + Sqrt[-a^2 + b^2])*(a + b*Sin[e + f*x])*((a*Tan[(e + f*x)/2])/(-b

+ Sqrt[-a^2 + b^2]))^(3/2)) - (2*b*Sqrt[-a^2 + b^2]*Cos[e + f*x]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*

x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sq

rt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a

^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[

(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*S

in[e + f*x])^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) + (Sqrt[-a^2 + b^2]*((a*b*Cos[e + f*x]*Sec[

(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2])/(a^2 - b^2))*(-(b*E

llipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b

^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f

*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b

+ Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Sec

[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) + (2*Sq

rt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-1/2*(b*EllipticE[ArcSin[Sqrt[(-

b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 +

b^2])]*Sec[(e + f*x)/2]^2) - (a^2*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2

+ b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-

b + Sqrt[-a^2 + b^2])])/(4*(b + Sqrt[-a^2 + b^2])*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a^2

*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 +

b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*(-b

+ Sqrt[-a^2 + b^2])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) + (a*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x

)/2]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])/(4*Sqrt[2]*Sqrt[-a^2 + b^

2]*Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan

[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]) + (a^2*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^

2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b

^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2

+ b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])])))/((a + b*Sin[e + f*x])

*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/96))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1987\) vs. \(2(329)=658\).

Time = 3.80 (sec) , antiderivative size = 1988, normalized size of antiderivative = 5.43


method	result	size

default	\(\text {Expression too large to display}\)	\(1988\)

[In]

int(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/12/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(10*cos(f*x+e)*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a

*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)

^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc

(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2

-5*cos(f*x+e)*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(

(a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-co

t(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2

^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2-10*cos(f*x+e)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-

a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2

)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*co

t(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b+10*cos(f*

x+e)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)

+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE

((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2

))/(-a^2+b^2)^(1/2))^(1/2))*b^3+10*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b

^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^

(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^

(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2-5*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+

b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)

-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)

^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/

2))^(1/2))*a^2-10*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*((a*cot(f*x+e)

-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1

/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-

a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b+10*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2

)^(1/2)+b))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1

/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1

/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^3+5*cos(f*x+e)*2^(1/2)*a^2*b+5*sin(

f*x+e)*2^(1/2)*a*b^2-2*sin(f*x+e)*tan(f*x+e)^3*2^(1/2)*b^3+5*2^(1/2)*a^2*b-5*tan(f*x+e)*2^(1/2)*a^3+tan(f*x+e)

*2^(1/2)*a*b^2-4*sec(f*x+e)*2^(1/2)*a^2*b-2*tan(f*x+e)*sec(f*x+e)^2*2^(1/2)*a^3-6*tan(f*x+e)*sec(f*x+e)^2*2^(1

/2)*a*b^2-6*sec(f*x+e)^3*2^(1/2)*a^2*b)*2^(1/2)

Fricas [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{4}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((2*a*b*sec(f*x + e)^4*sin(f*x + e) - (b^2*cos(f*x + e)^2 - a^2 - b^2)*sec(f*x + e)^4)*sqrt(b*sin(f*x

+ e) + a)*sqrt(d*sin(f*x + e))/(d*sin(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate(sec(f*x+e)**4*(a+b*sin(f*x+e))**(5/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/sqrt(d*sin(f*x + e)), x)

Giac [F]

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/sqrt(d*sin(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\cos \left (e+f\,x\right )}^4\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int((a + b*sin(e + f*x))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)),x)

[Out]

int((a + b*sin(e + f*x))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)), x)

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