\(\int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1515]

Optimal result

Integrand size = 35, antiderivative size = 502 \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=-\frac {3 a b \left (-2 a^2+b^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{5 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}-\frac {3 a \sec ^3(e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)} \left (-a \left (7 a^2+b^2\right )+2 b \left (-7 a^2+b^2\right ) \sin (e+f x)+5 a \left (a^2-b^2\right ) \sin ^2(e+f x)+\left (8 a^2 b-4 b^3\right ) \sin ^3(e+f x)\right )}{20 d f}-\frac {3 a (a+b)^{3/2} \left (5 a^2+3 a b-4 b^2\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{20 \sqrt {d} f}-\frac {3 b \left (2 a^4-3 a^2 b^2+b^4\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} E\left (\arcsin \left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|1-\frac {2 a}{a+b}\right ) \sqrt {d \sin (e+f x)} \sqrt {-\frac {a \csc ^2(e+f x) (1+\sin (e+f x)) (a+b \sin (e+f x))}{(a-b)^2}} \tan (e+f x)}{5 d f \sqrt {a+b \sin (e+f x)}} \]

[Out]

1/5*sec(f*x+e)^5*(a+b*sin(f*x+e))^(9/2)*(d*sin(f*x+e))^(1/2)/d/f-3/5*a*b*(-2*a^2+b^2)*cos(f*x+e)*(a+b*sin(f*x+
e))^(1/2)/f/(d*sin(f*x+e))^(1/2)-3/20*a*sec(f*x+e)^3*(-a*(7*a^2+b^2)+2*b*(-7*a^2+b^2)*sin(f*x+e)+5*a*(a^2-b^2)
*sin(f*x+e)^2+(8*a^2*b-4*b^3)*sin(f*x+e)^3)*(d*sin(f*x+e))^(1/2)*(a+b*sin(f*x+e))^(1/2)/d/f-3/20*a*(a+b)^(3/2)
*(5*a^2+3*a*b-4*b^2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(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)-3/5*b*(2*a^4-3*a^2
*b^2+b^4)*EllipticE(((-b-a*csc(f*x+e))/(a-b))^(1/2),(1-2*a/(a+b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(d*s
in(f*x+e))^(1/2)*(-a*csc(f*x+e)^2*(1+sin(f*x+e))*(a+b*sin(f*x+e))/(a-b)^2)^(1/2)*tan(f*x+e)/d/f/(a+b*sin(f*x+e
))^(1/2)

Rubi [F]

\[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx \]

[In]

Int[(Sec[e + f*x]^6*(a + b*Sin[e + f*x])^(9/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sec[e + f*x]^5*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2))/(5*d*f) + (9*a*Defer[Int][(Sec[e + f*x]^4*(a
+ b*Sin[e + f*x])^(7/2))/Sqrt[d*Sin[e + f*x]], x])/10

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}+\frac {1}{10} (9 a) \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{7/2}}{\sqrt {d \sin (e+f x)}} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.52 (sec) , antiderivative size = 1600, normalized size of antiderivative = 3.19 \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sin (e+f x) \sqrt {a+b \sin (e+f x)} \left (\frac {1}{20} \sec (e+f x) \left (15 a^4-15 a^2 b^2+4 b^4+24 a^3 b \sin (e+f x)-12 a b^3 \sin (e+f x)\right )+\frac {1}{10} \sec ^3(e+f x) \left (3 a^4-3 a^2 b^2-4 b^4+9 a^3 b \sin (e+f x)-5 a b^3 \sin (e+f x)\right )+\frac {1}{5} \sec ^5(e+f x) \left (a^4+6 a^2 b^2+b^4+4 a^3 b \sin (e+f x)+4 a b^3 \sin (e+f x)\right )\right )}{f \sqrt {d \sin (e+f x)}}+\frac {3 a \sqrt {\sin (e+f x)} \left (\frac {4 a \left (5 a^4-9 a^2 b^2+4 b^4\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+4 a \left (-8 a^3 b+4 a b^3\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )+2 \left (8 a^2 b^2-4 b^4\right ) \left (\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\sin (e+f x)}}+\frac {i \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x) E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\sin (e+f x)}}\right )|-\frac {2 a}{-a-b}\right ) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x)} \sqrt {\frac {\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{b}\right )\right )}{40 f \sqrt {d \sin (e+f x)}} \]

[In]

Integrate[(Sec[e + f*x]^6*(a + b*Sin[e + f*x])^(9/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]*(15*a^4 - 15*a^2*b^2 + 4*b^4 + 24*a^3*b*Sin[e + f*x] - 1
2*a*b^3*Sin[e + f*x]))/20 + (Sec[e + f*x]^3*(3*a^4 - 3*a^2*b^2 - 4*b^4 + 9*a^3*b*Sin[e + f*x] - 5*a*b^3*Sin[e
+ f*x]))/10 + (Sec[e + f*x]^5*(a^4 + 6*a^2*b^2 + b^4 + 4*a^3*b*Sin[e + f*x] + 4*a*b^3*Sin[e + f*x]))/5))/(f*Sq
rt[d*Sin[e + f*x]]) + (3*a*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot[(-e + Pi/2 -
f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2
*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])
/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e +
f*x]]) + 4*a*(-8*a^3*b + 4*a*b^3)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt
[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 -
f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a +
b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f
*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2
]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e
+ f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e
+ f*x]])) + 2*(8*a^2*b^2 - 4*b^4)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e
+ Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a -
b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e
+ f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[
(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/
2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[
e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/
2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]],
(-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*
x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*
x]])))/b)))/(40*f*Sqrt[d*Sin[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4485\) vs. \(2(461)=922\).

Time = 5.27 (sec) , antiderivative size = 4486, normalized size of antiderivative = 8.94

method	result	size
default	\(\text {Expression too large to display}\)	\(4486\)

[In]

int(sec(f*x+e)^6*(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/40/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(-48*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*csc(f*x+e)-a*cot(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*cs
c(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^2-3*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*csc(f*x+e)-a*cot(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*b^2+24*cos(f*x+e)*(1/(b+(-a^2+b^2)^(1/2))*(a*c
sc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(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^5+
24*(-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*csc(f*x
+e)-a*cot(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^4+15*(-a^2+b^2)^(1/2)*(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)*(-(a*csc(f*x+e)-a*cot(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^4+48*(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*csc(f*x+e)-a*cot(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^4*b-72*(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*csc(f
*x+e)-a*cot(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^3-9*(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*csc(f*x+e)-a*cot(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^4*b+9*(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*csc(f*x+e)-a*cot(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*b^3+15*tan(f*x+e)*2^(1/2)*a^5+24*(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*csc(f*x+e)-a*cot(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^5-4*tan(f*x+e)*sec(f*x+e)^2*2^(1/2)*a^3*b^2-3
4*tan(f*x+e)*sec(f*x+e)^2*2^(1/2)*a*b^4+40*tan(f*x+e)*sec(f*x+e)^4*2^(1/2)*a^3*b^2+20*tan(f*x+e)*sec(f*x+e)^4*
2^(1/2)*a*b^4-3*(-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*csc(f*x+e)-a*cot(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*b^2+24*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*csc(f*x+e)-a*cot(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^4+15*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*csc(f*x+e)-a*cot(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^4+48*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*csc(f*x+e)-a*cot(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^4*b-72*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*csc(f*
x+e)-a*cot(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^3-9*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*csc(f*x+e)-a*cot(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^4*b+9*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*csc(f*x+e)-a*cot(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*b^3-48*(-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*csc(f*x+e)-a*cot(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^2-24*2^(1/2)*a^4*b+12*2^(1/2)
*a^2*b^3+40*sec(f*x+e)^5*2^(1/2)*a^2*b^3+15*sec(f*x+e)*2^(1/2)*a^4*b-11*sec(f*x+e)*2^(1/2)*a^2*b^3+6*tan(f*x+e
)*sec(f*x+e)^2*2^(1/2)*a^5+4*sec(f*x+e)^3*2^(1/2)*a^4*b-56*sec(f*x+e)^3*2^(1/2)*a^2*b^3-15*2^(1/2)*cos(f*x+e)*
a^4*b+15*2^(1/2)*cos(f*x+e)*a^2*b^3-24*2^(1/2)*sin(f*x+e)*a^3*b^2+12*2^(1/2)*sin(f*x+e)*a*b^4-9*tan(f*x+e)*2^(
1/2)*a^3*b^2+2*tan(f*x+e)*2^(1/2)*a*b^4+4*sin(f*x+e)*tan(f*x+e)^5*2^(1/2)*b^5+4*tan(f*x+e)*sec(f*x+e)^4*2^(1/2
)*a^5+20*sec(f*x+e)^5*2^(1/2)*a^4*b)*2^(1/2)

Fricas [F]

\[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(sec(f*x+e)^6*(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(4*(a*b^3*cos(f*x + e)^2 - a^3*b - a*b^3)*sec(f*x + e)^6*sin(f*x + e) - (b^4*cos(f*x + e)^4 + a^4 +
6*a^2*b^2 + b^4 - 2*(3*a^2*b^2 + b^4)*cos(f*x + e)^2)*sec(f*x + e)^6)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x
+ e))/(d*sin(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate(sec(f*x+e)**6*(a+b*sin(f*x+e))**(9/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(sec(f*x+e)^6*(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(9/2)*sec(f*x + e)^6/sqrt(d*sin(f*x + e)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate(sec(f*x+e)^6*(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\cos \left (e+f\,x\right )}^6\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int((a + b*sin(e + f*x))^(9/2)/(cos(e + f*x)^6*(d*sin(e + f*x))^(1/2)),x)

[Out]

int((a + b*sin(e + f*x))^(9/2)/(cos(e + f*x)^6*(d*sin(e + f*x))^(1/2)), x)