3.1515 \(\int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx\)
Optimal. Leaf size=502 \[ -\frac{3 a b \left (b^2-2 a^2\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{5 f \sqrt{d \sin (e+f x)}}-\frac{3 a \sec ^3(e+f x) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)} \left (\left (8 a^2 b-4 b^3\right ) \sin ^3(e+f x)+5 a \left (a^2-b^2\right ) \sin ^2(e+f x)+2 b \left (b^2-7 a^2\right ) \sin (e+f x)-a \left (7 a^2+b^2\right )\right )}{20 d f}-\frac{3 b \left (-3 a^2 b^2+2 a^4+b^4\right ) \tan (e+f x) \sqrt{d \sin (e+f x)} \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{-\frac{a (\sin (e+f x)+1) \csc ^2(e+f x) (a+b \sin (e+f x))}{(a-b)^2}} E\left (\sin ^{-1}\left (\sqrt{-\frac{b+a \csc (e+f x)}{a-b}}\right )|1-\frac{2 a}{a+b}\right )}{5 d f \sqrt{a+b \sin (e+f x)}}-\frac{3 a (a+b)^{3/2} \left (5 a^2+3 a b-4 b^2\right ) \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{20 \sqrt{d} f}+\frac{\sec ^5(e+f x) \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f} \]
[Out]
(-3*a*b*(-2*a^2 + b^2)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(5*f*Sqrt[d*Sin[e + f*x]]) + (Sec[e + f*x]^5*Sqr
t[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2))/(5*d*f) - (3*a*Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Si
n[e + f*x]]*(-(a*(7*a^2 + b^2)) + 2*b*(-7*a^2 + b^2)*Sin[e + f*x] + 5*a*(a^2 - b^2)*Sin[e + f*x]^2 + (8*a^2*b
- 4*b^3)*Sin[e + f*x]^3))/(20*d*f) - (3*a*(a + b)^(3/2)*(5*a^2 + 3*a*b - 4*b^2)*Sqrt[-((a*(-1 + Csc[e + f*x]))
/(a + b))]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a +
b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(20*Sqrt[d]*f) - (3*b*(2*a^4 - 3*a^2*b^2 + b^4)*S
qrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*EllipticE[ArcSin[Sqrt[-((b + a*Csc[e + f*x])/(a - b))]], 1 - (2*a)/(a
+ b)]*Sqrt[d*Sin[e + f*x]]*Sqrt[-((a*Csc[e + f*x]^2*(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]])
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Rubi [F] time = 0.370213, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx \]
Verification is Not applicable to the result.
[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*} \int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx &=\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] time = 23.9746, size = 1600, normalized size = 3.19 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[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] time = 0.783, size = 5578, normalized size = 11.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(f*x+e)^6*(a+b*sin(f*x+e))^(9/2)/(d*sin(f*x+e))^(1/2),x)
[Out]
result too large to display
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (4 \,{\left (a b^{3} \cos \left (f x + e\right )^{2} - a^{3} b - a b^{3}\right )} \sec \left (f x + e\right )^{6} \sin \left (f x + e\right ) -{\left (b^{4} \cos \left (f x + e\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \,{\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sec \left (f x + e\right )^{6}\right )} \sqrt{b \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right )}}{d \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
integrate((b*sin(f*x + e) + a)^(9/2)*sec(f*x + e)^6/sqrt(d*sin(f*x + e)), x)