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 \[ -\frac {3 a b \left (b^2-2 a^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{5 f \sqrt {d \sin (e+f x)}}-\frac {3 a (a+b)^{3/2} \left (5 a^2+3 a b-4 b^2\right ) \tan (e+f x) \sqrt {-\frac {a (\csc (e+f x)-1)}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{20 \sqrt {d} f}-\frac {3 a \sec ^3(e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)} \left (\left (8 a^2 b-4 b^3\right ) \sin ^3(e+f x)+5 a \left (a^2-b^2\right ) \sin ^2(e+f x)+2 b \left (b^2-7 a^2\right ) \sin (e+f x)-a \left (7 a^2+b^2\right )\right )}{20 d f}-\frac {3 b \left (2 a^4-3 a^2 b^2+b^4\right ) \tan (e+f x) \sqrt {d \sin (e+f x)} \sqrt {-\frac {a (\csc (e+f x)-1)}{a+b}} \sqrt {-\frac {a (\sin (e+f x)+1) \csc ^2(e+f x) (a+b \sin (e+f x))}{(a-b)^2}} E\left (\sin ^{-1}\left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|1-\frac {2 a}{a+b}\right )}{5 d f \sqrt {a+b \sin (e+f x)}}+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f} \]

[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)

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [C]  time = 9.99, 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]])

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (4 \, {\left (a b^{3} \cos \left (f x + e\right )^{2} - a^{3} b - a b^{3}\right )} \sec \left (f x + e\right )^{6} \sin \left (f x + e\right ) - {\left (b^{4} \cos \left (f x + e\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sec \left (f x + e\right )^{6}\right )} \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right )}}{d \sin \left (f x + e\right )}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}}\,{d x} \]

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)

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maple [B]  time = 1.10, size = 5578, normalized size = 11.11 \[ \text {output too large to display} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\cos \left (e+f\,x\right )}^6\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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]

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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