3.5.81 \(\int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx\) [481]

Optimal. Leaf size=137 \[ -\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{21 a^4 b^2 f \sqrt {a \sin (e+f x)}} \]

[Out]

-2/7/a/b/f/(a*sin(f*x+e))^(7/2)/(b*sec(f*x+e))^(1/2)+2/21/a^3/b/f/(a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(1/2)+2/
21*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*(b*sec(f*x+e))^(1/2)*sin
(2*f*x+2*e)^(1/2)/a^4/b^2/f/(a*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2661, 2664, 2665, 2653, 2720} \begin {gather*} -\frac {2 \sqrt {\sin (2 e+2 f x)} F\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {b \sec (e+f x)}}{21 a^4 b^2 f \sqrt {a \sin (e+f x)}}+\frac {2}{21 a^3 b f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(9/2)),x]

[Out]

-2/(7*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2)) + 2/(21*a^3*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])
^(3/2)) - (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(21*a^4*b^2*f*Sqrt[a*Si
n[e + f*x]])

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2661

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a*Sin[e + f*
x])^(m + 1)*((b*Sec[e + f*x])^(n + 1)/(a*b*f*(m + 1))), x] - Dist[(n + 1)/(a^2*b^2*(m + 1)), Int[(a*Sin[e + f*
x])^(m + 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && LtQ[m, -1] && Integers
Q[2*m, 2*n]

Rule 2664

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(a*Sin[e +
f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1)/(a*f*(m + 1))), x] + Dist[(m - n + 2)/(a^2*(m + 1)), Int[(a*Sin[e + f*
x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2665

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(b*Cos[e + f*
x])^n*(b*Sec[e + f*x])^n, Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&
IntegerQ[m - 1/2] && IntegerQ[n - 1/2]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx &=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}-\frac {\int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx}{7 a^2 b^2}\\ &=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{21 a^4 b^2}\\ &=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {\left (2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}} \, dx}{21 a^4 b^2}\\ &=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {\left (2 \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{21 a^4 b^2 \sqrt {a \sin (e+f x)}}\\ &=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{21 a^4 b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.64, size = 119, normalized size = 0.87 \begin {gather*} \frac {\cos (2 (e+f x)) \csc ^4(e+f x) \sqrt {a \sin (e+f x)} \left ((5+\cos (2 (e+f x))) \sec ^2(e+f x)-2 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{7/4}\right )}{21 a^5 b f \sqrt {b \sec (e+f x)} \left (-2+\sec ^2(e+f x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(9/2)),x]

[Out]

(Cos[2*(e + f*x)]*Csc[e + f*x]^4*Sqrt[a*Sin[e + f*x]]*((5 + Cos[2*(e + f*x)])*Sec[e + f*x]^2 - 2*Hypergeometri
c2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(7/4)))/(21*a^5*b*f*Sqrt[b*Sec[e + f*x]]*(-2 + Sec[e + f
*x]^2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(538\) vs. \(2(142)=284\).
time = 0.23, size = 539, normalized size = 3.93

method	result	size
default	\(-\frac {\left (-2 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+2 \cos \left (f x +e \right ) \sqrt {2}\right ) \sin \left (f x +e \right ) \sqrt {2}}{21 f \cos \left (f x +e \right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (a \sin \left (f x +e \right )\right )^{\frac {9}{2}}}\)	\(539\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/21/f*(-2*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos
(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*sin(f*x+e)*cos(
f*x+e)^3-2*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(
f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*sin(f*x+e)*cos(f
*x+e)^2+2*cos(f*x+e)*sin(f*x+e)*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*
x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1
/2))+2*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+
e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*sin(f*x+e)+cos(f*x+e
)^3*2^(1/2)+2*cos(f*x+e)*2^(1/2))*sin(f*x+e)/cos(f*x+e)^2/(b/cos(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2)*2^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(9/2)), x)

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 0.12, size = 190, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ), -1\right ) + {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ), -1\right ) - {\left (\cos \left (f x + e\right )^{3} + 2 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{21 \, {\left (a^{5} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{5} b^{2} f \cos \left (f x + e\right )^{2} + a^{5} b^{2} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

2/21*((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(I*a*b)*ellipticF(cos(f*x + e) + I*sin(f*x + e), -1) + (cos(
f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-I*a*b)*ellipticF(cos(f*x + e) - I*sin(f*x + e), -1) - (cos(f*x + e)^3
+ 2*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)))/(a^5*b^2*f*cos(f*x + e)^4 - 2*a^5*b^2*f*cos(f*x
+ e)^2 + a^5*b^2*f)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))**(3/2)/(a*sin(f*x+e))**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(9/2)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(3/2)),x)

[Out]

int(1/((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(3/2)), x)

________________________________________________________________________________________