\(\int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx\) [482]

Optimal result

Integrand size = 25, antiderivative size = 174 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=-\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {4}{77 a^5 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{77 a^6 b^2 f \sqrt {a \sin (e+f x)}} \]

[Out]

-2/11/a/b/f/(a*sin(f*x+e))^(11/2)/(b*sec(f*x+e))^(1/2)+2/77/a^3/b/f/(a*sin(f*x+e))^(7/2)/(b*sec(f*x+e))^(1/2)+
4/77/a^5/b/f/(a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(1/2)+4/77*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*Elli
pticF(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^6/b^2/f/(a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2661, 2664, 2665, 2653, 2720} \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=-\frac {4 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{77 a^6 b^2 f \sqrt {a \sin (e+f x)}}+\frac {4}{77 a^5 b f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}+\frac {2}{77 a^3 b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}-\frac {2}{11 a b f (a \sin (e+f x))^{11/2} \sqrt {b \sec (e+f x)}} \]

[In]

Int[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(13/2)),x]

[Out]

-2/(11*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(11/2)) + 2/(77*a^3*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x
])^(7/2)) + 4/(77*a^5*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2)) - (4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[
b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(77*a^6*b^2*f*Sqrt[a*Sin[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*} \text {integral}& = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}-\frac {\int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx}{11 a^2 b^2} \\ & = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}-\frac {6 \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx}{77 a^4 b^2} \\ & = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {4}{77 a^5 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {4 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{77 a^6 b^2} \\ & = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {4}{77 a^5 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {\left (4 \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}{77 a^6 b^2} \\ & = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {4}{77 a^5 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {\left (4 \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}{77 a^6 b^2 \sqrt {a \sin (e+f x)}} \\ & = -\frac {2}{11 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{11/2}}+\frac {2}{77 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {4}{77 a^5 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{77 a^6 b^2 f \sqrt {a \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\frac {2 \cot (2 (e+f x)) \csc (2 (e+f x)) \sqrt {a \sin (e+f x)} \left ((23+6 \cos (2 (e+f x))-\cos (4 (e+f x))) \csc ^4(e+f x)+8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{77 a^7 b f \sqrt {b \sec (e+f x)} \left (-2+\sec ^2(e+f x)\right )} \]

[In]

Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(13/2)),x]

[Out]

(2*Cot[2*(e + f*x)]*Csc[2*(e + f*x)]*Sqrt[a*Sin[e + f*x]]*((23 + 6*Cos[2*(e + f*x)] - Cos[4*(e + f*x)])*Csc[e
+ f*x]^4 + 8*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(77*a^7*b*f*Sqrt[b*Sec
[e + f*x]]*(-2 + Sec[e + f*x]^2))

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+4 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )-2 \sqrt {2}\, \left (\cot ^{4}\left (f x +e \right )\right ) \csc \left (f x +e \right )+5 \sqrt {2}\, \left (\cot ^{2}\left (f x +e \right )\right ) \left (\csc ^{3}\left (f x +e \right )\right )+4 \sqrt {2}\, \left (\csc ^{5}\left (f x +e \right )\right )\right )}{77 f \sqrt {a \sin \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}\, a^{6} b}\)	\(253\)

[In]

int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(13/2),x,method=_RETURNVERBOSE)

[Out]

-1/77/f*2^(1/2)/(a*sin(f*x+e))^(1/2)/(b*sec(f*x+e))^(1/2)/a^6/b*(4*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e
)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))+4*
(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot
(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*sec(f*x+e)-2*2^(1/2)*cot(f*x+e)^4*csc(f*x+e)+5*2^(1/2)*cot(f*x+e)^2*c
sc(f*x+e)^3+4*2^(1/2)*csc(f*x+e)^5)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\frac {2 \, {\left (2 \, {\left (\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + 2 \, {\left (\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - {\left (2 \, \cos \left (f x + e\right )^{5} - 5 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{77 \, {\left (a^{7} b^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{7} b^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{7} b^{2} f \cos \left (f x + e\right )^{2} - a^{7} b^{2} f\right )}} \]

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(13/2),x, algorithm="fricas")

[Out]

2/77*(2*(cos(f*x + e)^6 - 3*cos(f*x + e)^4 + 3*cos(f*x + e)^2 - 1)*sqrt(I*a*b)*elliptic_f(arcsin(cos(f*x + e)
+ I*sin(f*x + e)), -1) + 2*(cos(f*x + e)^6 - 3*cos(f*x + e)^4 + 3*cos(f*x + e)^2 - 1)*sqrt(-I*a*b)*elliptic_f(
arcsin(cos(f*x + e) - I*sin(f*x + e)), -1) - (2*cos(f*x + e)^5 - 5*cos(f*x + e)^3 - 4*cos(f*x + e))*sqrt(a*sin
(f*x + e))*sqrt(b/cos(f*x + e)))/(a^7*b^2*f*cos(f*x + e)^6 - 3*a^7*b^2*f*cos(f*x + e)^4 + 3*a^7*b^2*f*cos(f*x
+ e)^2 - a^7*b^2*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*sec(f*x+e))**(3/2)/(a*sin(f*x+e))**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {13}{2}}} \,d x } \]

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(13/2),x, algorithm="maxima")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(13/2)), x)

Giac [F]

\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {13}{2}}} \,d x } \]

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(13/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(13/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx=\int \frac {1}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{13/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/((a*sin(e + f*x))^(13/2)*(b/cos(e + f*x))^(3/2)),x)

[Out]

int(1/((a*sin(e + f*x))^(13/2)*(b/cos(e + f*x))^(3/2)), x)