\(\int \frac {\sec ^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [271]

Optimal result

Integrand size = 19, antiderivative size = 93 \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

2/3*sec(a+b*ln(c*x^n))^(3/2)*sin(a+b*ln(c*x^n))/b/n+2/3*(cos(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cos(1/2*a+1/2*b*l
n(c*x^n))*EllipticF(sin(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))*cos(a+b*ln(c*x^n))^(1/2)*sec(a+b*ln(c*x^n))^(1/2)/b/n

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3853, 3856, 2720} \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(3*b*n) + (2*Se
c[a + b*Log[c*x^n]]^(3/2)*Sin[a + b*Log[c*x^n]])/(3*b*n)

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]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sec ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int \sqrt {\sec (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = \frac {2 \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\left (\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = \frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )+\sin \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(2*Sec[a + b*Log[c*x^n]]^(3/2)*(Cos[a + b*Log[c*x^n]]^(3/2)*EllipticF[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log
[c*x^n]]))/(3*b*n)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(119)=238\).

Time = 20.80 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.13

method	result	size
derivativedivides	\(-\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right )}^{\frac {3}{2}} \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(291\)
default	\(-\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right )}^{\frac {3}{2}} \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(291\)

[In]

int(sec(a+b*ln(c*x^n))^(5/2)/x,x,method=_RETURNVERBOSE)

[Out]

-2/3/n*(-2*(sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-1+2*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*EllipticF(cos(1/2*a+
1/2*b*ln(c*x^n)),2^(1/2))*sin(1/2*a+1/2*b*ln(c*x^n))^2-2*sin(1/2*a+1/2*b*ln(c*x^n))^2*cos(1/2*a+1/2*b*ln(c*x^n
))+(sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-1+2*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*EllipticF(cos(1/2*a+1/2*b*ln
(c*x^n)),2^(1/2)))*((2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/(-2*sin(1/2*a+1/2*b
*ln(c*x^n))^4+sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/(2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)^(3/2)/sin(1/2*a+1/2*b*ln(
c*x^n))/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.56 \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + \frac {2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sqrt {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{3 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]

[In]

integrate(sec(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")

[Out]

1/3*(-I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)*weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) + I*s
in(b*n*log(x) + b*log(c) + a)) + I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)*weierstrassPInverse(-4, 0, cos(b*n*l
og(x) + b*log(c) + a) - I*sin(b*n*log(x) + b*log(c) + a)) + 2*sin(b*n*log(x) + b*log(c) + a)/sqrt(cos(b*n*log(
x) + b*log(c) + a)))/(b*n*cos(b*n*log(x) + b*log(c) + a))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

integrate(sec(a+b*ln(c*x**n))**(5/2)/x,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^(5/2)/x,x, algorithm="maxima")

[Out]

integrate(sec(b*log(c*x^n) + a)^(5/2)/x, x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

integrate(sec(a+b*log(c*x^n))^(5/2)/x,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \]

[In]

int((1/cos(a + b*log(c*x^n)))^(5/2)/x,x)

[Out]

int((1/cos(a + b*log(c*x^n)))^(5/2)/x, x)