\(\int \frac {\sec ^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\) [269]

Optimal result

Integrand size = 19, antiderivative size = 89 \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

2*sin(a+b*ln(c*x^n))*sec(a+b*ln(c*x^n))^(1/2)/b/n-2*(cos(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cos(1/2*a+1/2*b*ln(c*
x^n))*EllipticE(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 = 89, 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, 2719} \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

(-2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(b*n) + (2*Sqr
t[Sec[a + b*Log[c*x^n]]]*Sin[a + b*Log[c*x^n]])/(b*n)

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\sec (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{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 \sqrt {\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \left (-\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(119)=238\).

Time = 1.85 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.81

method	result	size
derivativedivides	\(-\frac {2 \left (-2 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(250\)
default	\(-\frac {2 \left (-2 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(250\)

[In]

int(sec(a+b*ln(c*x^n))^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

-2/n*(-2*cos(1/2*a+1/2*b*ln(c*x^n))*(-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)*sin(1
/2*a+1/2*b*ln(c*x^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)*(-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)*EllipticE(cos(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)/sin(1/2*a+1/2*b*ln(c*x^n))/(2*cos(1/2*a+1/
2*b*ln(c*x^n))^2-1)^(1/2)/b

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \frac {2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sqrt {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{b n} \]

[In]

integrate(sec(a+b*log(c*x^n))^(3/2)/x,x, algorithm="fricas")

[Out]

(-I*sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*log(x
) + b*log(c) + a))) + I*sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*n*log(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)

Sympy [F]

\[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sec ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(sec(a+b*ln(c*x**n))**(3/2)/x,x)

[Out]

Integral(sec(a + b*log(c*x**n))**(3/2)/x, x)

Maxima [F]

\[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^(3/2)/x,x, algorithm="maxima")

[Out]

integrate(sec(b*log(c*x^n) + a)^(3/2)/x, x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

integrate(sec(a+b*log(c*x^n))^(3/2)/x,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]

[In]

int((1/cos(a + b*log(c*x^n)))^(3/2)/x,x)

[Out]

int((1/cos(a + b*log(c*x^n)))^(3/2)/x, x)