3.3.77 \(\int \frac {1}{x \sec ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [277]

Optimal. Leaf size=93 \[ \frac {6 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2/5*sin(a+b*ln(c*x^n))/b/n/sec(a+b*ln(c*x^n))^(3/2)+6/5*(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))*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

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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3854, 3856, 2719} \begin {gather*} \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 \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 )}{5 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sec[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(6*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]]])/(5*b*n) + (2*Si
n[a + b*Log[c*x^n]])/(5*b*n*Sec[a + b*Log[c*x^n]]^(3/2))

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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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*} \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sec ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {\sec (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\left (3 \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 )}{5 n}\\ &=\frac {6 \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 )}}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 83, normalized size = 0.89 \begin {gather*} \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \left (12 \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 )+\sin \left (3 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{10 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sec[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(Sqrt[Sec[a + b*Log[c*x^n]]]*(12*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Lo
g[c*x^n]] + Sin[3*(a + b*Log[c*x^n])]))/(10*b*n)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(119)=238\).
time = 0.36, size = 280, normalized size = 3.01

method	result	size
derivativedivides	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{6}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right ) \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{5 n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(280\)
default	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{6}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right ) \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{5 n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/sec(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/n*((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)*(-8*cos(1/2*a+1/2*b*ln(c*x^n))*
sin(1/2*a+1/2*b*ln(c*x^n))^6+8*cos(1/2*a+1/2*b*ln(c*x^n))*sin(1/2*a+1/2*b*ln(c*x^n))^4-2*sin(1/2*a+1/2*b*ln(c*
x^n))^2*cos(1/2*a+1/2*b*ln(c*x^n))-3*(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))^2-1)^(
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

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.70, size = 113, normalized size = 1.22 \begin {gather*} \frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{\frac {3}{2}} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 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 ) - 3 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 )}{5 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sec(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

1/5*(2*cos(b*n*log(x) + b*log(c) + a)^(3/2)*sin(b*n*log(x) + b*log(c) + a) + 3*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))) - 3*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))))/(b*n)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sec(a+b*ln(c*x**n))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5008 deep

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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