3.196 \(\int \frac {\text {sech}^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=97 \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]

[Out]

2/3*sech(a+b*ln(c*x^n))^(3/2)*sinh(a+b*ln(c*x^n))/b/n-2/3*I*(cosh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cosh(1/2*a+1
/2*b*ln(c*x^n))*EllipticF(I*sinh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))*cosh(a+b*ln(c*x^n))^(1/2)*sech(a+b*ln(c*x^n))
^(1/2)/b/n

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Rubi [A]  time = 0.06, antiderivative size = 97, 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 = {3768, 3771, 2641} \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(((-2*I)/3)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(I/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/
(b*n) + (2*Sech[a + b*Log[c*x^n]]^(3/2)*Sinh[a + b*Log[c*x^n]])/(3*b*n)

Rule 2641

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

Rule 3768

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 3771

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 {\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\operatorname {Subst}\left (\int \sqrt {\text {sech}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\left (\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 74, normalized size = 0.76 \[ \frac {2 \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\sinh \left (a+b \log \left (c x^n\right )\right )-i \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sech[a + b*Log[c*x^n]]^(3/2)*((-I)*Cosh[a + b*Log[c*x^n]]^(3/2)*EllipticF[(I/2)*(a + b*Log[c*x^n]), 2] + Si
nh[a + b*Log[c*x^n]]))/(3*b*n)

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sech(b*log(c*x^n) + a)^(5/2)/x, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.73, size = 295, normalized size = 3.04 \[ \frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))^(5/2)/x,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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