3.197 \(\int \frac {\text {sech}^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=93 \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[Out]

2*sinh(a+b*ln(c*x^n))*sech(a+b*ln(c*x^n))^(1/2)/b/n+2*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))*EllipticE(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 = 0.07, 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 = {3768, 3771, 2639} \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

Rule 2639

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

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Mathematica [A]  time = 0.09, size = 72, normalized size = 0.77 \[ \frac {2 \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \left (\sinh \left (a+b \log \left (c x^n\right )\right )+i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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maple [A]  time = 0.64, size = 141, normalized size = 1.52 \[ \frac {2 \EllipticE \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}+4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))^(3/2)/x,x)

[Out]

2/n*(EllipticE(cosh(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)*(-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))*sinh(1/2*a+1/2*b*ln(c*x^n))^2)/sinh(1/2*a+1/2*b*ln(c*x^n
))/(2*cosh(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 \[ \int \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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