3.252 \(\int \frac {\cosh ^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=67 \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]

[Out]

-6/5*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))/b/n+2/5*cosh(a+b*ln(c*x^n))^(3/2)*sinh(a+b*ln(c*x^n))/b/n

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2635, 2639} \[ \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(((-6*I)/5)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2])/(b*n) + (2*Cosh[a + b*Log[c*x^n]]^(3/2)*Sinh[a + b*Log[c*x
^n]])/(5*b*n)

Rule 2635

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

Rubi steps

\begin {align*} \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac {3 \operatorname {Subst}\left (\int \sqrt {\cosh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac {6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 62, normalized size = 0.93 \[ \frac {\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right ) \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}-6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*I)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2] + Sqrt[Cosh[a + b*Log[c*x^n]]]*Sinh[2*(a + b*Log[c*x^n])])/(5*b
*n)

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.43, size = 256, normalized size = 3.82 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (8 \left (\cosh ^{7}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-16 \left (\cosh ^{5}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+10 \left (\cosh ^{3}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-3 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticE \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{5 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/n*((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)*(8*cosh(1/2*a+1/2*b*ln(c*x^n))
^7-16*cosh(1/2*a+1/2*b*ln(c*x^n))^5+10*cosh(1/2*a+1/2*b*ln(c*x^n))^3-3*(-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)^(1/2)*EllipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))-2*cosh(1/2*a+1/2*b*l
n(c*x^n)))/(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)/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 \[ \int \frac {\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*log(c*x^n))^(5/2)/x,x)

[Out]

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

[Out]

Timed out

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