3.3.53 \(\int \frac {\cosh ^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\) [253]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2715, 2720} \begin {gather*} \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}-\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

Rule 2715

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] && Integ
erQ[2*n]

Rule 2720

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

Rubi steps

\begin {align*} \int \frac {\cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \cosh ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \sqrt {\cosh \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 {\text {Subst}\left (\int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}+\frac {2 \sqrt {\cosh \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 [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.09, size = 114, normalized size = 1.70 \begin {gather*} \frac {\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sqrt {1+\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}}{3 b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sinh[2*(a + b*Log[c*x^n])] + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*(a + b*Log[c*x^n])] - Sinh[2*(a + b*L
og[c*x^n])]]*Sqrt[1 + Cosh[2*(a + b*Log[c*x^n])] + Sinh[2*(a + b*Log[c*x^n])]])/(3*b*n*Sqrt[Cosh[a + b*Log[c*x
^n]]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(236\) vs. \(2(97)=194\).
time = 3.60, size = 237, normalized size = 3.54

method	result	size
derivativedivides	\(\frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (4 \left (\cosh ^{5}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-6 \left (\cosh ^{3}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(237\)
default	\(\frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (4 \left (\cosh ^{5}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-6 \left (\cosh ^{3}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b}\)	\(237\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/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)*(4*cosh(1/2*a+1/2*b*ln(c*x^n))
^5-6*cosh(1/2*a+1/2*b*ln(c*x^n))^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)*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)))/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.19, size = 170, normalized size = 2.54 \begin {gather*} \frac {2 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*log(x) + b*log(c) + a))*weierstrassPInverse
(-4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)) + (cosh(b*n*log(x) + b*log(c) + a)^
2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 1)
*sqrt(cosh(b*n*log(x) + b*log(c) + a)))/(b*n*cosh(b*n*log(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c)
+ a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(cosh(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 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*log(c*x^n))^(3/2)/x,x)

[Out]

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

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