3.3.57 \(\int \frac {1}{x \cosh ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [257]

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

[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))/b/n/cosh(a+b*ln(c*x^n))^(3/2)

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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 = {2716, 2720} \begin {gather*} \frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\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[1/(x*Cosh[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

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

Rule 2716

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.06, size = 122, normalized size = 1.82 \begin {gather*} \frac {2 \left (\sinh \left (a+b \log \left (c x^n\right )\right )+\cosh \left (a+b \log \left (c x^n\right )\right ) \, _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 )}\right )}{3 b n \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(97)=194\).
time = 3.49, size = 295, normalized size = 4.40

method	result	size
derivativedivides	\(\frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(295\)
default	\(\frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}}{3 n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(295\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+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)^(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))^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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*cosh(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 = 0.16, size = 501, normalized size = 7.48 \begin {gather*} \frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \sqrt {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 )^{3} + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sqrt {2}\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2}\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 ) + 2 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \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 )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sqrt {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, b n \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 )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \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(1/x/cosh(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5007 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/cosh(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

integrate(1/(x*cosh(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\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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