3.169 \(\int \frac {\text {csch}^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=89 \[ -\frac {3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

[Out]

-3/8*arctanh(cosh(a+b*ln(c*x^n)))/b/n+3/8*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))/b/n-1/4*coth(a+b*ln(c*x^n))*
csch(a+b*ln(c*x^n))^3/b/n

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Rubi [A]  time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ -\frac {3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^5/x,x]

[Out]

(-3*ArcTanh[Cosh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(8*b*n) - (Co
th[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^3)/(4*b*n)

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {csch}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \operatorname {Subst}\left (\int \text {csch}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \operatorname {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=-\frac {3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 135, normalized size = 1.52 \[ \frac {3 \log \left (\tanh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}+\frac {3 \text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}-\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}+\frac {3 \text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^5/x,x]

[Out]

(3*Csch[(a + b*Log[c*x^n])/2]^2)/(32*b*n) - Csch[(a + b*Log[c*x^n])/2]^4/(64*b*n) + (3*Log[Tanh[(a + b*Log[c*x
^n])/2]])/(8*b*n) + (3*Sech[(a + b*Log[c*x^n])/2]^2)/(32*b*n) + Sech[(a + b*Log[c*x^n])/2]^4/(64*b*n)

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fricas [B]  time = 0.51, size = 1806, normalized size = 20.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(6*cosh(b*n*log(x) + b*log(c) + a)^7 + 42*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^
6 + 6*sinh(b*n*log(x) + b*log(c) + a)^7 + 2*(63*cosh(b*n*log(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*lo
g(c) + a)^5 - 22*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*(21*cosh(b*n*log(x) + b*log(c) + a)^3 - 11*cosh(b*n*lo
g(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(105*cosh(b*n*log(x) + b*log(c) + a)^4 - 110*cosh(
b*n*log(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*log(c) + a)^3 - 22*cosh(b*n*log(x) + b*log(c) + a)^3 +
2*(63*cosh(b*n*log(x) + b*log(c) + a)^5 - 110*cosh(b*n*log(x) + b*log(c) + a)^3 - 33*cosh(b*n*log(x) + b*log(c
) + a))*sinh(b*n*log(x) + b*log(c) + a)^2 - 3*(cosh(b*n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c
) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 + 4*(7*cosh(b*n*log(x) + b*log(c)
+ a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^6 - 4*cosh(b*n*log(x) + b*log(c) + a)^6 + 8*(7*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)^5 + 2*(35*cosh(b*n*log(x)
+ b*log(c) + a)^4 - 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x) + b*log(c) + a)^4 + 6*cosh(b*n*
log(x) + b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*c
osh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^6 - 1
5*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)
^2 - 4*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7 - 3*cosh(b*n*log(x) + b*log(c)
+ a)^5 + 3*cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) +
a) + 1)*log(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a) + 1) + 3*(cosh(b*n*log(x) + b*lo
g(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c)
+ a)^8 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^6 - 4*cosh(b*n*log(x) +
b*log(c) + a)^6 + 8*(7*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)^5 + 2*(35*cosh(b*n*log(x) + b*log(c) + a)^4 - 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b
*n*log(x) + b*log(c) + a)^4 + 6*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 - 1
0*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)^3 + 4
*(7*cosh(b*n*log(x) + b*log(c) + a)^6 - 15*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) +
a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 4*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log
(c) + a)^7 - 3*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*l
og(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log
(c) + a) - 1) + 2*(21*cosh(b*n*log(x) + b*log(c) + a)^6 - 55*cosh(b*n*log(x) + b*log(c) + a)^4 - 33*cosh(b*n*l
og(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x) + b*log(c) + a) + 6*cosh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*
n*log(x) + b*log(c) + a)^8 + 8*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sin
h(b*n*log(x) + b*log(c) + a)^8 - 4*b*n*cosh(b*n*log(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c)
+ a)^2 - b*n)*sinh(b*n*log(x) + b*log(c) + a)^6 + 6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n
*log(x) + b*log(c) + a)^3 - 3*b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b
*n*cosh(b*n*log(x) + b*log(c) + a)^4 - 30*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*l
og(c) + a)^4 - 4*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*b*n*c
osh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 +
4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^6 - 15*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*b*n*cosh(b*n*log(x)
+ b*log(c) + a)^2 - b*n)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 -
3*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - b*n*cosh(b*n*log(x) + b*lo
g(c) + a))*sinh(b*n*log(x) + b*log(c) + a))

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giac [B]  time = 0.22, size = 248, normalized size = 2.79 \[ -\frac {1}{8} \, c^{5 \, b} {\left (\frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {2 \, {\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} - 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*c^(5*b)*(3*c^b*e^(-5*a)*log(sqrt(2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*pi*b)*e^a + x^(2*b*n)*abs(
c)^(2*b)*e^(2*a) + 1))/(b*c^(6*b)*n) - 3*c^b*e^(-5*a)*log(sqrt(-2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*
pi*b)*e^a + x^(2*b*n)*abs(c)^(2*b)*e^(2*a) + 1))/(b*c^(6*b)*n) - 2*(3*c^(6*b)*x^(7*b*n)*e^(6*a) - 11*c^(4*b)*x
^(5*b*n)*e^(4*a) - 11*c^(2*b)*x^(3*b*n)*e^(2*a) + 3*x^(b*n))*e^(-4*a)/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^4*b*c^(
4*b)*n))*e^(5*a)

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maple [A]  time = 0.31, size = 84, normalized size = 0.94 \[ -\frac {\coth \left (a +b \ln \left (c \,x^{n}\right )\right ) \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{4 b n}+\frac {3 \coth \left (a +b \ln \left (c \,x^{n}\right )\right ) \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8 b n}-\frac {3 \arctanh \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))^3/b/n+3/8*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))/b/n-3/4/b/n*arc
tanh(exp(a+b*ln(c*x^n)))

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maxima [B]  time = 0.35, size = 232, normalized size = 2.61 \[ \frac {3 \, c^{7 \, b} e^{\left (7 \, b \log \left (x^{n}\right ) + 7 \, a\right )} - 11 \, c^{5 \, b} e^{\left (5 \, b \log \left (x^{n}\right ) + 5 \, a\right )} - 11 \, c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + 3 \, c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{4 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} - 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} + \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*c^(7*b)*e^(7*b*log(x^n) + 7*a) - 11*c^(5*b)*e^(5*b*log(x^n) + 5*a) - 11*c^(3*b)*e^(3*b*log(x^n) + 3*a)
+ 3*c^b*e^(b*log(x^n) + a))/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c
^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 3/8*log((c^b*e^(b*log(x^n) + a
) + 1)*e^(-a)/c^b)/(b*n) + 3/8*log((c^b*e^(b*log(x^n) + a) - 1)*e^(-a)/c^b)/(b*n)

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mupad [B]  time = 1.48, size = 318, normalized size = 3.57 \[ \frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {-b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n-\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*sinh(a + b*log(c*x^n))^5),x)

[Out]

(2*exp(-a))/((c*x^n)^b*(b*n - (3*b*n*exp(-2*a))/(c*x^n)^(2*b) + (3*b*n*exp(-4*a))/(c*x^n)^(4*b) - (b*n*exp(-6*
a))/(c*x^n)^(6*b))) - (3*atan((exp(-a)*(-b^2*n^2)^(1/2))/(b*n*(c*x^n)^b)))/(4*(-b^2*n^2)^(1/2)) - (3*exp(-a))/
(4*(c*x^n)^b*(b*n - (b*n*exp(-2*a))/(c*x^n)^(2*b))) - (4*exp(-3*a))/((c*x^n)^(3*b)*(b*n - (4*b*n*exp(-2*a))/(c
*x^n)^(2*b) + (6*b*n*exp(-4*a))/(c*x^n)^(4*b) - (4*b*n*exp(-6*a))/(c*x^n)^(6*b) + (b*n*exp(-8*a))/(c*x^n)^(8*b
))) - exp(-a)/(2*(c*x^n)^b*(b*n - (2*b*n*exp(-2*a))/(c*x^n)^(2*b) + (b*n*exp(-4*a))/(c*x^n)^(4*b)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(a + b*log(c*x**n))**5/x, x)

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