\(\int \frac {\text {csch}^3(a+b \log (c x^n))}{x} \, dx\) [167]

Optimal result

Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

1/2*arctanh(cosh(a+b*ln(c*x^n)))/b/n-1/2*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))/b/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855} \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Int[Csch[a + b*Log[c*x^n]]^3/x,x]

[Out]

ArcTanh[Cosh[a + b*Log[c*x^n]]]/(2*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(2*b*n)

Rule 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {csch}^3(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}\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\text {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = \frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.95 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\log \left (\cosh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac {\log \left (\sinh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac {\text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \]

[In]

Integrate[Csch[a + b*Log[c*x^n]]^3/x,x]

[Out]

-1/8*Csch[(a + b*Log[c*x^n])/2]^2/(b*n) + Log[Cosh[(a + b*Log[c*x^n])/2]]/(2*b*n) - Log[Sinh[(a + b*Log[c*x^n]
)/2]]/(2*b*n) - Sech[(a + b*Log[c*x^n])/2]^2/(8*b*n)

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\)	\(45\)
default	\(\frac {-\frac {\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\)	\(45\)
parallelrisch	\(\frac {-{\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+{\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}-4 \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )}{8 b n}\)	\(64\)
risch	\(-\frac {c^{b} \left (x^{n}\right )^{b} \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}+{\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}\right )}{b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}-1\right )}^{2}}-\frac {\ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}-1\right )}{2 b n}+\frac {\ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}+1\right )}{2 b n}\)	\(534\)

[In]

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

[Out]

1/n/b*(-1/2*csch(a+b*ln(c*x^n))*coth(a+b*ln(c*x^n))+arctanh(exp(a+b*ln(c*x^n))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (51) = 102\).

Time = 0.28 (sec) , antiderivative size = 643, normalized size of antiderivative = 11.69 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(2*cosh(b*n*log(x) + b*log(c) + a)^3 + 6*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^
2 + 2*sinh(b*n*log(x) + b*log(c) + a)^3 - (cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) +
a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a
)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(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) + (cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*lo
g(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log
(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(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) + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*l
og(x) + b*log(c) + a) + 2*cosh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh
(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*log(
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))

Sympy [F]

\[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (51) = 102\).

Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.73 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} - \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} \]

[In]

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

[Out]

-(c^(3*b)*e^(3*b*log(x^n) + 3*a) + c^b*e^(b*log(x^n) + a))/(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 2*b*c^(2*b)*n
*e^(2*b*log(x^n) + 2*a) + b*n) + 1/2*log((c^b*e^(b*log(x^n) + a) + 1)*e^(-a)/c^b)/(b*n) - 1/2*log((c^b*e^(b*lo
g(x^n) + a) - 1)*e^(-a)/c^b)/(b*n)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (51) = 102\).

Time = 0.34 (sec) , antiderivative size = 210, normalized size of antiderivative = 3.82 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{2} \, c^{3 \, b} {\left (\frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {2 \, {\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \]

[In]

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

[Out]

1/2*c^(3*b)*(c^b*e^(-3*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^(4*b)*n) - c^b*e^(-3*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^(4*b)*n) - 2*(c^(2*b)*x^(3*b*n)*e^(2*a) + x^(b*n))*e^(-2*a)/(
(c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^2*b*c^(2*b)*n))*e^(3*a)

Mupad [B] (verification not implemented)

Time = 2.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.55 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}}+\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]

[In]

int(1/(x*sinh(a + b*log(c*x^n))^3),x)

[Out]

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