Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \] Output:
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
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.95 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\log \left (\cosh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac {\log \left (\sinh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac {\text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \] Input:
Integrate[Csch[a + b*Log[c*x^n]]^3/x,x]
Output:
-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)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3039, 3042, 26, 4255, 26, 3042, 26, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \text {csch}^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i \csc \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \csc \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \int -i \text {csch}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-\frac {1}{2} i \int \text {csch}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (-\frac {1}{2} i \int i \csc \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \int \csc \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {i \left (\frac {i \text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
Input:
Int[Csch[a + b*Log[c*x^n]]^3/x,x]
Output:
((-I)*(((I/2)*ArcTanh[Cosh[a + b*Log[c*x^n]]])/b - ((I/2)*Coth[a + b*Log[c *x^n]]*Csch[a + b*Log[c*x^n]])/b))/n
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.95 (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}\) | \(534\) |
Input:
int(csch(a+b*ln(c*x^n))^3/x,x,method=_RETURNVERBOSE)
Output:
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))))
Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (51) = 102\).
Time = 0.10 (sec) , antiderivative size = 643, normalized size of antiderivative = 11.69 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="fricas")
Output:
-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*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) + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(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*lo...
\[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(csch(a+b*ln(c*x**n))**3/x,x)
Output:
Integral(csch(a + b*log(c*x**n))**3/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (51) = 102\).
Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.73 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} - \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} \] Input:
integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="maxima")
Output:
-(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*l og((c^b*e^(b*log(x^n) + a) + 1)*e^(-a)/c^b)/(b*n) - 1/2*log((c^b*e^(b*log( x^n) + a) - 1)*e^(-a)/c^b)/(b*n)
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (51) = 102\).
Time = 0.22 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.33 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{2} \, c^{3 \, b} {\left (\frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {2 \, {\left | c \right |}^{b} {\left | x \right |}^{b n} \cos \left (\frac {1}{2} \, \pi b n \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, \pi b n + \frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) - \frac {1}{2} \, \pi b\right ) e^{a} + {\left | c \right |}^{2 \, b} {\left | x \right |}^{2 \, b n} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {-2 \, {\left | c \right |}^{b} {\left | x \right |}^{b n} \cos \left (\frac {1}{2} \, \pi b n \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, \pi b n + \frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) - \frac {1}{2} \, \pi b\right ) e^{a} + {\left | c \right |}^{2 \, b} {\left | x \right |}^{2 \, b n} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {2 \, {\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \] Input:
integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="giac")
Output:
1/2*c^(3*b)*(c^b*e^(-3*a)*log(sqrt(2*abs(c)^b*abs(x)^(b*n)*cos(1/2*pi*b*n* sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*e^a + abs(c)^(2*b)*abs(x )^(2*b*n)*e^(2*a) + 1))/(b*c^(4*b)*n) - c^b*e^(-3*a)*log(sqrt(-2*abs(c)^b* abs(x)^(b*n)*cos(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi *b)*e^a + abs(c)^(2*b)*abs(x)^(2*b*n)*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)
Time = 2.61 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.55 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}}+\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \] Input:
int(1/(x*sinh(a + b*log(c*x^n))^3),x)
Output:
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)) )
Time = 0.23 (sec) , antiderivative size = 272, normalized size of antiderivative = 4.95 \[ \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-2 x^{3 b n} e^{3 a} c^{3 b}-2 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+2 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-2 x^{b n} e^{a} c^{b}+\mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-\mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )}{2 b n \left (x^{4 b n} e^{4 a} c^{4 b}-2 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:
int(csch(a+b*log(c*x^n))^3/x,x)
Output:
(x**(4*b*n)*e**(4*a)*c**(4*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) - x**(4*b *n)*e**(4*a)*c**(4*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) - 2*x**(3*b*n)*e* *(3*a)*c**(3*b) - 2*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b ) + c**b) + 2*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b) - c* *b) - 2*x**(b*n)*e**a*c**b + log(x**(b*n)*e**a*c**(2*b) + c**b) - log(x**( b*n)*e**a*c**(2*b) - c**b))/(2*b*n*(x**(4*b*n)*e**(4*a)*c**(4*b) - 2*x**(2 *b*n)*e**(2*a)*c**(2*b) + 1))