Integrand size = 9, antiderivative size = 26 \[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\frac {\coth ^{1+n}(x)}{1+n}-\frac {\coth ^{3+n}(x)}{3+n} \] Output:
coth(x)^(1+n)/(1+n)-coth(x)^(3+n)/(3+n)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\frac {(-2-n+\cosh (2 x)) \coth ^{1+n}(x) \text {csch}^2(x)}{(1+n) (3+n)} \] Input:
Integrate[Coth[x]^n*Csch[x]^4,x]
Output:
((-2 - n + Cosh[2*x])*Coth[x]^(1 + n)*Csch[x]^2)/((1 + n)*(3 + n))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3087, 244, 2009}
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.
\(\displaystyle \int \text {csch}^4(x) \coth ^n(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec \left (-\frac {\pi }{2}+i x\right )^4 \left (-i \tan \left (-\frac {\pi }{2}+i x\right )\right )^ndx\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -i \int \coth ^n(x) \left (1-\coth ^2(x)\right )d(i \coth (x))\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -i \int \left (\coth ^n(x)-\coth ^{n+2}(x)\right )d(i \coth (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i \left (\frac {i \coth ^{n+1}(x)}{n+1}-\frac {i \coth ^{n+3}(x)}{n+3}\right )\) |
Input:
Int[Coth[x]^n*Csch[x]^4,x]
Output:
(-I)*((I*Coth[x]^(1 + n))/(1 + n) - (I*Coth[x]^(3 + n))/(3 + n))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Time = 15.37 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\coth \left (x \right ) {\mathrm e}^{n \ln \left (\coth \left (x \right )\right )}}{1+n}-\frac {\coth \left (x \right )^{3} {\mathrm e}^{n \ln \left (\coth \left (x \right )\right )}}{3+n}\) | \(33\) |
default | \(\frac {\coth \left (x \right ) {\mathrm e}^{n \ln \left (\coth \left (x \right )\right )}}{1+n}-\frac {\coth \left (x \right )^{3} {\mathrm e}^{n \ln \left (\coth \left (x \right )\right )}}{3+n}\) | \(33\) |
risch | \(-\frac {2 \left (-{\mathrm e}^{6 x}+2 n \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{4 x}+2 n \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x}-1\right ) \left ({\mathrm e}^{x}-1\right )^{-n} \left ({\mathrm e}^{x}+1\right )^{-n} \left ({\mathrm e}^{2 x}+1\right )^{n} {\mathrm e}^{-\frac {i \pi n \left (-\operatorname {csgn}\left (i \left ({\mathrm e}^{2 x}+1\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right )}^{2}+\operatorname {csgn}\left (i \left ({\mathrm e}^{2 x}+1\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}+1}\right )+{\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right )}^{3}-{\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right )}^{2} \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}+1}\right )-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}\right )}^{2}+\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{{\mathrm e}^{x}+1}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right )+{\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}\right )}^{3}-{\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+1\right )}{\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right )\right )}{2}}}{\left (1+n \right ) \left (3+n \right ) \left ({\mathrm e}^{2 x}-1\right )^{3}}\) | \(352\) |
Input:
int(coth(x)^n*csch(x)^4,x,method=_RETURNVERBOSE)
Output:
1/(1+n)*coth(x)*exp(n*ln(coth(x)))-1/(3+n)*coth(x)^3*exp(n*ln(coth(x)))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.38 \[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\frac {2 \, {\left ({\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - {\left (2 \, n + 3\right )} \cosh \left (x\right )\right )} \cosh \left (n \log \left (\frac {\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right ) + {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - {\left (2 \, n + 3\right )} \cosh \left (x\right )\right )} \sinh \left (n \log \left (\frac {\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right )\right )}}{{\left (n^{2} + 4 \, n + 3\right )} \sinh \left (x\right )^{3} + 3 \, {\left ({\left (n^{2} + 4 \, n + 3\right )} \cosh \left (x\right )^{2} - n^{2} - 4 \, n - 3\right )} \sinh \left (x\right )} \] Input:
integrate(coth(x)^n*csch(x)^4,x, algorithm="fricas")
Output:
2*((cosh(x)^3 + 3*cosh(x)*sinh(x)^2 - (2*n + 3)*cosh(x))*cosh(n*log(cosh(x )/sinh(x))) + (cosh(x)^3 + 3*cosh(x)*sinh(x)^2 - (2*n + 3)*cosh(x))*sinh(n *log(cosh(x)/sinh(x))))/((n^2 + 4*n + 3)*sinh(x)^3 + 3*((n^2 + 4*n + 3)*co sh(x)^2 - n^2 - 4*n - 3)*sinh(x))
\[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\int \coth ^{n}{\left (x \right )} \operatorname {csch}^{4}{\left (x \right )}\, dx \] Input:
integrate(coth(x)**n*csch(x)**4,x)
Output:
Integral(coth(x)**n*csch(x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 368, normalized size of antiderivative = 14.15 \[ \int \coth ^n(x) \text {csch}^4(x) \, dx=-\frac {2 \, {\left (2 \, n + 3\right )} e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 2 \, x\right )}}{n^{2} - 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} - {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} - \frac {2 \, {\left (2 \, n + 3\right )} e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 4 \, x\right )}}{n^{2} - 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} - {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} + \frac {2 \, e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 6 \, x\right )}}{n^{2} - 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} - {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} + \frac {2 \, e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )}}{n^{2} - 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} - {\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} \] Input:
integrate(coth(x)^n*csch(x)^4,x, algorithm="maxima")
Output:
-2*(2*n + 3)*e^(-n*log(e^(-x) + 1) - n*log(-e^(-x) + 1) + n*log(e^(-2*x) + 1) - 2*x)/(n^2 - 3*(n^2 + 4*n + 3)*e^(-2*x) + 3*(n^2 + 4*n + 3)*e^(-4*x) - (n^2 + 4*n + 3)*e^(-6*x) + 4*n + 3) - 2*(2*n + 3)*e^(-n*log(e^(-x) + 1) - n*log(-e^(-x) + 1) + n*log(e^(-2*x) + 1) - 4*x)/(n^2 - 3*(n^2 + 4*n + 3) *e^(-2*x) + 3*(n^2 + 4*n + 3)*e^(-4*x) - (n^2 + 4*n + 3)*e^(-6*x) + 4*n + 3) + 2*e^(-n*log(e^(-x) + 1) - n*log(-e^(-x) + 1) + n*log(e^(-2*x) + 1) - 6*x)/(n^2 - 3*(n^2 + 4*n + 3)*e^(-2*x) + 3*(n^2 + 4*n + 3)*e^(-4*x) - (n^2 + 4*n + 3)*e^(-6*x) + 4*n + 3) + 2*e^(-n*log(e^(-x) + 1) - n*log(-e^(-x) + 1) + n*log(e^(-2*x) + 1))/(n^2 - 3*(n^2 + 4*n + 3)*e^(-2*x) + 3*(n^2 + 4 *n + 3)*e^(-4*x) - (n^2 + 4*n + 3)*e^(-6*x) + 4*n + 3)
\[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\int { \coth \left (x\right )^{n} \operatorname {csch}\left (x\right )^{4} \,d x } \] Input:
integrate(coth(x)^n*csch(x)^4,x, algorithm="giac")
Output:
integrate(coth(x)^n*csch(x)^4, x)
Time = 1.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.35 \[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\frac {\left (\frac {4\,\mathrm {cosh}\left (3\,x\right )}{n^2+4\,n+3}-\frac {2\,\mathrm {cosh}\left (x\right )\,\left (4\,n+6\right )}{n^2+4\,n+3}\right )\,{\left (\frac {{\mathrm {e}}^{2\,x}+1}{{\mathrm {e}}^{2\,x}-1}\right )}^n}{2\,\mathrm {sinh}\left (3\,x\right )-\frac {2\,\mathrm {sinh}\left (x\right )\,\left (3\,n^2+12\,n+9\right )}{n^2+4\,n+3}} \] Input:
int(coth(x)^n/sinh(x)^4,x)
Output:
(((4*cosh(3*x))/(4*n + n^2 + 3) - (2*cosh(x)*(4*n + 6))/(4*n + n^2 + 3))*( (exp(2*x) + 1)/(exp(2*x) - 1))^n)/(2*sinh(3*x) - (2*sinh(x)*(12*n + 3*n^2 + 9))/(4*n + n^2 + 3))
\[ \int \coth ^n(x) \text {csch}^4(x) \, dx=\int \coth \left (x \right )^{n} \mathrm {csch}\left (x \right )^{4}d x \] Input:
int(coth(x)^n*csch(x)^4,x)
Output:
int(coth(x)**n*csch(x)**4,x)