Integrand size = 13, antiderivative size = 47 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))-4 i \coth (x)+\frac {4}{3} i \coth ^3(x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)} \] Output:
3/2*arctanh(cosh(x))-4*I*coth(x)+4/3*I*coth(x)^3-3/2*coth(x)*csch(x)+coth( x)*csch(x)^2/(I+sinh(x))
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.13 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {1}{6} \text {sech}(x) \left (-9+9 \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}-8 i \text {csch}(x)-3 \text {csch}^2(x)+2 i \text {csch}^3(x)-16 i \sinh (x)\right ) \] Input:
Integrate[Csch[x]^4/(I + Sinh[x]),x]
Output:
(Sech[x]*(-9 + 9*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2] - (8*I)*Csch[x] - 3*Csch[x]^2 + (2*I)*Csch[x]^3 - (16*I)*Sinh[x]))/6
Time = 0.47 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3247, 25, 3042, 3227, 26, 3042, 26, 4254, 2009, 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.
\(\displaystyle \int \frac {\text {csch}^4(x)}{\sinh (x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i-i \sin (i x)) \sin (i x)^4}dx\) |
\(\Big \downarrow \) 3247 |
\(\displaystyle \int -\text {csch}^4(x) (4 i-3 \sinh (x))dx+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}-\int \text {csch}^4(x) (4 i-3 \sinh (x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}-\int \frac {3 i \sin (i x)+4 i}{\sin (i x)^4}dx\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle -4 i \int \text {csch}^4(x)dx-3 i \int i \text {csch}^3(x)dx+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -4 i \int \text {csch}^4(x)dx+3 \int \text {csch}^3(x)dx+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int -i \csc (i x)^3dx-4 i \int \csc (i x)^4dx+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 i \int \csc (i x)^3dx-4 i \int \csc (i x)^4dx+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -3 i \int \csc (i x)^3dx+4 \int \left (1-\coth ^2(x)\right )d(-i \coth (x))+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 i \int \csc (i x)^3dx+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -3 i \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 i \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -3 i \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 i \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -3 i \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+4 \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i}\) |
Input:
Int[Csch[x]^4/(I + Sinh[x]),x]
Output:
4*((-I)*Coth[x] + (I/3)*Coth[x]^3) - (3*I)*((I/2)*ArcTanh[Cosh[x]] - (I/2) *Coth[x]*Csch[x]) + (Coth[x]*Csch[x]^2)/(I + Sinh[x])
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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 = 0.71 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.51
method | result | size |
default | \(-\frac {7 i \tanh \left (\frac {x}{2}\right )}{8}+\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {7 i}{8 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}\) | \(71\) |
risch | \(-\frac {9 i {\mathrm e}^{5 x}-24 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{6 x}-24 i {\mathrm e}^{3 x}+39 \,{\mathrm e}^{2 x}+7 i {\mathrm e}^{x}-16}{3 \left ({\mathrm e}^{2 x}-1\right )^{3} \left ({\mathrm e}^{x}+i\right )}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{2}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{2}\) | \(72\) |
parallelrisch | \(\frac {\left (-36 \tanh \left (\frac {x}{2}\right )-36 i\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )+i \tanh \left (\frac {x}{2}\right )^{4}-2 i \coth \left (\frac {x}{2}\right )^{2}-18 i \tanh \left (\frac {x}{2}\right )^{2}-\coth \left (\frac {x}{2}\right )^{3}+2 \tanh \left (\frac {x}{2}\right )^{3}-90 i+18 \coth \left (\frac {x}{2}\right )}{24 \tanh \left (\frac {x}{2}\right )+24 i}\) | \(80\) |
Input:
int(csch(x)^4/(I+sinh(x)),x,method=_RETURNVERBOSE)
Output:
-7/8*I*tanh(1/2*x)+1/24*I*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2+1/24*I/tanh(1/2* x)^3-7/8*I/tanh(1/2*x)-1/8/tanh(1/2*x)^2-3/2*ln(tanh(1/2*x))-2*I/(tanh(1/2 *x)+I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (35) = 70\).
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.70 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {9 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 9 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 18 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} + 48 \, e^{\left (4 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 78 \, e^{\left (2 \, x\right )} - 14 i \, e^{x} + 32}{6 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )}} \] Input:
integrate(csch(x)^4/(I+sinh(x)),x, algorithm="fricas")
Output:
1/6*(9*(e^(7*x) + I*e^(6*x) - 3*e^(5*x) - 3*I*e^(4*x) + 3*e^(3*x) + 3*I*e^ (2*x) - e^x - I)*log(e^x + 1) - 9*(e^(7*x) + I*e^(6*x) - 3*e^(5*x) - 3*I*e ^(4*x) + 3*e^(3*x) + 3*I*e^(2*x) - e^x - I)*log(e^x - 1) - 18*e^(6*x) - 18 *I*e^(5*x) + 48*e^(4*x) + 48*I*e^(3*x) - 78*e^(2*x) - 14*I*e^x + 32)/(e^(7 *x) + I*e^(6*x) - 3*e^(5*x) - 3*I*e^(4*x) + 3*e^(3*x) + 3*I*e^(2*x) - e^x - I)
\[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \] Input:
integrate(csch(x)**4/(I+sinh(x)),x)
Output:
Integral(csch(x)**4/(sinh(x) + I), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (35) = 70\).
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.19 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {-7 i \, e^{\left (-x\right )} + 39 \, e^{\left (-2 \, x\right )} + 24 i \, e^{\left (-3 \, x\right )} - 24 \, e^{\left (-4 \, x\right )} - 9 i \, e^{\left (-5 \, x\right )} + 9 \, e^{\left (-6 \, x\right )} - 16}{3 \, {\left (e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i\right )}} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate(csch(x)^4/(I+sinh(x)),x, algorithm="maxima")
Output:
1/3*(-7*I*e^(-x) + 39*e^(-2*x) + 24*I*e^(-3*x) - 24*e^(-4*x) - 9*I*e^(-5*x ) + 9*e^(-6*x) - 16)/(e^(-x) + 3*I*e^(-2*x) - 3*e^(-3*x) - 3*I*e^(-4*x) + 3*e^(-5*x) + I*e^(-6*x) - e^(-7*x) - I) + 3/2*log(e^(-x) + 1) - 3/2*log(e^ (-x) - 1)
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.23 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=-\frac {2}{e^{x} + i} - \frac {3 \, e^{\left (5 \, x\right )} + 6 i \, e^{\left (4 \, x\right )} - 24 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 10 i}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(csch(x)^4/(I+sinh(x)),x, algorithm="giac")
Output:
-2/(e^x + I) - 1/3*(3*e^(5*x) + 6*I*e^(4*x) - 24*I*e^(2*x) - 3*e^x + 10*I) /(e^(2*x) - 1)^3 + 3/2*log(e^x + 1) - 3/2*log(abs(e^x - 1))
Time = 1.97 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \] Input:
int(1/(sinh(x)^4*(sinh(x) + 1i)),x)
Output:
(3*log(3*exp(x) + 3))/2 - (3*log(3*exp(x) - 3))/2 - exp(x)/(exp(2*x) - 1) - (2*exp(x))/(exp(2*x) - 1)^2 - 2/(exp(x) + 1i) - 2i/(exp(2*x) - 1) + 4i/( exp(2*x) - 1)^2 + 8i/(3*(exp(2*x) - 1)^3)
\[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\int \frac {\mathrm {csch}\left (x \right )^{4}}{\sinh \left (x \right )+i}d x \] Input:
int(csch(x)^4/(I+sinh(x)),x)
Output:
int(csch(x)**4/(sinh(x) + i),x)