Integrand size = 13, antiderivative size = 36 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {3 i x}{2}+2 \cosh (x)-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh (x)}{i+\text {csch}(x)} \] Output:
3/2*I*x+2*cosh(x)-3/2*I*cosh(x)*sinh(x)-cosh(x)*sinh(x)/(I+csch(x))
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\cosh (x)+\frac {1}{4} i \left (6 x-\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}-\sinh (2 x)\right ) \] Input:
Integrate[Sinh[x]^2/(I + Csch[x]),x]
Output:
Cosh[x] + (I/4)*(6*x - (8*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2]) - Sinh[2*x] )
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 25, 26, 4306, 3042, 25, 4274, 25, 26, 3042, 25, 26, 3115, 24, 3118}
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.
\(\displaystyle \int \frac {\sinh ^2(x)}{\text {csch}(x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{(i \csc (i x)+i) \csc (i x)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {i}{\csc (i x)^2 (\csc (i x)+1)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\csc (i x)^2 (\csc (i x)+1)}dx\) |
\(\Big \downarrow \) 4306 |
\(\displaystyle i \left (\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}-\int (2 i \text {csch}(x)+3) \sinh ^2(x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}-\int -\frac {3-2 \csc (i x)}{\csc (i x)^2}dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (\int \frac {3-2 \csc (i x)}{\csc (i x)^2}dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle i \left (3 \int -\sinh ^2(x)dx-2 \int i \sinh (x)dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-3 \int \sinh ^2(x)dx-2 \int i \sinh (x)dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-3 \int \sinh ^2(x)dx-2 i \int \sinh (x)dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (-2 i \int -i \sin (i x)dx-3 \int -\sin (i x)^2dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-2 i \int -i \sin (i x)dx+3 \int \sin (i x)^2dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-2 \int \sin (i x)dx+3 \int \sin (i x)^2dx+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle i \left (-2 \int \sin (i x)dx+3 \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle i \left (-2 \int \sin (i x)dx+3 \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle i \left (-2 i \cosh (x)+3 \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {\sinh (x) \cosh (x)}{1-i \text {csch}(x)}\right )\) |
Input:
Int[Sinh[x]^2/(I + Csch[x]),x]
Output:
I*((-2*I)*Cosh[x] + (Cosh[x]*Sinh[x])/(1 - I*Csch[x]) + 3*(x/2 - (Cosh[x]* Sinh[x])/2))
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x]))), x] - Simp[1/a^2 Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0 ]
Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {3 i x}{2}-\frac {i {\mathrm e}^{2 x}}{8}+\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {i {\mathrm e}^{-2 x}}{8}+\frac {2}{{\mathrm e}^{x}-i}\) | \(39\) |
default | \(-\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-1-\frac {i}{2}}{\tanh \left (\frac {x}{2}\right )-1}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1-\frac {i}{2}}{\tanh \left (\frac {x}{2}\right )+1}\) | \(80\) |
parallelrisch | \(\frac {\left (12 i \sinh \left (\frac {x}{2}\right )+12 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\left (-12 i \sinh \left (\frac {x}{2}\right )-12 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+8 i \cosh \left (\frac {x}{2}\right )+3 i \cosh \left (\frac {3 x}{2}\right )+i \cosh \left (\frac {5 x}{2}\right )+16 \sinh \left (\frac {x}{2}\right )-3 \sinh \left (\frac {3 x}{2}\right )+\sinh \left (\frac {5 x}{2}\right )}{8 i \cosh \left (\frac {x}{2}\right )-8 \sinh \left (\frac {x}{2}\right )}\) | \(100\) |
Input:
int(sinh(x)^2/(I+csch(x)),x,method=_RETURNVERBOSE)
Output:
3/2*I*x-1/8*I*exp(x)^2+1/2*exp(x)+1/2/exp(x)+1/8*I/exp(x)^2+2/(exp(x)-I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=-\frac {4 \, {\left (-3 i \, x + i\right )} e^{\left (3 \, x\right )} - 4 \, {\left (3 \, x + 5\right )} e^{\left (2 \, x\right )} + i \, e^{\left (5 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 i \, e^{x} - 1}{8 \, {\left (e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )}\right )}} \] Input:
integrate(sinh(x)^2/(I+csch(x)),x, algorithm="fricas")
Output:
-1/8*(4*(-3*I*x + I)*e^(3*x) - 4*(3*x + 5)*e^(2*x) + I*e^(5*x) - 3*e^(4*x) + 3*I*e^x - 1)/(e^(3*x) - I*e^(2*x))
\[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{2}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \] Input:
integrate(sinh(x)**2/(I+csch(x)),x)
Output:
Integral(sinh(x)**2/(csch(x) + I), x)
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {3}{2} i \, x + \frac {3 i \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} + 1}{4 \, {\left (2 i \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )}\right )}} + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} \] Input:
integrate(sinh(x)^2/(I+csch(x)),x, algorithm="maxima")
Output:
3/2*I*x + 1/4*(3*I*e^(-x) + 20*e^(-2*x) + 1)/(2*I*e^(-2*x) + 2*e^(-3*x)) + 1/2*e^(-x) + 1/8*I*e^(-2*x)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {3}{2} i \, x + \frac {{\left (20 \, e^{\left (2 \, x\right )} - 3 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )}}{8 \, {\left (e^{x} - i\right )}} - \frac {1}{8} i \, e^{\left (2 \, x\right )} + \frac {1}{2} \, e^{x} \] Input:
integrate(sinh(x)^2/(I+csch(x)),x, algorithm="giac")
Output:
3/2*I*x + 1/8*(20*e^(2*x) - 3*I*e^x + 1)*e^(-2*x)/(e^x - I) - 1/8*I*e^(2*x ) + 1/2*e^x
Time = 2.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {x\,3{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^x}{2}+\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \] Input:
int(sinh(x)^2/(1/sinh(x) + 1i),x)
Output:
(x*3i)/2 + exp(-x)/2 + (exp(-2*x)*1i)/8 - (exp(2*x)*1i)/8 + exp(x)/2 + 2/( exp(x) - 1i)
\[ \int \frac {\sinh ^2(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh \left (x \right )^{2}}{\mathrm {csch}\left (x \right )+i}d x \] Input:
int(sinh(x)^2/(I+csch(x)),x)
Output:
int(sinh(x)**2/(csch(x) + i),x)