Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 x}{2}+4 i \cosh (x)-\frac {4}{3} i \cosh ^3(x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)} \] Output:
-3/2*x+4*I*cosh(x)-4/3*I*cosh(x)^3+3/2*cosh(x)*sinh(x)-cosh(x)*sinh(x)^2/( I+csch(x))
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}+\sinh (2 x)\right )\right ) \] Input:
Integrate[Sinh[x]^3/(I + Csch[x]),x]
Output:
((21*I)*Cosh[x] - I*Cosh[3*x] + 3*(-6*x + (8*Sinh[x/2])/(Cosh[x/2] + I*Sin h[x/2]) + Sinh[2*x]))/12
Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 26, 26, 4306, 26, 3042, 26, 4274, 25, 26, 3042, 25, 26, 3113, 2009, 3115, 24}
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.
\(\displaystyle \int \frac {\sinh ^3(x)}{\text {csch}(x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{(i \csc (i x)+i) \csc (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int -\frac {i}{\csc (i x)^3 (\csc (i x)+1)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {1}{\csc (i x)^3 (1+\csc (i x))}dx\) |
\(\Big \downarrow \) 4306 |
\(\displaystyle \frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}-\int i (3 i \text {csch}(x)+4) \sinh ^3(x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}-i \int (3 i \text {csch}(x)+4) \sinh ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}-i \int \frac {i (4-3 \csc (i x))}{\csc (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {4-3 \csc (i x)}{\csc (i x)^3}dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle 4 \int -i \sinh ^3(x)dx-3 \int -\sinh ^2(x)dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 \int -i \sinh ^3(x)dx+3 \int \sinh ^2(x)dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -4 i \int \sinh ^3(x)dx+3 \int \sinh ^2(x)dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int -\sin (i x)^2dx-4 i \int i \sin (i x)^3dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 \int \sin (i x)^2dx-4 i \int i \sin (i x)^3dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 \int \sin (i x)^2dx+4 \int \sin (i x)^3dx+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -3 \int \sin (i x)^2dx+4 i \int \left (1-\cosh ^2(x)\right )d\cosh (x)+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \sin (i x)^2dx+4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -3 \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-3 \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {i \sinh ^2(x) \cosh (x)}{1-i \text {csch}(x)}\) |
Input:
Int[Sinh[x]^3/(I + Csch[x]),x]
Output:
(4*I)*(Cosh[x] - Cosh[x]^3/3) + (I*Cosh[x]*Sinh[x]^2)/(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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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 = 0.74 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {3 x}{2}-\frac {i {\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{8}+\frac {7 i {\mathrm e}^{x}}{8}+\frac {7 i {\mathrm e}^{-x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {i {\mathrm e}^{-3 x}}{24}+\frac {2 i}{{\mathrm e}^{x}-i}\) | \(53\) |
default | \(-\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {1}{2}+\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )+1}+\frac {-\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2}{\tanh \left (\frac {x}{2}\right )-i}+\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\frac {1}{2}-\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}\) | \(101\) |
parallelrisch | \(\frac {\left (36 i \sinh \left (\frac {x}{2}\right )+36 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\left (-36 i \sinh \left (\frac {x}{2}\right )-36 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+85 i \cosh \left (\frac {x}{2}\right )+18 i \cosh \left (\frac {3 x}{2}\right )+2 i \cosh \left (\frac {5 x}{2}\right )-i \cosh \left (\frac {7 x}{2}\right )+5 \sinh \left (\frac {x}{2}\right )-18 \sinh \left (\frac {3 x}{2}\right )+2 \sinh \left (\frac {5 x}{2}\right )+\sinh \left (\frac {7 x}{2}\right )}{24 \cosh \left (\frac {x}{2}\right )+24 i \sinh \left (\frac {x}{2}\right )}\) | \(113\) |
Input:
int(sinh(x)^3/(I+csch(x)),x,method=_RETURNVERBOSE)
Output:
-3/2*x-1/24*I*exp(x)^3+1/8*exp(x)^2+7/8*I*exp(x)+7/8*I/exp(x)-1/8/exp(x)^2 -1/24*I/exp(x)^3+2*I/(exp(x)-I)
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 \, {\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} + 3 \, {\left (-12 i \, x - 23 i\right )} e^{\left (3 \, x\right )} + i \, e^{\left (7 \, x\right )} - 2 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1}{24 \, {\left (e^{\left (4 \, x\right )} - i \, e^{\left (3 \, x\right )}\right )}} \] Input:
integrate(sinh(x)^3/(I+csch(x)),x, algorithm="fricas")
Output:
-1/24*(3*(12*x - 7)*e^(4*x) + 3*(-12*I*x - 23*I)*e^(3*x) + I*e^(7*x) - 2*e ^(6*x) - 18*I*e^(5*x) - 18*e^(2*x) - 2*I*e^x + 1)/(e^(4*x) - I*e^(3*x))
\[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \] Input:
integrate(sinh(x)**3/(I+csch(x)),x)
Output:
Integral(sinh(x)**3/(csch(x) + I), x)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x + \frac {2 i \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} + 69 i \, e^{\left (-3 \, x\right )} + 1}{8 \, {\left (3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac {7}{8} i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} \] Input:
integrate(sinh(x)^3/(I+csch(x)),x, algorithm="maxima")
Output:
-3/2*x + 1/8*(2*I*e^(-x) - 18*e^(-2*x) + 69*I*e^(-3*x) + 1)/(3*I*e^(-3*x) + 3*e^(-4*x)) + 7/8*I*e^(-x) - 1/8*e^(-2*x) - 1/24*I*e^(-3*x)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x - \frac {{\left (-69 i \, e^{\left (3 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} - i\right )}} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {7}{8} i \, e^{x} \] Input:
integrate(sinh(x)^3/(I+csch(x)),x, algorithm="giac")
Output:
-3/2*x - 1/24*(-69*I*e^(3*x) - 18*e^(2*x) - 2*I*e^x + 1)*e^(-3*x)/(e^x - I ) - 1/24*I*e^(3*x) + 1/8*e^(2*x) + 7/8*I*e^x
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-x}\,7{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,7{}\mathrm {i}}{8}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \] Input:
int(sinh(x)^3/(1/sinh(x) + 1i),x)
Output:
(exp(-x)*7i)/8 - (3*x)/2 - exp(-2*x)/8 + exp(2*x)/8 - (exp(-3*x)*1i)/24 - (exp(3*x)*1i)/24 + (exp(x)*7i)/8 + 2i/(exp(x) - 1i)
\[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh \left (x \right )^{3}}{\mathrm {csch}\left (x \right )+i}d x \] Input:
int(sinh(x)^3/(I+csch(x)),x)
Output:
int(sinh(x)**3/(csch(x) + i),x)