Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3 i x}{2}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)} \] Output:
3/2*I*x-4*cosh(x)+4/3*cosh(x)^3-3/2*I*cosh(x)*sinh(x)-cosh(x)*sinh(x)^3/(I +sinh(x))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(134\) vs. \(2(46)=92\).
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.91 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {\cosh (x) \left (-16 i \left (\arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+\sqrt {\cosh ^2(x)}\right )-\left (16 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+7 \sqrt {\cosh ^2(x)}\right ) \sinh (x)-i \sqrt {\cosh ^2(x)} \sinh ^2(x)+2 \sqrt {\cosh ^2(x)} \sinh ^3(x)+i \text {arcsinh}(\sinh (x)) (i+\sinh (x))\right )}{6 \sqrt {\cosh ^2(x)} (i+\sinh (x))} \] Input:
Integrate[Sinh[x]^4/(I + Sinh[x]),x]
Output:
(Cosh[x]*((-16*I)*(ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]] + Sqrt[Cosh[x]^2]) - (16*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]] + 7*Sqrt[Cosh[x]^2])*Sinh[x] - I *Sqrt[Cosh[x]^2]*Sinh[x]^2 + 2*Sqrt[Cosh[x]^2]*Sinh[x]^3 + I*ArcSinh[Sinh[ x]]*(I + Sinh[x])))/(6*Sqrt[Cosh[x]^2]*(I + Sinh[x]))
Time = 0.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3246, 3042, 25, 26, 3227, 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.
\(\displaystyle \int \frac {\sinh ^4(x)}{\sinh (x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (i x)^4}{i-i \sin (i x)}dx\) |
\(\Big \downarrow \) 3246 |
\(\displaystyle -\int (3 i-4 \sinh (x)) \sinh ^2(x)dx-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -\left ((4 i \sin (i x)+3 i) \sin (i x)^2\right )dx-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int i \sin (i x)^2 (4 \sin (i x)+3)dx-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin (i x)^2 (4 \sin (i x)+3)dx-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle i \left (3 \int -\sinh ^2(x)dx+4 \int -i \sinh ^3(x)dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-3 \int \sinh ^2(x)dx+4 \int -i \sinh ^3(x)dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-3 \int \sinh ^2(x)dx-4 i \int \sinh ^3(x)dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (-3 \int -\sin (i x)^2dx-4 i \int i \sin (i x)^3dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (3 \int \sin (i x)^2dx-4 i \int i \sin (i x)^3dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (3 \int \sin (i x)^2dx+4 \int \sin (i x)^3dx\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle i \left (3 \int \sin (i x)^2dx+4 i \int \left (1-\cosh ^2(x)\right )d\cosh (x)\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (3 \int \sin (i x)^2dx+4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle i \left (3 \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle i \left (3 \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+4 i \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )\right )-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}\) |
Input:
Int[Sinh[x]^4/(I + Sinh[x]),x]
Output:
-((Cosh[x]*Sinh[x]^3)/(I + Sinh[x])) + I*((4*I)*(Cosh[x] - Cosh[x]^3/3) + 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[((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*c - a*d))*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(a + b*Sin[e + f*x]))), x] - Simp[d/(a*b) Int[(c + d* Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[ e + f*x], x], x], x] /; FreeQ[{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] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])
Time = 0.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {3 i x}{2}+\frac {{\mathrm e}^{3 x}}{24}-\frac {i {\mathrm e}^{2 x}}{8}-\frac {7 \,{\mathrm e}^{x}}{8}-\frac {7 \,{\mathrm e}^{-x}}{8}+\frac {i {\mathrm e}^{-2 x}}{8}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {2}{{\mathrm e}^{x}+i}\) | \(51\) |
default | \(-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {\frac {3}{2}-\frac {i}{2}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {-\frac {1}{2}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {-\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {-\frac {3}{2}-\frac {i}{2}}{\tanh \left (\frac {x}{2}\right )+1}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(102\) |
parallelrisch | \(\frac {\left (-36 i \sinh \left (\frac {x}{2}\right )+36 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\left (36 i \sinh \left (\frac {x}{2}\right )-36 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-85 i \cosh \left (\frac {x}{2}\right )-18 i \cosh \left (\frac {3 x}{2}\right )-2 i \cosh \left (\frac {5 x}{2}\right )+i \cosh \left (\frac {7 x}{2}\right )+5 \sinh \left (\frac {x}{2}\right )-18 \sinh \left (\frac {3 x}{2}\right )+2 \sinh \left (\frac {5 x}{2}\right )+\sinh \left (\frac {7 x}{2}\right )}{24 i \cosh \left (\frac {x}{2}\right )+24 \sinh \left (\frac {x}{2}\right )}\) | \(113\) |
Input:
int(sinh(x)^4/(I+sinh(x)),x,method=_RETURNVERBOSE)
Output:
3/2*I*x+1/24*exp(x)^3-1/8*I*exp(x)^2-7/8*exp(x)-7/8/exp(x)+1/8*I/exp(x)^2+ 1/24/exp(x)^3-2/(exp(x)+I)
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=-\frac {3 \, {\left (-12 i \, x + 7 i\right )} e^{\left (4 \, x\right )} + 3 \, {\left (12 \, x + 23\right )} e^{\left (3 \, x\right )} - e^{\left (7 \, x\right )} + 2 i \, e^{\left (6 \, x\right )} + 18 \, e^{\left (5 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i}{24 \, {\left (e^{\left (4 \, x\right )} + i \, e^{\left (3 \, x\right )}\right )}} \] Input:
integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="fricas")
Output:
-1/24*(3*(-12*I*x + 7*I)*e^(4*x) + 3*(12*x + 23)*e^(3*x) - e^(7*x) + 2*I*e ^(6*x) + 18*e^(5*x) + 18*I*e^(2*x) + 2*e^x - I)/(e^(4*x) + I*e^(3*x))
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3 i x}{2} + \frac {e^{3 x}}{24} - \frac {i e^{2 x}}{8} - \frac {7 e^{x}}{8} - \frac {7 e^{- x}}{8} + \frac {i e^{- 2 x}}{8} + \frac {e^{- 3 x}}{24} - \frac {2}{e^{x} + i} \] Input:
integrate(sinh(x)**4/(I+sinh(x)),x)
Output:
3*I*x/2 + exp(3*x)/24 - I*exp(2*x)/8 - 7*exp(x)/8 - 7*exp(-x)/8 + I*exp(-2 *x)/8 + exp(-3*x)/24 - 2/(exp(x) + I)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} i \, x - \frac {2 \, e^{\left (-x\right )} - 18 i \, e^{\left (-2 \, x\right )} + 69 \, e^{\left (-3 \, x\right )} + i}{8 \, {\left (-3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} \] Input:
integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="maxima")
Output:
3/2*I*x - 1/8*(2*e^(-x) - 18*I*e^(-2*x) + 69*e^(-3*x) + I)/(-3*I*e^(-3*x) + 3*e^(-4*x)) - 7/8*e^(-x) + 1/8*I*e^(-2*x) + 1/24*e^(-3*x)
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} i \, x - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} + i\right )}} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} \] Input:
integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="giac")
Output:
3/2*I*x - 1/24*(69*e^(3*x) + 18*I*e^(2*x) + 2*e^x - I)*e^(-3*x)/(e^x + I) + 1/24*e^(3*x) - 1/8*I*e^(2*x) - 7/8*e^x
Time = 1.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {x\,3{}\mathrm {i}}{2}-\frac {7\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \] Input:
int(sinh(x)^4/(sinh(x) + 1i),x)
Output:
(x*3i)/2 - (7*exp(-x))/8 + (exp(-2*x)*1i)/8 - (exp(2*x)*1i)/8 + exp(-3*x)/ 24 + exp(3*x)/24 - (7*exp(x))/8 - 2/(exp(x) + 1i)
\[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\int \frac {\sinh \left (x \right )^{4}}{\sinh \left (x \right )+i}d x \] Input:
int(sinh(x)^4/(I+sinh(x)),x)
Output:
int(sinh(x)**4/(sinh(x) + i),x)