\(\int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx\) [40]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3 i x}{2}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)} \]

[Out]

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))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2846, 2827, 2715, 8, 2713} \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3 i x}{2}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}-\frac {3}{2} i \sinh (x) \cosh (x) \]

[In]

Int[Sinh[x]^4/(I + Sinh[x]),x]

[Out]

((3*I)/2)*x - 4*Cosh[x] + (4*Cosh[x]^3)/3 - ((3*I)/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x]^3)/(I + Sinh[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2846

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] - Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\int \sinh ^2(x) (-3 i+4 \sinh (x)) \, dx \\ & = -\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}-3 i \int \sinh ^2(x) \, dx+4 \int \sinh ^3(x) \, dx \\ & = -\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\frac {3}{2} i \int 1 \, dx-4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right ) \\ & = \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)} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(134\) vs. \(2(46)=92\).

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.91 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {\cosh (x) \left (-16 i \left (\arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+\sqrt {\cosh ^2(x)}\right )-\left (16 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+7 \sqrt {\cosh ^2(x)}\right ) \sinh (x)-i \sqrt {\cosh ^2(x)} \sinh ^2(x)+2 \sqrt {\cosh ^2(x)} \sinh ^3(x)+i \text {arcsinh}(\sinh (x)) (i+\sinh (x))\right )}{6 \sqrt {\cosh ^2(x)} (i+\sinh (x))} \]

[In]

Integrate[Sinh[x]^4/(I + Sinh[x]),x]

[Out]

(Cosh[x]*((-16*I)*(ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]] + Sqrt[Cosh[x]^2]) - (16*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqr
t[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[Sin
h[x]]*(I + Sinh[x])))/(6*Sqrt[Cosh[x]^2]*(I + Sinh[x]))

Maple [A] (verified)

Time = 3.49 (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	\(\frac {\left (-36 i \cosh \left (x \right )+36 \sinh \left (x \right )+36 i\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+\left (36 i \cosh \left (x \right )-36 i-36 \sinh \left (x \right )\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )-3 i \sinh \left (3 x \right )+i \sinh \left (4 x \right )-67 i \sinh \left (x \right )-16 i \sinh \left (2 x \right )+23 \cosh \left (x \right )-20 \cosh \left (2 x \right )+\cosh \left (3 x \right )+\cosh \left (4 x \right )-5}{24 i \sinh \left (x \right )+24 \cosh \left (x \right )-24}\)	\(105\)

[In]

int(sinh(x)^4/(I+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=-\frac {3 \, {\left (-12 i \, x + 7 i\right )} e^{\left (4 \, x\right )} + 3 \, {\left (12 \, x + 23\right )} e^{\left (3 \, x\right )} - e^{\left (7 \, x\right )} + 2 i \, e^{\left (6 \, x\right )} + 18 \, e^{\left (5 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i}{24 \, {\left (e^{\left (4 \, x\right )} + i \, e^{\left (3 \, x\right )}\right )}} \]

[In]

integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="fricas")

[Out]

-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))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3 i x}{2} + \frac {e^{3 x}}{24} - \frac {i e^{2 x}}{8} - \frac {7 e^{x}}{8} - \frac {7 e^{- x}}{8} + \frac {i e^{- 2 x}}{8} + \frac {e^{- 3 x}}{24} - \frac {2}{e^{x} + i} \]

[In]

integrate(sinh(x)**4/(I+sinh(x)),x)

[Out]

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
)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} i \, x - \frac {2 \, e^{\left (-x\right )} - 18 i \, e^{\left (-2 \, x\right )} + 69 \, e^{\left (-3 \, x\right )} + i}{8 \, {\left (-3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} \]

[In]

integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="maxima")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} i \, x - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} + i\right )}} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} \]

[In]

integrate(sinh(x)^4/(I+sinh(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx=\frac {x\,3{}\mathrm {i}}{2}-\frac {7\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \]

[In]

int(sinh(x)^4/(sinh(x) + 1i),x)

[Out]

(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)