\(\int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx\) [171]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)-\frac {5}{8} \cosh (x) \sinh (x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))} \]

[Out]

-5/8*x-5/12*I*cosh(x)^3-5/8*cosh(x)*sinh(x)+1/4*cosh(x)^5/(I+sinh(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2758, 2761, 2715, 8} \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)+\frac {\cosh ^5(x)}{4 (\sinh (x)+i)}-\frac {5}{8} \sinh (x) \cosh (x) \]

[In]

Int[Cosh[x]^6/(I + Sinh[x])^2,x]

[Out]

(-5*x)/8 - ((5*I)/12)*Cosh[x]^3 - (5*Cosh[x]*Sinh[x])/8 + Cosh[x]^5/(4*(I + Sinh[x]))

Rule 8

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

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 2758

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(a*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p,
0] && IntegersQ[2*m, 2*p]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5}{4} i \int \frac {\cosh ^4(x)}{i+\sinh (x)} \, dx \\ & = -\frac {5}{12} i \cosh ^3(x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5}{4} \int \cosh ^2(x) \, dx \\ & = -\frac {5}{12} i \cosh ^3(x)-\frac {5}{8} \cosh (x) \sinh (x)+\frac {\cosh ^5(x)}{4 (i+\sinh (x))}-\frac {5 \int 1 \, dx}{8} \\ & = -\frac {5 x}{8}-\frac {5}{12} i \cosh ^3(x)-\frac {5}{8} \cosh (x) \sinh (x)+\frac {\cosh ^5(x)}{4 (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. \(121\) vs. \(2(40)=80\).

Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.02 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {i \cosh ^7(x) \left (16+\frac {30 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sqrt {1-i \sinh (x)}}{\sqrt {1+i \sinh (x)}}-25 i \sinh (x)+7 \sinh ^2(x)-10 i \sinh ^3(x)+6 \sinh ^4(x)\right )}{24 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^8 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^6} \]

[In]

Integrate[Cosh[x]^6/(I + Sinh[x])^2,x]

[Out]

((-1/24*I)*Cosh[x]^7*(16 + (30*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]]*Sqrt[1 - I*Sinh[x]])/Sqrt[1 + I*Sinh[x]] -
(25*I)*Sinh[x] + 7*Sinh[x]^2 - (10*I)*Sinh[x]^3 + 6*Sinh[x]^4))/((Cosh[x/2] - I*Sinh[x/2])^8*(Cosh[x/2] + I*Si
nh[x/2])^6)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (30 ) = 60\).

Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.80

\[\frac {\frac {1}{2}+\frac {2 i}{3}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {-\frac {1}{8}+i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-\frac {3}{8}+i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {\frac {1}{2}-\frac {2 i}{3}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {1}{8}+i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {-\frac {3}{8}-i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}\]

[In]

int(cosh(x)^6/(I+sinh(x))^2,x)

[Out]

(1/2+2/3*I)/(tanh(1/2*x)-1)^3+(-1/8+I)/(tanh(1/2*x)-1)^2+(-3/8+I)/(tanh(1/2*x)-1)+1/4/(tanh(1/2*x)-1)^4+5/8*ln
(tanh(1/2*x)-1)+(1/2-2/3*I)/(tanh(1/2*x)+1)^3+(1/8+I)/(tanh(1/2*x)+1)^2-(3/8+I)/(tanh(1/2*x)+1)-1/4/(tanh(1/2*
x)+1)^4-5/8*ln(tanh(1/2*x)+1)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.38 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{192} \, {\left (120 \, x e^{\left (4 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 16 i \, e^{\left (7 \, x\right )} + 24 \, e^{\left (6 \, x\right )} + 48 i \, e^{\left (5 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} \]

[In]

integrate(cosh(x)^6/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

-1/192*(120*x*e^(4*x) - 3*e^(8*x) + 16*I*e^(7*x) + 24*e^(6*x) + 48*I*e^(5*x) + 48*I*e^(3*x) - 24*e^(2*x) + 16*
I*e^x + 3)*e^(-4*x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.62 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=- \frac {5 x}{8} + \frac {e^{4 x}}{64} - \frac {i e^{3 x}}{12} - \frac {e^{2 x}}{8} - \frac {i e^{x}}{4} - \frac {i e^{- x}}{4} + \frac {e^{- 2 x}}{8} - \frac {i e^{- 3 x}}{12} - \frac {e^{- 4 x}}{64} \]

[In]

integrate(cosh(x)**6/(I+sinh(x))**2,x)

[Out]

-5*x/8 + exp(4*x)/64 - I*exp(3*x)/12 - exp(2*x)/8 - I*exp(x)/4 - I*exp(-x)/4 + exp(-2*x)/8 - I*exp(-3*x)/12 -
exp(-4*x)/64

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{192} \, {\left (16 i \, e^{\left (-x\right )} + 24 \, e^{\left (-2 \, x\right )} + 48 i \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )} - \frac {5}{8} \, x - \frac {1}{4} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{12} i \, e^{\left (-3 \, x\right )} - \frac {1}{64} \, e^{\left (-4 \, x\right )} \]

[In]

integrate(cosh(x)^6/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

-1/192*(16*I*e^(-x) + 24*e^(-2*x) + 48*I*e^(-3*x) - 3)*e^(4*x) - 5/8*x - 1/4*I*e^(-x) + 1/8*e^(-2*x) - 1/12*I*
e^(-3*x) - 1/64*e^(-4*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{192} \, {\left (48 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} - \frac {5}{8} \, x + \frac {1}{64} \, e^{\left (4 \, x\right )} - \frac {1}{12} i \, e^{\left (3 \, x\right )} - \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{4} i \, e^{x} \]

[In]

integrate(cosh(x)^6/(I+sinh(x))^2,x, algorithm="giac")

[Out]

-1/192*(48*I*e^(3*x) - 24*e^(2*x) + 16*I*e^x + 3)*e^(-4*x) - 5/8*x + 1/64*e^(4*x) - 1/12*I*e^(3*x) - 1/8*e^(2*
x) - 1/4*I*e^x

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh ^6(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{4}-\frac {5\,x}{8}-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{12}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{12}-\frac {{\mathrm {e}}^{-4\,x}}{64}+\frac {{\mathrm {e}}^{4\,x}}{64}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4} \]

[In]

int(cosh(x)^6/(sinh(x) + 1i)^2,x)

[Out]

exp(-2*x)/8 - (exp(-x)*1i)/4 - (5*x)/8 - exp(2*x)/8 - (exp(-3*x)*1i)/12 - (exp(3*x)*1i)/12 - exp(-4*x)/64 + ex
p(4*x)/64 - (exp(x)*1i)/4