\(\int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx\) [161]

Optimal result

Integrand size = 13, antiderivative size = 33 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=-i \sinh (x)+\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x)+\frac {\sinh ^4(x)}{4} \]

[Out]

-I*sinh(x)+1/2*sinh(x)^2-1/3*I*sinh(x)^3+1/4*sinh(x)^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 45} \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=\frac {\sinh ^4(x)}{4}-\frac {1}{3} i \sinh ^3(x)+\frac {\sinh ^2(x)}{2}-i \sinh (x) \]

[In]

Int[Cosh[x]^5/(I + Sinh[x]),x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=\frac {1}{12} \sinh (x) \left (-12 i+6 \sinh (x)-4 i \sinh ^2(x)+3 \sinh ^3(x)\right ) \]

[In]

Integrate[Cosh[x]^5/(I + Sinh[x]),x]

[Out]

(Sinh[x]*(-12*I + 6*Sinh[x] - (4*I)*Sinh[x]^2 + 3*Sinh[x]^3))/12

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

\[-i \sinh \left (x \right )+\frac {\sinh \left (x \right )^{2}}{2}-\frac {i \sinh \left (x \right )^{3}}{3}+\frac {\sinh \left (x \right )^{4}}{4}\]

[In]

int(cosh(x)^5/(I+sinh(x)),x)

[Out]

-I*sinh(x)+1/2*sinh(x)^2-1/3*I*sinh(x)^3+1/4*sinh(x)^4

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).

Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=\frac {1}{192} \, {\left (3 \, e^{\left (8 \, x\right )} - 8 i \, e^{\left (7 \, x\right )} + 12 \, e^{\left (6 \, x\right )} - 72 i \, e^{\left (5 \, x\right )} + 72 i \, e^{\left (3 \, x\right )} + 12 \, e^{\left (2 \, x\right )} + 8 i \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} \]

[In]

integrate(cosh(x)^5/(I+sinh(x)),x, algorithm="fricas")

[Out]

1/192*(3*e^(8*x) - 8*I*e^(7*x) + 12*e^(6*x) - 72*I*e^(5*x) + 72*I*e^(3*x) + 12*e^(2*x) + 8*I*e^x + 3)*e^(-4*x)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=\frac {e^{4 x}}{64} - \frac {i e^{3 x}}{24} + \frac {e^{2 x}}{16} - \frac {3 i e^{x}}{8} + \frac {3 i e^{- x}}{8} + \frac {e^{- 2 x}}{16} + \frac {i e^{- 3 x}}{24} + \frac {e^{- 4 x}}{64} \]

[In]

integrate(cosh(x)**5/(I+sinh(x)),x)

[Out]

exp(4*x)/64 - I*exp(3*x)/24 + exp(2*x)/16 - 3*I*exp(x)/8 + 3*I*exp(-x)/8 + exp(-2*x)/16 + I*exp(-3*x)/24 + exp
(-4*x)/64

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=-\frac {1}{192} \, {\left (8 i \, e^{\left (-x\right )} - 12 \, e^{\left (-2 \, x\right )} + 72 i \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )} + \frac {3}{8} i \, e^{\left (-x\right )} + \frac {1}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{24} i \, e^{\left (-3 \, x\right )} + \frac {1}{64} \, e^{\left (-4 \, x\right )} \]

[In]

integrate(cosh(x)^5/(I+sinh(x)),x, algorithm="maxima")

[Out]

-1/192*(8*I*e^(-x) - 12*e^(-2*x) + 72*I*e^(-3*x) - 3)*e^(4*x) + 3/8*I*e^(-x) + 1/16*e^(-2*x) + 1/24*I*e^(-3*x)
+ 1/64*e^(-4*x)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=-\frac {1}{192} \, {\left (-72 i \, e^{\left (3 \, x\right )} - 12 \, e^{\left (2 \, x\right )} - 8 i \, e^{x} - 3\right )} e^{\left (-4 \, x\right )} + \frac {1}{64} \, e^{\left (4 \, x\right )} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} - \frac {3}{8} i \, e^{x} \]

[In]

integrate(cosh(x)^5/(I+sinh(x)),x, algorithm="giac")

[Out]

-1/192*(-72*I*e^(3*x) - 12*e^(2*x) - 8*I*e^x - 3)*e^(-4*x) + 1/64*e^(4*x) - 1/24*I*e^(3*x) + 1/16*e^(2*x) - 3/
8*I*e^x

Mupad [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {\cosh ^5(x)}{i+\sinh (x)} \, dx=\frac {{\mathrm {e}}^{-x}\,3{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-2\,x}}{16}+\frac {{\mathrm {e}}^{2\,x}}{16}+\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{-4\,x}}{64}+\frac {{\mathrm {e}}^{4\,x}}{64}-\frac {{\mathrm {e}}^x\,3{}\mathrm {i}}{8} \]

[In]

int(cosh(x)^5/(sinh(x) + 1i),x)

[Out]

(exp(-x)*3i)/8 + exp(-2*x)/16 + exp(2*x)/16 + (exp(-3*x)*1i)/24 - (exp(3*x)*1i)/24 + exp(-4*x)/64 + exp(4*x)/6
4 - (exp(x)*3i)/8