\(\int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx\) [159]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=-(i-\sinh (x))^4-\frac {4}{5} i (i-\sinh (x))^5+\frac {1}{6} (i-\sinh (x))^6 \]

[Out]

-(I-sinh(x))^4-4/5*I*(I-sinh(x))^5+1/6*(I-sinh(x))^6

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, 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} \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=\frac {1}{6} (-\sinh (x)+i)^6-\frac {4}{5} i (-\sinh (x)+i)^5-(-\sinh (x)+i)^4 \]

[In]

Int[Cosh[x]^7/(I + Sinh[x]),x]

[Out]

-(I - Sinh[x])^4 - ((4*I)/5)*(I - Sinh[x])^5 + (I - Sinh[x])^6/6

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)^3 (i+x)^2 \, dx,x,\sinh (x)\right ) \\ & = -\text {Subst}\left (\int \left (-4 (i-x)^3-4 i (i-x)^4+(i-x)^5\right ) \, dx,x,\sinh (x)\right ) \\ & = -(i-\sinh (x))^4-\frac {4}{5} i (i-\sinh (x))^5+\frac {1}{6} (i-\sinh (x))^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=\frac {1}{30} \sinh (x) \left (-30 i+15 \sinh (x)-20 i \sinh ^2(x)+15 \sinh ^3(x)-6 i \sinh ^4(x)+5 \sinh ^5(x)\right ) \]

[In]

Integrate[Cosh[x]^7/(I + Sinh[x]),x]

[Out]

(Sinh[x]*(-30*I + 15*Sinh[x] - (20*I)*Sinh[x]^2 + 15*Sinh[x]^3 - (6*I)*Sinh[x]^4 + 5*Sinh[x]^5))/30

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91

\[-i \sinh \left (x \right )+\frac {\sinh \left (x \right )^{6}}{6}-\frac {i \sinh \left (x \right )^{5}}{5}+\frac {\sinh \left (x \right )^{4}}{2}-\frac {2 i \sinh \left (x \right )^{3}}{3}+\frac {\sinh \left (x \right )^{2}}{2}\]

[In]

int(cosh(x)^7/(I+sinh(x)),x)

[Out]

-I*sinh(x)+1/6*sinh(x)^6-1/5*I*sinh(x)^5+1/2*sinh(x)^4-2/3*I*sinh(x)^3+1/2*sinh(x)^2

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. 72 vs. \(2 (25) = 50\).

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=\frac {1}{1920} \, {\left (5 \, e^{\left (12 \, x\right )} - 12 i \, e^{\left (11 \, x\right )} + 30 \, e^{\left (10 \, x\right )} - 100 i \, e^{\left (9 \, x\right )} + 75 \, e^{\left (8 \, x\right )} - 600 i \, e^{\left (7 \, x\right )} + 600 i \, e^{\left (5 \, x\right )} + 75 \, e^{\left (4 \, x\right )} + 100 i \, e^{\left (3 \, x\right )} + 30 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} + 5\right )} e^{\left (-6 \, x\right )} \]

[In]

integrate(cosh(x)^7/(I+sinh(x)),x, algorithm="fricas")

[Out]

1/1920*(5*e^(12*x) - 12*I*e^(11*x) + 30*e^(10*x) - 100*I*e^(9*x) + 75*e^(8*x) - 600*I*e^(7*x) + 600*I*e^(5*x)
+ 75*e^(4*x) + 100*I*e^(3*x) + 30*e^(2*x) + 12*I*e^x + 5)*e^(-6*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. 100 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.33 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=\frac {e^{6 x}}{384} - \frac {i e^{5 x}}{160} + \frac {e^{4 x}}{64} - \frac {5 i e^{3 x}}{96} + \frac {5 e^{2 x}}{128} - \frac {5 i e^{x}}{16} + \frac {5 i e^{- x}}{16} + \frac {5 e^{- 2 x}}{128} + \frac {5 i e^{- 3 x}}{96} + \frac {e^{- 4 x}}{64} + \frac {i e^{- 5 x}}{160} + \frac {e^{- 6 x}}{384} \]

[In]

integrate(cosh(x)**7/(I+sinh(x)),x)

[Out]

exp(6*x)/384 - I*exp(5*x)/160 + exp(4*x)/64 - 5*I*exp(3*x)/96 + 5*exp(2*x)/128 - 5*I*exp(x)/16 + 5*I*exp(-x)/1
6 + 5*exp(-2*x)/128 + 5*I*exp(-3*x)/96 + exp(-4*x)/64 + I*exp(-5*x)/160 + exp(-6*x)/384

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. 75 vs. \(2 (25) = 50\).

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=-\frac {1}{1920} \, {\left (12 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} + 100 i \, e^{\left (-3 \, x\right )} - 75 \, e^{\left (-4 \, x\right )} + 600 i \, e^{\left (-5 \, x\right )} - 5\right )} e^{\left (6 \, x\right )} + \frac {5}{16} i \, e^{\left (-x\right )} + \frac {5}{128} \, e^{\left (-2 \, x\right )} + \frac {5}{96} i \, e^{\left (-3 \, x\right )} + \frac {1}{64} \, e^{\left (-4 \, x\right )} + \frac {1}{160} i \, e^{\left (-5 \, x\right )} + \frac {1}{384} \, e^{\left (-6 \, x\right )} \]

[In]

integrate(cosh(x)^7/(I+sinh(x)),x, algorithm="maxima")

[Out]

-1/1920*(12*I*e^(-x) - 30*e^(-2*x) + 100*I*e^(-3*x) - 75*e^(-4*x) + 600*I*e^(-5*x) - 5)*e^(6*x) + 5/16*I*e^(-x
) + 5/128*e^(-2*x) + 5/96*I*e^(-3*x) + 1/64*e^(-4*x) + 1/160*I*e^(-5*x) + 1/384*e^(-6*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. 71 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=-\frac {1}{1920} \, {\left (-600 i \, e^{\left (5 \, x\right )} - 75 \, e^{\left (4 \, x\right )} - 100 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} - 12 i \, e^{x} - 5\right )} e^{\left (-6 \, x\right )} + \frac {1}{384} \, e^{\left (6 \, x\right )} - \frac {1}{160} i \, e^{\left (5 \, x\right )} + \frac {1}{64} \, e^{\left (4 \, x\right )} - \frac {5}{96} i \, e^{\left (3 \, x\right )} + \frac {5}{128} \, e^{\left (2 \, x\right )} - \frac {5}{16} i \, e^{x} \]

[In]

integrate(cosh(x)^7/(I+sinh(x)),x, algorithm="giac")

[Out]

-1/1920*(-600*I*e^(5*x) - 75*e^(4*x) - 100*I*e^(3*x) - 30*e^(2*x) - 12*I*e^x - 5)*e^(-6*x) + 1/384*e^(6*x) - 1
/160*I*e^(5*x) + 1/64*e^(4*x) - 5/96*I*e^(3*x) + 5/128*e^(2*x) - 5/16*I*e^x

Mupad [B] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.79 \[ \int \frac {\cosh ^7(x)}{i+\sinh (x)} \, dx=\frac {{\mathrm {e}}^{-x}\,5{}\mathrm {i}}{16}+\frac {5\,{\mathrm {e}}^{-2\,x}}{128}+\frac {5\,{\mathrm {e}}^{2\,x}}{128}+\frac {{\mathrm {e}}^{-3\,x}\,5{}\mathrm {i}}{96}-\frac {{\mathrm {e}}^{3\,x}\,5{}\mathrm {i}}{96}+\frac {{\mathrm {e}}^{-4\,x}}{64}+\frac {{\mathrm {e}}^{4\,x}}{64}+\frac {{\mathrm {e}}^{-5\,x}\,1{}\mathrm {i}}{160}-\frac {{\mathrm {e}}^{5\,x}\,1{}\mathrm {i}}{160}+\frac {{\mathrm {e}}^{-6\,x}}{384}+\frac {{\mathrm {e}}^{6\,x}}{384}-\frac {{\mathrm {e}}^x\,5{}\mathrm {i}}{16} \]

[In]

int(cosh(x)^7/(sinh(x) + 1i),x)

[Out]

(exp(-x)*5i)/16 + (5*exp(-2*x))/128 + (5*exp(2*x))/128 + (exp(-3*x)*5i)/96 - (exp(3*x)*5i)/96 + exp(-4*x)/64 +
exp(4*x)/64 + (exp(-5*x)*1i)/160 - (exp(5*x)*1i)/160 + exp(-6*x)/384 + exp(6*x)/384 - (exp(x)*5i)/16