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

Optimal. Leaf size=43 \[ \frac{1}{6} (-\sinh (x)+i)^6-\frac{4}{5} i (-\sinh (x)+i)^5-(-\sinh (x)+i)^4 \]

[Out]

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

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Rubi [A]  time = 0.0451925, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \frac{1}{6} (-\sinh (x)+i)^6-\frac{4}{5} i (-\sinh (x)+i)^5-(-\sinh (x)+i)^4 \]

Antiderivative was successfully verified.

[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 2667

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])

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\cosh ^7(x)}{i+\sinh (x)} \, dx &=-\operatorname{Subst}\left (\int (i-x)^3 (i+x)^2 \, dx,x,\sinh (x)\right )\\ &=-\operatorname{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]  time = 0.029949, size = 42, normalized size = 0.98 \[ \frac{1}{30} \sinh (x) \left (5 \sinh ^5(x)-6 i \sinh ^4(x)+15 \sinh ^3(x)-20 i \sinh ^2(x)+15 \sinh (x)-30 i\right ) \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.057, size = 142, normalized size = 3.3 \begin{align*}{{\frac{11}{16}}-{\frac{7\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{{\frac{7}{8}}-{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}-{{\frac{1}{2}}-{\frac{i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-{{\frac{5}{16}}-i \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{{\frac{11}{12}}-{\frac{11\,i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-6}}+{{\frac{11}{12}}+{\frac{11\,i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{{\frac{11}{16}}+{\frac{7\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{{\frac{7}{8}}+{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-4}}+{{\frac{1}{2}}+{\frac{i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-5}}+{{\frac{5}{16}}+i \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11/16-7/8*I)/(tanh(1/2*x)+1)^2+(7/8-1/2*I)/(tanh(1/2*x)+1)^4+(-1/2+1/5*I)/(tanh(1/2*x)+1)^5+(-5/16+I)/(tanh(1
/2*x)+1)+(-11/12+11/12*I)/(tanh(1/2*x)+1)^3+1/6/(tanh(1/2*x)+1)^6+(11/12+11/12*I)/(tanh(1/2*x)-1)^3+(11/16+7/8
*I)/(tanh(1/2*x)-1)^2+(7/8+1/2*I)/(tanh(1/2*x)-1)^4+(1/2+1/5*I)/(tanh(1/2*x)-1)^5+(5/16+I)/(tanh(1/2*x)-1)+1/6
/(tanh(1/2*x)-1)^6

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Maxima [B]  time = 1.24185, size = 101, normalized size = 2.35 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.85776, size = 240, normalized size = 5.58 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.59726, size = 100, normalized size = 2.33 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.18407, size = 96, normalized size = 2.23 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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