3.2.58 \(\int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx\) [158]

Optimal. Leaf size=50 \[ -\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x) \]

[Out]

-5/16*I*x+1/7*cosh(x)^7-5/16*I*cosh(x)*sinh(x)-5/24*I*cosh(x)^3*sinh(x)-1/6*I*cosh(x)^5*sinh(x)

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Rubi [A]
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2761, 2715, 8} \begin {gather*} -\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {1}{6} i \sinh (x) \cosh ^5(x)-\frac {5}{24} i \sinh (x) \cosh ^3(x)-\frac {5}{16} i \sinh (x) \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^8/(I + Sinh[x]),x]

[Out]

((-5*I)/16)*x + Cosh[x]^7/7 - ((5*I)/16)*Cosh[x]*Sinh[x] - ((5*I)/24)*Cosh[x]^3*Sinh[x] - (I/6)*Cosh[x]^5*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 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*} \int \frac {\cosh ^8(x)}{i+\sinh (x)} \, dx &=\frac {\cosh ^7(x)}{7}-i \int \cosh ^6(x) \, dx\\ &=\frac {\cosh ^7(x)}{7}-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{6} i \int \cosh ^4(x) \, dx\\ &=\frac {\cosh ^7(x)}{7}-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{8} i \int \cosh ^2(x) \, dx\\ &=\frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x)-\frac {5}{16} i \int 1 \, dx\\ &=-\frac {5 i x}{16}+\frac {\cosh ^7(x)}{7}-\frac {5}{16} i \cosh (x) \sinh (x)-\frac {5}{24} i \cosh ^3(x) \sinh (x)-\frac {1}{6} i \cosh ^5(x) \sinh (x)\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(50)=100\).
time = 0.13, size = 219, normalized size = 4.38 \begin {gather*} \frac {\cosh ^9(x) \left (6 i \left (35 \text {ArcSin}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sqrt {1-i \sinh (x)}+8 \sqrt {1+i \sinh (x)}\right )+279 \sqrt {1+i \sinh (x)} \sinh (x)-87 i \sqrt {1+i \sinh (x)} \sinh ^2(x)+326 \sqrt {1+i \sinh (x)} \sinh ^3(x)-38 i \sqrt {1+i \sinh (x)} \sinh ^4(x)+200 \sqrt {1+i \sinh (x)} \sinh ^5(x)-8 i \sqrt {1+i \sinh (x)} \sinh ^6(x)+48 \sqrt {1+i \sinh (x)} \sinh ^7(x)\right )}{336 \sqrt {1+i \sinh (x)} (-i+\sinh (x))^4 (i+\sinh (x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^8/(I + Sinh[x]),x]

[Out]

(Cosh[x]^9*((6*I)*(35*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]]*Sqrt[1 - I*Sinh[x]] + 8*Sqrt[1 + I*Sinh[x]]) + 279*S
qrt[1 + I*Sinh[x]]*Sinh[x] - (87*I)*Sqrt[1 + I*Sinh[x]]*Sinh[x]^2 + 326*Sqrt[1 + I*Sinh[x]]*Sinh[x]^3 - (38*I)
*Sqrt[1 + I*Sinh[x]]*Sinh[x]^4 + 200*Sqrt[1 + I*Sinh[x]]*Sinh[x]^5 - (8*I)*Sqrt[1 + I*Sinh[x]]*Sinh[x]^6 + 48*
Sqrt[1 + I*Sinh[x]]*Sinh[x]^7))/(336*Sqrt[1 + I*Sinh[x]]*(-I + Sinh[x])^4*(I + Sinh[x])^5)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (36 ) = 72\).
time = 0.60, size = 186, normalized size = 3.72

method	result	size
risch	\(-\frac {5 i x}{16}+\frac {{\mathrm e}^{7 x}}{896}-\frac {i {\mathrm e}^{6 x}}{384}+\frac {{\mathrm e}^{5 x}}{128}-\frac {3 i {\mathrm e}^{4 x}}{128}+\frac {3 \,{\mathrm e}^{3 x}}{128}-\frac {15 i {\mathrm e}^{2 x}}{128}+\frac {5 \,{\mathrm e}^{x}}{128}+\frac {5 \,{\mathrm e}^{-x}}{128}+\frac {15 i {\mathrm e}^{-2 x}}{128}+\frac {3 \,{\mathrm e}^{-3 x}}{128}+\frac {3 i {\mathrm e}^{-4 x}}{128}+\frac {{\mathrm e}^{-5 x}}{128}+\frac {i {\mathrm e}^{-6 x}}{384}+\frac {{\mathrm e}^{-7 x}}{896}\)	\(94\)
default	\(\frac {-\frac {5}{16}-\frac {11 i}{16}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {-\frac {1}{2}-\frac {i}{6}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}+\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}+\frac {-\frac {1}{2}+\frac {i}{6}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}+\frac {-\frac {9}{8}-\frac {7 i}{6}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {-\frac {5}{4}+i}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {-\frac {5}{4}-i}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{7}}-\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}+\frac {1-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {-1-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {-\frac {11}{16}+\frac {19 i}{16}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {-\frac {11}{16}-\frac {19 i}{16}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\frac {9}{8}-\frac {7 i}{6}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {5}{16}-\frac {11 i}{16}}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{7}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^8/(I+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

(-5/16-11/16*I)/(tanh(1/2*x)-1)-(1/2+1/6*I)/(tanh(1/2*x)-1)^6+5/16*I*ln(tanh(1/2*x)-1)+(-1/2+1/6*I)/(tanh(1/2*
x)+1)^6-(9/8+7/6*I)/(tanh(1/2*x)-1)^3+(-5/4+I)/(tanh(1/2*x)+1)^4-(5/4+I)/(tanh(1/2*x)-1)^4+1/7/(tanh(1/2*x)+1)
^7-5/16*I*ln(tanh(1/2*x)+1)+(1-1/2*I)/(tanh(1/2*x)+1)^5-(1+1/2*I)/(tanh(1/2*x)-1)^5+(-11/16+19/16*I)/(tanh(1/2
*x)+1)^2-(11/16+19/16*I)/(tanh(1/2*x)-1)^2+(9/8-7/6*I)/(tanh(1/2*x)+1)^3+(5/16-11/16*I)/(tanh(1/2*x)+1)-1/7/(t
anh(1/2*x)-1)^7

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (32) = 64\).
time = 0.29, size = 90, normalized size = 1.80 \begin {gather*} -\frac {1}{2688} \, {\left (7 i \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 63 i \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 315 i \, e^{\left (-5 \, x\right )} - 105 \, e^{\left (-6 \, x\right )} - 3\right )} e^{\left (7 \, x\right )} - \frac {5}{16} i \, x + \frac {5}{128} \, e^{\left (-x\right )} + \frac {15}{128} i \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-3 \, x\right )} + \frac {3}{128} i \, e^{\left (-4 \, x\right )} + \frac {1}{128} \, e^{\left (-5 \, x\right )} + \frac {1}{384} i \, e^{\left (-6 \, x\right )} + \frac {1}{896} \, e^{\left (-7 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="maxima")

[Out]

-1/2688*(7*I*e^(-x) - 21*e^(-2*x) + 63*I*e^(-3*x) - 63*e^(-4*x) + 315*I*e^(-5*x) - 105*e^(-6*x) - 3)*e^(7*x) -
5/16*I*x + 5/128*e^(-x) + 15/128*I*e^(-2*x) + 3/128*e^(-3*x) + 3/128*I*e^(-4*x) + 1/128*e^(-5*x) + 1/384*I*e^
(-6*x) + 1/896*e^(-7*x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (32) = 64\).
time = 0.36, size = 91, normalized size = 1.82 \begin {gather*} \frac {1}{2688} \, {\left (-840 i \, x e^{\left (7 \, x\right )} + 3 \, e^{\left (14 \, x\right )} - 7 i \, e^{\left (13 \, x\right )} + 21 \, e^{\left (12 \, x\right )} - 63 i \, e^{\left (11 \, x\right )} + 63 \, e^{\left (10 \, x\right )} - 315 i \, e^{\left (9 \, x\right )} + 105 \, e^{\left (8 \, x\right )} + 105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="fricas")

[Out]

1/2688*(-840*I*x*e^(7*x) + 3*e^(14*x) - 7*I*e^(13*x) + 21*e^(12*x) - 63*I*e^(11*x) + 63*e^(10*x) - 315*I*e^(9*
x) + 105*e^(8*x) + 105*e^(6*x) + 315*I*e^(5*x) + 63*e^(4*x) + 63*I*e^(3*x) + 21*e^(2*x) + 7*I*e^x + 3)*e^(-7*x
)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
time = 0.14, size = 124, normalized size = 2.48 \begin {gather*} - \frac {5 i x}{16} + \frac {e^{7 x}}{896} - \frac {i e^{6 x}}{384} + \frac {e^{5 x}}{128} - \frac {3 i e^{4 x}}{128} + \frac {3 e^{3 x}}{128} - \frac {15 i e^{2 x}}{128} + \frac {5 e^{x}}{128} + \frac {5 e^{- x}}{128} + \frac {15 i e^{- 2 x}}{128} + \frac {3 e^{- 3 x}}{128} + \frac {3 i e^{- 4 x}}{128} + \frac {e^{- 5 x}}{128} + \frac {i e^{- 6 x}}{384} + \frac {e^{- 7 x}}{896} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**8/(I+sinh(x)),x)

[Out]

-5*I*x/16 + exp(7*x)/896 - I*exp(6*x)/384 + exp(5*x)/128 - 3*I*exp(4*x)/128 + 3*exp(3*x)/128 - 15*I*exp(2*x)/1
28 + 5*exp(x)/128 + 5*exp(-x)/128 + 15*I*exp(-2*x)/128 + 3*exp(-3*x)/128 + 3*I*exp(-4*x)/128 + exp(-5*x)/128 +
I*exp(-6*x)/384 + exp(-7*x)/896

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (32) = 64\).
time = 0.43, size = 86, normalized size = 1.72 \begin {gather*} \frac {1}{2688} \, {\left (105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} - \frac {5}{16} i \, x + \frac {1}{896} \, e^{\left (7 \, x\right )} - \frac {1}{384} i \, e^{\left (6 \, x\right )} + \frac {1}{128} \, e^{\left (5 \, x\right )} - \frac {3}{128} i \, e^{\left (4 \, x\right )} + \frac {3}{128} \, e^{\left (3 \, x\right )} - \frac {15}{128} i \, e^{\left (2 \, x\right )} + \frac {5}{128} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="giac")

[Out]

1/2688*(105*e^(6*x) + 315*I*e^(5*x) + 63*e^(4*x) + 63*I*e^(3*x) + 21*e^(2*x) + 7*I*e^x + 3)*e^(-7*x) - 5/16*I*
x + 1/896*e^(7*x) - 1/384*I*e^(6*x) + 1/128*e^(5*x) - 3/128*I*e^(4*x) + 3/128*e^(3*x) - 15/128*I*e^(2*x) + 5/1
28*e^x

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Mupad [B]
time = 0.78, size = 93, normalized size = 1.86 \begin {gather*} \frac {5\,{\mathrm {e}}^{-x}}{128}+\frac {3\,{\mathrm {e}}^{-3\,x}}{128}+\frac {3\,{\mathrm {e}}^{3\,x}}{128}+\frac {{\mathrm {e}}^{-5\,x}}{128}+\frac {{\mathrm {e}}^{5\,x}}{128}+\frac {{\mathrm {e}}^{-7\,x}}{896}+\frac {{\mathrm {e}}^{7\,x}}{896}+\frac {5\,{\mathrm {e}}^x}{128}-\frac {x\,5{}\mathrm {i}}{16}+\frac {{\mathrm {e}}^{-2\,x}\,15{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{2\,x}\,15{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-4\,x}\,3{}\mathrm {i}}{128}-\frac {{\mathrm {e}}^{4\,x}\,3{}\mathrm {i}}{128}+\frac {{\mathrm {e}}^{-6\,x}\,1{}\mathrm {i}}{384}-\frac {{\mathrm {e}}^{6\,x}\,1{}\mathrm {i}}{384} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^8/(sinh(x) + 1i),x)

[Out]

(5*exp(-x))/128 - (x*5i)/16 + (exp(-2*x)*15i)/128 - (exp(2*x)*15i)/128 + (3*exp(-3*x))/128 + (3*exp(3*x))/128
+ (exp(-4*x)*3i)/128 - (exp(4*x)*3i)/128 + exp(-5*x)/128 + exp(5*x)/128 + (exp(-6*x)*1i)/384 - (exp(6*x)*1i)/3
84 + exp(-7*x)/896 + exp(7*x)/896 + (5*exp(x))/128

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