3.208 \(\int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac {1}{5} i \tanh ^5(x)-\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]

[Out]

-sech(x)+2/3*sech(x)^3-1/5*sech(x)^5-1/5*I*tanh(x)^5

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Rubi [A]  time = 0.08, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac {1}{5} i \tanh ^5(x)-\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^4/(I + Sinh[x]),x]

[Out]

-Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5 - (I/5)*Tanh[x]^5

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [B]  time = 0.14, size = 96, normalized size = 3.10 \[ -\frac {64 i \sinh (x)+178 i \sinh (2 x)-192 i \sinh (3 x)+89 i \sinh (4 x)-534 \cosh (x)+288 \cosh (2 x)-178 \cosh (3 x)+24 \cosh (4 x)+200}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^5 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^4/(I + Sinh[x]),x]

[Out]

-1/960*(200 - 534*Cosh[x] + 288*Cosh[2*x] - 178*Cosh[3*x] + 24*Cosh[4*x] + (64*I)*Sinh[x] + (178*I)*Sinh[2*x]
- (192*I)*Sinh[3*x] + (89*I)*Sinh[4*x])/((Cosh[x/2] - I*Sinh[x/2])^5*(Cosh[x/2] + I*Sinh[x/2])^3)

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fricas [B]  time = 0.66, size = 88, normalized size = 2.84 \[ -\frac {30 \, e^{\left (7 \, x\right )} + 30 i \, e^{\left (6 \, x\right )} + 10 \, e^{\left (5 \, x\right )} + 50 i \, e^{\left (4 \, x\right )} + 26 \, e^{\left (3 \, x\right )} + 42 i \, e^{\left (2 \, x\right )} - 18 \, e^{x} + 6 i}{15 \, e^{\left (8 \, x\right )} + 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} + 90 i \, e^{\left (5 \, x\right )} + 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} + 30 i \, e^{x} - 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(I+sinh(x)),x, algorithm="fricas")

[Out]

-(30*e^(7*x) + 30*I*e^(6*x) + 10*e^(5*x) + 50*I*e^(4*x) + 26*e^(3*x) + 42*I*e^(2*x) - 18*e^x + 6*I)/(15*e^(8*x
) + 30*I*e^(7*x) + 30*e^(6*x) + 90*I*e^(5*x) + 90*I*e^(3*x) - 30*e^(2*x) + 30*I*e^x - 15)

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giac [B]  time = 0.28, size = 53, normalized size = 1.71 \[ -\frac {15 \, e^{\left (2 \, x\right )} - 24 i \, e^{x} - 13}{24 \, {\left (e^{x} - i\right )}^{3}} - \frac {165 \, e^{\left (4 \, x\right )} + 480 i \, e^{\left (3 \, x\right )} - 650 \, e^{\left (2 \, x\right )} - 400 i \, e^{x} + 113}{120 \, {\left (e^{x} + i\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(I+sinh(x)),x, algorithm="giac")

[Out]

-1/24*(15*e^(2*x) - 24*I*e^x - 13)/(e^x - I)^3 - 1/120*(165*e^(4*x) + 480*I*e^(3*x) - 650*e^(2*x) - 400*I*e^x
+ 113)/(e^x + I)^5

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maple [B]  time = 0.10, size = 93, normalized size = 3.00 \[ \frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^4/(I+sinh(x)),x)

[Out]

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

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maxima [B]  time = 0.32, size = 413, normalized size = 13.32 \[ \frac {18 \, e^{\left (-x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {42 i \, e^{\left (-2 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {26 \, e^{\left (-3 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {50 i \, e^{\left (-4 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {10 \, e^{\left (-5 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {30 i \, e^{\left (-6 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {30 \, e^{\left (-7 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {6 i}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^4/(I+sinh(x)),x, algorithm="maxima")

[Out]

18*e^(-x)/(-30*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8
*x) - 15) + 42*I*e^(-2*x)/(-30*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(
-7*x) + 15*e^(-8*x) - 15) - 26*e^(-3*x)/(-30*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6
*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15) + 50*I*e^(-4*x)/(-30*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(
-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15) - 10*e^(-5*x)/(-30*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3
*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15) + 30*I*e^(-6*x)/(-30*I*e^(-x) - 30*e^(-2
*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15) - 30*e^(-7*x)/(-30*I*e^(
-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15) + 6*I/(-30
*I*e^(-x) - 30*e^(-2*x) - 90*I*e^(-3*x) - 90*I*e^(-5*x) + 30*e^(-6*x) - 30*I*e^(-7*x) + 15*e^(-8*x) - 15)

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mupad [B]  time = 1.17, size = 231, normalized size = 7.45 \[ -\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {\frac {11\,{\mathrm {e}}^x}{40}+\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{2\,x}}{40}-\frac {17}{120}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {1{}\mathrm {i}}{4\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {5}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {11}{40\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {\frac {{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}}{8}+\frac {11\,{\mathrm {e}}^{3\,x}}{40}-\frac {17\,{\mathrm {e}}^x}{40}-\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{4\,x}}{40}-\frac {17\,{\mathrm {e}}^{2\,x}}{20}+\frac {11}{40}+\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^4/(sinh(x) + 1i),x)

[Out]

1i/(4*(exp(x)*2i - exp(2*x) + 1)) - ((11*exp(x))/40 + 1i/8)/(exp(2*x) + exp(x)*2i - 1) - ((11*exp(2*x))/40 + (
exp(x)*1i)/4 - 17/120)/(exp(2*x)*3i + exp(3*x) - 3*exp(x) - 1i) - 1/(6*(exp(2*x)*3i - exp(3*x) + 3*exp(x) - 1i
)) - 5/(8*(exp(x) - 1i)) - 11/(40*(exp(x) + 1i)) - ((exp(2*x)*3i)/8 + (11*exp(3*x))/40 - (17*exp(x))/40 - 1i/8
)/(exp(3*x)*4i - 6*exp(2*x) + exp(4*x) - exp(x)*4i + 1) - ((exp(3*x)*1i)/2 - (17*exp(2*x))/20 + (11*exp(4*x))/
40 - (exp(x)*1i)/2 + 11/40)/(exp(4*x)*5i - 10*exp(3*x) - exp(2*x)*10i + exp(5*x) + 5*exp(x) + 1i)

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sympy [B]  time = 0.29, size = 107, normalized size = 3.45 \[ \frac {30 e^{7 x} + 30 i e^{6 x} + 10 e^{5 x} + 50 i e^{4 x} + 26 e^{3 x} + 42 i e^{2 x} - 18 e^{x} + 6 i}{- 15 e^{8 x} - 30 i e^{7 x} - 30 e^{6 x} - 90 i e^{5 x} - 90 i e^{3 x} + 30 e^{2 x} - 30 i e^{x} + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**4/(I+sinh(x)),x)

[Out]

(30*exp(7*x) + 30*I*exp(6*x) + 10*exp(5*x) + 50*I*exp(4*x) + 26*exp(3*x) + 42*I*exp(2*x) - 18*exp(x) + 6*I)/(-
15*exp(8*x) - 30*I*exp(7*x) - 30*exp(6*x) - 90*I*exp(5*x) - 90*I*exp(3*x) + 30*exp(2*x) - 30*I*exp(x) + 15)

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