\(\int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx\) [170]

Optimal result

Integrand size = 13, antiderivative size = 80 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=-\frac {5}{16} i \arctan (\sinh (x))-\frac {1}{32 (i-\sinh (x))^2}+\frac {i}{8 (i-\sinh (x))}+\frac {i}{24 (i+\sinh (x))^3}+\frac {3}{32 (i+\sinh (x))^2}-\frac {3 i}{16 (i+\sinh (x))} \]

[Out]

-5/16*I*arctan(sinh(x))-1/32/(I-sinh(x))^2+1/8*I/(I-sinh(x))+1/24*I/(I+sinh(x))^3+3/32/(I+sinh(x))^2-3/16*I/(I
+sinh(x))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 209} \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=-\frac {5}{16} i \arctan (\sinh (x))+\frac {i}{8 (-\sinh (x)+i)}-\frac {3 i}{16 (\sinh (x)+i)}-\frac {1}{32 (-\sinh (x)+i)^2}+\frac {3}{32 (\sinh (x)+i)^2}+\frac {i}{24 (\sinh (x)+i)^3} \]

[In]

Int[Sech[x]^5/(I + Sinh[x]),x]

[Out]

((-5*I)/16)*ArcTan[Sinh[x]] - 1/(32*(I - Sinh[x])^2) + (I/8)/(I - Sinh[x]) + (I/24)/(I + Sinh[x])^3 + 3/(32*(I
+ Sinh[x])^2) - ((3*I)/16)/(I + Sinh[x])

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 \frac {1}{(i-x)^3 (i+x)^4} \, dx,x,\sinh (x)\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {1}{16 (-i+x)^3}-\frac {i}{8 (-i+x)^2}+\frac {i}{8 (i+x)^4}+\frac {3}{16 (i+x)^3}-\frac {3 i}{16 (i+x)^2}+\frac {5 i}{16 \left (1+x^2\right )}\right ) \, dx,x,\sinh (x)\right ) \\ & = -\frac {1}{32 (i-\sinh (x))^2}+\frac {i}{8 (i-\sinh (x))}+\frac {i}{24 (i+\sinh (x))^3}+\frac {3}{32 (i+\sinh (x))^2}-\frac {3 i}{16 (i+\sinh (x))}-\frac {5}{16} i \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right ) \\ & = -\frac {5}{16} i \arctan (\sinh (x))-\frac {1}{32 (i-\sinh (x))^2}+\frac {i}{8 (i-\sinh (x))}+\frac {i}{24 (i+\sinh (x))^3}+\frac {3}{32 (i+\sinh (x))^2}-\frac {3 i}{16 (i+\sinh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=-\frac {i \text {sech}^4(x) \left (8+15 i \arctan (\sinh (x))+5 (5 i+3 \arctan (\sinh (x))) \sinh (x)+5 (5+6 i \arctan (\sinh (x))) \sinh ^2(x)+15 (i+2 \arctan (\sinh (x))) \sinh ^3(x)+15 (1+i \arctan (\sinh (x))) \sinh ^4(x)+15 \arctan (\sinh (x)) \sinh ^5(x)\right )}{48 (i+\sinh (x))} \]

[In]

Integrate[Sech[x]^5/(I + Sinh[x]),x]

[Out]

((-1/48*I)*Sech[x]^4*(8 + (15*I)*ArcTan[Sinh[x]] + 5*(5*I + 3*ArcTan[Sinh[x]])*Sinh[x] + 5*(5 + (6*I)*ArcTan[S
inh[x]])*Sinh[x]^2 + 15*(I + 2*ArcTan[Sinh[x]])*Sinh[x]^3 + 15*(1 + I*ArcTan[Sinh[x]])*Sinh[x]^4 + 15*ArcTan[S
inh[x]]*Sinh[x]^5))/(I + Sinh[x])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (59 ) = 118\).

Time = 2.35 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.71

\[\frac {3 i}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )}-\frac {i}{4 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{3}}+\frac {1}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{4}}-\frac {1}{2 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {5 \ln \left (-i+\tanh \left (\frac {x}{2}\right )\right )}{16}+\frac {i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}+\frac {i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {25 i}{12 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}-\frac {15}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {15}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{16}\]

[In]

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

[Out]

3/8*I/(-I+tanh(1/2*x))-1/4*I/(-I+tanh(1/2*x))^3+1/8/(-I+tanh(1/2*x))^4-1/2/(-I+tanh(1/2*x))^2-5/16*ln(-I+tanh(
1/2*x))+I/(tanh(1/2*x)+I)^5+I/(tanh(1/2*x)+I)-25/12*I/(tanh(1/2*x)+I)^3+1/3/(tanh(1/2*x)+I)^6-15/8/(tanh(1/2*x
)+I)^4+15/8/(tanh(1/2*x)+I)^2+5/16*ln(tanh(1/2*x)+I)

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. 245 vs. \(2 (46) = 92\).

Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.06 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=\frac {15 \, {\left (e^{\left (10 \, x\right )} + 2 i \, e^{\left (9 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 8 i \, e^{\left (7 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 12 i \, e^{\left (5 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{x} + i\right ) - 15 \, {\left (e^{\left (10 \, x\right )} + 2 i \, e^{\left (9 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 8 i \, e^{\left (7 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 12 i \, e^{\left (5 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{x} - i\right ) - 30 i \, e^{\left (9 \, x\right )} + 60 \, e^{\left (8 \, x\right )} - 80 i \, e^{\left (7 \, x\right )} + 220 \, e^{\left (6 \, x\right )} - 36 i \, e^{\left (5 \, x\right )} - 220 \, e^{\left (4 \, x\right )} - 80 i \, e^{\left (3 \, x\right )} - 60 \, e^{\left (2 \, x\right )} - 30 i \, e^{x}}{48 \, {\left (e^{\left (10 \, x\right )} + 2 i \, e^{\left (9 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 8 i \, e^{\left (7 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 12 i \, e^{\left (5 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )}} \]

[In]

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

[Out]

1/48*(15*(e^(10*x) + 2*I*e^(9*x) + 3*e^(8*x) + 8*I*e^(7*x) + 2*e^(6*x) + 12*I*e^(5*x) - 2*e^(4*x) + 8*I*e^(3*x
) - 3*e^(2*x) + 2*I*e^x - 1)*log(e^x + I) - 15*(e^(10*x) + 2*I*e^(9*x) + 3*e^(8*x) + 8*I*e^(7*x) + 2*e^(6*x) +
12*I*e^(5*x) - 2*e^(4*x) + 8*I*e^(3*x) - 3*e^(2*x) + 2*I*e^x - 1)*log(e^x - I) - 30*I*e^(9*x) + 60*e^(8*x) -
80*I*e^(7*x) + 220*e^(6*x) - 36*I*e^(5*x) - 220*e^(4*x) - 80*I*e^(3*x) - 60*e^(2*x) - 30*I*e^x)/(e^(10*x) + 2*
I*e^(9*x) + 3*e^(8*x) + 8*I*e^(7*x) + 2*e^(6*x) + 12*I*e^(5*x) - 2*e^(4*x) + 8*I*e^(3*x) - 3*e^(2*x) + 2*I*e^x
- 1)

Sympy [F]

\[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {sech}^{5}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \]

[In]

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

[Out]

Integral(sech(x)**5/(sinh(x) + I), x)

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. 140 vs. \(2 (46) = 92\).

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.75 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=\frac {32 \, {\left (15 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} + 40 i \, e^{\left (-3 \, x\right )} - 110 \, e^{\left (-4 \, x\right )} + 18 i \, e^{\left (-5 \, x\right )} + 110 \, e^{\left (-6 \, x\right )} + 40 i \, e^{\left (-7 \, x\right )} + 30 \, e^{\left (-8 \, x\right )} + 15 i \, e^{\left (-9 \, x\right )}\right )}}{-1536 i \, e^{\left (-x\right )} - 2304 \, e^{\left (-2 \, x\right )} - 6144 i \, e^{\left (-3 \, x\right )} - 1536 \, e^{\left (-4 \, x\right )} - 9216 i \, e^{\left (-5 \, x\right )} + 1536 \, e^{\left (-6 \, x\right )} - 6144 i \, e^{\left (-7 \, x\right )} + 2304 \, e^{\left (-8 \, x\right )} - 1536 i \, e^{\left (-9 \, x\right )} + 768 \, e^{\left (-10 \, x\right )} - 768} - \frac {5}{16} \, \log \left (e^{\left (-x\right )} + i\right ) + \frac {5}{16} \, \log \left (e^{\left (-x\right )} - i\right ) \]

[In]

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

[Out]

32*(15*I*e^(-x) - 30*e^(-2*x) + 40*I*e^(-3*x) - 110*e^(-4*x) + 18*I*e^(-5*x) + 110*e^(-6*x) + 40*I*e^(-7*x) +
30*e^(-8*x) + 15*I*e^(-9*x))/(-1536*I*e^(-x) - 2304*e^(-2*x) - 6144*I*e^(-3*x) - 1536*e^(-4*x) - 9216*I*e^(-5*
x) + 1536*e^(-6*x) - 6144*I*e^(-7*x) + 2304*e^(-8*x) - 1536*I*e^(-9*x) + 768*e^(-10*x) - 768) - 5/16*log(e^(-x
) + I) + 5/16*log(e^(-x) - I)

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. 118 vs. \(2 (46) = 92\).

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=\frac {15 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 76 i \, e^{\left (-x\right )} - 76 i \, e^{x} - 100}{64 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{2}} - \frac {55 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 402 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 1020 \, e^{\left (-x\right )} + 1020 \, e^{x} + 936 i}{192 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}^{3}} + \frac {5}{32} \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {5}{32} \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]

[In]

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

[Out]

1/64*(15*(e^(-x) - e^x)^2 + 76*I*e^(-x) - 76*I*e^x - 100)/(e^(-x) - e^x + 2*I)^2 - 1/192*(55*(e^(-x) - e^x)^3
- 402*I*(e^(-x) - e^x)^2 - 1020*e^(-x) + 1020*e^x + 936*I)/(e^(-x) - e^x - 2*I)^3 + 5/32*log(-e^(-x) + e^x + 2
*I) - 5/32*log(-e^(-x) + e^x - 2*I)

Mupad [B] (verification not implemented)

Time = 2.66 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.11 \[ \int \frac {\text {sech}^5(x)}{i+\sinh (x)} \, dx=\frac {5\,\ln \left (-\frac {5}{8}+\frac {{\mathrm {e}}^x\,5{}\mathrm {i}}{8}\right )}{16}-\frac {5\,\ln \left (\frac {5}{8}+\frac {{\mathrm {e}}^x\,5{}\mathrm {i}}{8}\right )}{16}-\frac {1{}\mathrm {i}}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}\right )}+\frac {1}{8\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {5}{8\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}\right )}-\frac {1}{8\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {3{}\mathrm {i}}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {1}{3\,\left (15\,{\mathrm {e}}^{2\,x}-15\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1-{\mathrm {e}}^{3\,x}\,20{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,6{}\mathrm {i}+{\mathrm {e}}^x\,6{}\mathrm {i}\right )}-\frac {5{}\mathrm {i}}{12\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )} \]

[In]

int(1/(cosh(x)^5*(sinh(x) + 1i)),x)

[Out]

(5*log((exp(x)*5i)/8 - 5/8))/16 - (5*log((exp(x)*5i)/8 + 5/8))/16 - 1i/(exp(4*x)*5i - 10*exp(3*x) - exp(2*x)*1
0i + exp(5*x) + 5*exp(x) + 1i) + 1i/(4*(exp(2*x)*3i - exp(3*x) + 3*exp(x) - 1i)) + 1/(8*(exp(4*x) - exp(3*x)*4
i - 6*exp(2*x) + exp(x)*4i + 1)) + 5/(8*(exp(3*x)*4i - 6*exp(2*x) + exp(4*x) - exp(x)*4i + 1)) - 1/(8*(exp(x)*
2i - exp(2*x) + 1)) - 1i/(4*(exp(x) - 1i)) - 3i/(8*(exp(x) + 1i)) - 1/(3*(15*exp(2*x) - exp(3*x)*20i - 15*exp(
4*x) + exp(5*x)*6i + exp(6*x) + exp(x)*6i - 1)) - 5i/(12*(exp(2*x)*3i + exp(3*x) - 3*exp(x) - 1i))