\(\int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx\) [90]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{8} i \arctan (\sinh (x))-\frac {\text {sech}^4(x)}{4}-\frac {1}{8} i \text {sech}(x) \tanh (x)+\frac {1}{4} i \text {sech}^3(x) \tanh (x) \]

[Out]

-1/8*I*arctan(sinh(x))-1/4*sech(x)^4-1/8*I*sech(x)*tanh(x)+1/4*I*sech(x)^3*tanh(x)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3957, 2914, 2686, 30, 2691, 3853, 3855} \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{8} i \arctan (\sinh (x))-\frac {1}{4} \text {sech}^4(x)+\frac {1}{4} i \tanh (x) \text {sech}^3(x)-\frac {1}{8} i \tanh (x) \text {sech}(x) \]

[In]

Int[Sech[x]^3/(I + Csch[x]),x]

[Out]

(-1/8*I)*ArcTan[Sinh[x]] - Sech[x]^4/4 - (I/8)*Sech[x]*Tanh[x] + (I/4)*Sech[x]^3*Tanh[x]

Rule 30

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

Rule 2686

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 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\text {sech}^2(x) \tanh (x)}{i-\sinh (x)} \, dx \\ & = -\left (i \int \text {sech}^3(x) \tanh ^2(x) \, dx\right )+\int \text {sech}^4(x) \tanh (x) \, dx \\ & = \frac {1}{4} i \text {sech}^3(x) \tanh (x)-\frac {1}{4} i \int \text {sech}^3(x) \, dx-\text {Subst}\left (\int x^3 \, dx,x,\text {sech}(x)\right ) \\ & = -\frac {1}{4} \text {sech}^4(x)-\frac {1}{8} i \text {sech}(x) \tanh (x)+\frac {1}{4} i \text {sech}^3(x) \tanh (x)-\frac {1}{8} i \int \text {sech}(x) \, dx \\ & = -\frac {1}{8} i \arctan (\sinh (x))-\frac {\text {sech}^4(x)}{4}-\frac {1}{8} i \text {sech}(x) \tanh (x)+\frac {1}{4} i \text {sech}^3(x) \tanh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{8} \left (-i \arctan (\sinh (x))+\frac {1}{(-i+\sinh (x))^2}-\frac {i}{i+\sinh (x)}\right ) \]

[In]

Integrate[Sech[x]^3/(I + Csch[x]),x]

[Out]

((-I)*ArcTan[Sinh[x]] + (-I + Sinh[x])^(-2) - I/(I + Sinh[x]))/8

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29 ) = 58\).

Time = 13.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.52

method	result	size
risch	\(-\frac {i {\mathrm e}^{x} \left (-2 i {\mathrm e}^{3 x}+{\mathrm e}^{4 x}+2 i {\mathrm e}^{x}-10 \,{\mathrm e}^{2 x}+1\right )}{4 \left ({\mathrm e}^{x}-i\right )^{4} \left ({\mathrm e}^{x}+i\right )^{2}}-\frac {\ln \left ({\mathrm e}^{x}-i\right )}{8}+\frac {\ln \left ({\mathrm e}^{x}+i\right )}{8}\)	\(61\)
default	\(\frac {i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{8}+\frac {i}{4 \tanh \left (\frac {x}{2}\right )+4 i}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{8}\)	\(89\)

[In]

int(sech(x)^3/(I+csch(x)),x,method=_RETURNVERBOSE)

[Out]

-1/4*I*exp(x)*(-2*I*exp(x)^3+exp(x)^4+2*I*exp(x)-10*exp(x)^2+1)/(exp(x)-I)^4/(exp(x)+I)^2-1/8*ln(exp(x)-I)+1/8
*ln(exp(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. 142 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.55 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )} \log \left (e^{x} + i\right ) - {\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )} \log \left (e^{x} - i\right ) - 2 i \, e^{\left (5 \, x\right )} - 4 \, e^{\left (4 \, x\right )} + 20 i \, e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} - 2 i \, e^{x}}{8 \, {\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )}} \]

[In]

integrate(sech(x)^3/(I+csch(x)),x, algorithm="fricas")

[Out]

1/8*((e^(6*x) - 2*I*e^(5*x) + e^(4*x) - 4*I*e^(3*x) - e^(2*x) - 2*I*e^x - 1)*log(e^x + I) - (e^(6*x) - 2*I*e^(
5*x) + e^(4*x) - 4*I*e^(3*x) - e^(2*x) - 2*I*e^x - 1)*log(e^x - I) - 2*I*e^(5*x) - 4*e^(4*x) + 20*I*e^(3*x) +
4*e^(2*x) - 2*I*e^x)/(e^(6*x) - 2*I*e^(5*x) + e^(4*x) - 4*I*e^(3*x) - e^(2*x) - 2*I*e^x - 1)

Sympy [F]

\[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(sech(x)**3/(I+csch(x)),x)

[Out]

Integral(sech(x)**3/(csch(x) + I), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(sech(x)^3/(I+csch(x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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. 94 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.35 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} + i \, e^{x} - 6}{16 \, {\left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right )}} + \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 i \, e^{\left (-x\right )} - 12 i \, e^{x} + 4}{32 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{2}} + \frac {1}{16} \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {1}{16} \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]

[In]

integrate(sech(x)^3/(I+csch(x)),x, algorithm="giac")

[Out]

-1/16*(-I*e^(-x) + I*e^x - 6)/(-I*e^(-x) + I*e^x - 2) + 1/32*(3*(e^(-x) - e^x)^2 + 12*I*e^(-x) - 12*I*e^x + 4)
/(e^(-x) - e^x + 2*I)^2 + 1/16*log(-e^(-x) + e^x + 2*I) - 1/16*log(-e^(-x) + e^x - 2*I)

Mupad [B] (verification not implemented)

Time = 2.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.05 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {\ln \left (-\frac {1}{4}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}\right )}{8}-\frac {\ln \left (\frac {1}{4}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}\right )}{8}-\frac {1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}\right )}-\frac {1}{2\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]

[In]

int(1/(cosh(x)^3*(1/sinh(x) + 1i)),x)

[Out]

log((exp(x)*1i)/4 - 1/4)/8 - log((exp(x)*1i)/4 + 1/4)/8 - 1i/(exp(2*x)*3i - exp(3*x) + 3*exp(x) - 1i) - 1/(4*(
exp(2*x) + exp(x)*2i - 1)) - 1/(2*(exp(4*x) - exp(3*x)*4i - 6*exp(2*x) + exp(x)*4i + 1)) - 1/(2*(exp(x)*2i - e
xp(2*x) + 1)) - 1i/(4*(exp(x) + 1i))