Integrand size = 13, antiderivative size = 40 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{8} i \arctan (\sinh (x))-\frac {\text {sech}^4(x)}{4}-\frac {1}{8} i \text {sech}(x) \tanh (x)+\frac {1}{4} i \text {sech}^3(x) \tanh (x) \]
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{8} \left (-i \arctan (\sinh (x))+\frac {1}{(-i+\sinh (x))^2}-\frac {i}{i+\sinh (x)}\right ) \]
Time = 0.53 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.462, Rules used = {3042, 4360, 26, 3042, 26, 26, 3314, 25, 26, 3042, 25, 26, 3086, 15, 3091, 3042, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\text {sech}^3(x)}{\text {csch}(x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (i x)^3 (i \csc (i x)+i)}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \frac {i \tanh (x) \text {sech}^2(x)}{-\sinh (x)+i}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\text {sech}^2(x) \tanh (x)}{i-\sinh (x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \int -\frac {i \sin (i x)}{\cos (i x)^3 (i \sin (i x)+i)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int -\frac {i \sin (i x)}{(1+\sin (i x)) \cos (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{\cos (i x)^3 (\sin (i x)+1)}dx\) |
\(\Big \downarrow \) 3314 |
\(\displaystyle -i \left (-\int -\text {sech}^3(x) \tanh ^2(x)dx+\int i \text {sech}^4(x) \tanh (x)dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -i \left (\int \text {sech}^3(x) \tanh ^2(x)dx+\int i \text {sech}^4(x) \tanh (x)dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\int \text {sech}^3(x) \tanh ^2(x)dx+i \int \text {sech}^4(x) \tanh (x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (i \int -i \sec (i x)^4 \tan (i x)dx+\int -\sec (i x)^3 \tan (i x)^2dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -i \left (i \int -i \sec (i x)^4 \tan (i x)dx-\int \sec (i x)^3 \tan (i x)^2dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\int \sec (i x)^4 \tan (i x)dx-\int \sec (i x)^3 \tan (i x)^2dx\right )\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -i \left (-i \int \text {sech}^3(x)d\text {sech}(x)-\int \sec (i x)^3 \tan (i x)^2dx\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -i \left (-\int \sec (i x)^3 \tan (i x)^2dx-\frac {1}{4} i \text {sech}^4(x)\right )\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -i \left (\frac {1}{4} \int \text {sech}^3(x)dx-\frac {1}{4} i \text {sech}^4(x)-\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {1}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^3dx-\frac {1}{4} i \text {sech}^4(x)-\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -i \left (\frac {1}{4} \left (\frac {\int \text {sech}(x)dx}{2}+\frac {1}{2} \tanh (x) \text {sech}(x)\right )-\frac {1}{4} i \text {sech}^4(x)-\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {1}{4} \left (\frac {1}{2} \tanh (x) \text {sech}(x)+\frac {1}{2} \int \csc \left (i x+\frac {\pi }{2}\right )dx\right )-\frac {1}{4} i \text {sech}^4(x)-\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -i \left (\frac {1}{4} \left (\frac {1}{2} \arctan (\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x)\right )-\frac {1}{4} i \text {sech}^4(x)-\frac {1}{4} \tanh (x) \text {sech}^3(x)\right )\) |
(-I)*((-1/4*I)*Sech[x]^4 - (Sech[x]^3*Tanh[x])/4 + (ArcTan[Sinh[x]]/2 + (S ech[x]*Tanh[x])/2)/4)
3.1.90.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[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])
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] - Simp[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] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/(( a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[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]))
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
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 | \(\frac {i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{8}+\frac {i}{4 \tanh \left (\frac {x}{2}\right )+4 i}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{8}\) | \(89\) |
-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)
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 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )} \log \left (e^{x} + i\right ) - {\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )} \log \left (e^{x} - i\right ) - 2 i \, e^{\left (5 \, x\right )} - 4 \, e^{\left (4 \, x\right )} + 20 i \, e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} - 2 i \, e^{x}}{8 \, {\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )}} \]
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)
\[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
Exception generated. \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} + i \, e^{x} - 6}{16 \, {\left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right )}} + \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 i \, e^{\left (-x\right )} - 12 i \, e^{x} + 4}{32 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{2}} + \frac {1}{16} \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {1}{16} \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
-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)
Time = 2.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.05 \[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx=\frac {\ln \left (-\frac {1}{4}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}\right )}{8}-\frac {\ln \left (\frac {1}{4}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}\right )}{8}-\frac {1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}\right )}-\frac {1}{2\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]