Integrand size = 11, antiderivative size = 40 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=i \log (i+\sinh (x))-\frac {2 i}{(1-i \sinh (x))^2}+\frac {4 i}{1-i \sinh (x)} \] Output:
I*ln(I+sinh(x))-2*I/(1-I*sinh(x))^2+4*I/(1-I*sinh(x))
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+i \log (\cosh (x))+\frac {2 i+4 \sinh (x)}{\left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^4} \] Input:
Integrate[(Sech[x] - I*Tanh[x])^(-5),x]
Output:
2*ArcTan[Tanh[x/2]] + I*Log[Cosh[x]] + (2*I + 4*Sinh[x])/(Cosh[x/2] - I*Si nh[x/2])^4
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4891, 3042, 3146, 49, 2009}
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.
\(\displaystyle \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\sec (i x)-\tan (i x))^5}dx\) |
\(\Big \downarrow \) 4891 |
\(\displaystyle \int \frac {\cosh ^5(x)}{(1-i \sinh (x))^5}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)^5}{(1-\sin (i x))^5}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle i \int \frac {(i \sinh (x)+1)^2}{(1-i \sinh (x))^3}d(-i \sinh (x))\) |
\(\Big \downarrow \) 49 |
\(\displaystyle i \int \left (\frac {1}{1-i \sinh (x)}-\frac {4}{(1-i \sinh (x))^2}+\frac {4}{(1-i \sinh (x))^3}\right )d(-i \sinh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (\frac {4}{1-i \sinh (x)}-\frac {2}{(1-i \sinh (x))^2}+\log (1-i \sinh (x))\right )\) |
Input:
Int[(Sech[x] - I*Tanh[x])^(-5),x]
Output:
I*(Log[1 - I*Sinh[x]] - 2/(1 - I*Sinh[x])^2 + 4/(1 - I*Sinh[x]))
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] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[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] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x _)]^(n_.))^(p_), x_Symbol] :> Int[ActivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a *Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 1.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.70
\[\frac {8 i}{\left (i+\tanh \left (\frac {x}{2}\right )\right )^{2}}+2 i \ln \left (i+\tanh \left (\frac {x}{2}\right )\right )-\frac {8 i}{\left (i+\tanh \left (\frac {x}{2}\right )\right )^{4}}+\frac {16}{\left (i+\tanh \left (\frac {x}{2}\right )\right )^{3}}-i \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\]
Input:
int(1/(sech(x)-I*tanh(x))^5,x)
Output:
8*I/(I+tanh(1/2*x))^2+2*I*ln(I+tanh(1/2*x))-8*I/(I+tanh(1/2*x))^4+16/(I+ta nh(1/2*x))^3-I*ln(1+tanh(1/2*x))-I*ln(tanh(1/2*x)-1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.35 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=\frac {-i \, x e^{\left (4 \, x\right )} + 4 \, {\left (x - 2\right )} e^{\left (3 \, x\right )} - 2 \, {\left (-3 i \, x + 4 i\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x - 2\right )} e^{x} - 2 \, {\left (-i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} - 4 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (4 \, x\right )} + 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1} \] Input:
integrate(1/(sech(x)-I*tanh(x))^5,x, algorithm="fricas")
Output:
(-I*x*e^(4*x) + 4*(x - 2)*e^(3*x) - 2*(-3*I*x + 4*I)*e^(2*x) - 4*(x - 2)*e ^x - 2*(-I*e^(4*x) + 4*e^(3*x) + 6*I*e^(2*x) - 4*e^x - I)*log(e^x + I) - I *x)/(e^(4*x) + 4*I*e^(3*x) - 6*e^(2*x) - 4*I*e^x + 1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (29) = 58\).
Time = 2.31 (sec) , antiderivative size = 1445, normalized size of antiderivative = 36.12 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(sech(x)-I*tanh(x))**5,x)
Output:
36*I*x*tanh(x)**4/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)* *2*sech(x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) - 144*x*tanh(x)* *3*sech(x)/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech (x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) - 216*I*x*tanh(x)**2*se ch(x)**2/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech(x )**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) + 144*x*tanh(x)*sech(x)** 3/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech(x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) + 36*I*x*sech(x)**4/(36*tanh(x)* *4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech(x)**2 - 144*I*tanh(x)* sech(x)**3 + 36*sech(x)**4) - 36*I*log(tanh(x) + 1)*tanh(x)**4/(36*tanh(x) **4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech(x)**2 - 144*I*tanh(x) *sech(x)**3 + 36*sech(x)**4) + 144*log(tanh(x) + 1)*tanh(x)**3*sech(x)/(36 *tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech(x)**2 - 144*I *tanh(x)*sech(x)**3 + 36*sech(x)**4) + 216*I*log(tanh(x) + 1)*tanh(x)**2*s ech(x)**2/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)**2*sech( x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) - 144*log(tanh(x) + 1)*t anh(x)*sech(x)**3/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x)* *2*sech(x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) - 36*I*log(tanh( x) + 1)*sech(x)**4/(36*tanh(x)**4 + 144*I*tanh(x)**3*sech(x) - 216*tanh(x) **2*sech(x)**2 - 144*I*tanh(x)*sech(x)**3 + 36*sech(x)**4) + 36*I*log(t...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (28) = 56\).
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=i \, x - \frac {8 \, {\left (e^{\left (-x\right )} + i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )}\right )}}{4 i \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} - 4 i \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 i \, \log \left (e^{\left (-x\right )} - i\right ) \] Input:
integrate(1/(sech(x)-I*tanh(x))^5,x, algorithm="maxima")
Output:
I*x - 8*(e^(-x) + I*e^(-2*x) - e^(-3*x))/(4*I*e^(-x) - 6*e^(-2*x) - 4*I*e^ (-3*x) + e^(-4*x) + 1) + 2*I*log(e^(-x) - I)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=-i \, x - \frac {8 \, {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x}\right )}}{{\left (e^{x} + i\right )}^{4}} + 2 i \, \log \left (e^{x} + i\right ) \] Input:
integrate(1/(sech(x)-I*tanh(x))^5,x, algorithm="giac")
Output:
-I*x - 8*(e^(3*x) + I*e^(2*x) - e^x)/(e^x + I)^4 + 2*I*log(e^x + I)
Time = 1.01 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.25 \[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=-x\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,2{}\mathrm {i}+\frac {16{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {8{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {8}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {16}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}} \] Input:
int(-1/(tanh(x)*1i - 1/cosh(x))^5,x)
Output:
log(exp(x) + 1i)*2i - x*1i + 16i/(exp(2*x) + exp(x)*2i - 1) - 8i/(exp(3*x) *4i - 6*exp(2*x) + exp(4*x) - exp(x)*4i + 1) - 8/(exp(x) + 1i) + 16/(exp(2 *x)*3i + exp(3*x) - 3*exp(x) - 1i)
\[ \int \frac {1}{(\text {sech}(x)-i \tanh (x))^5} \, dx=-40 \left (\int \frac {e^{8 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right ) i +240 \left (\int \frac {e^{7 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right )+640 \left (\int \frac {e^{6 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right ) i -1008 \left (\int \frac {e^{5 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right )-1040 \left (\int \frac {e^{4 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right ) i +720 \left (\int \frac {e^{3 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right )+320 \left (\int \frac {e^{2 x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right ) i -80 \left (\int \frac {e^{x}}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right )-8 \left (\int \frac {1}{e^{10 x}+10 e^{9 x} i -45 e^{8 x}-120 e^{7 x} i +210 e^{6 x}+252 e^{5 x} i -210 e^{4 x}-120 e^{3 x} i +45 e^{2 x}+10 e^{x} i -1}d x \right ) i +\mathrm {log}\left (e^{10 x} i -10 e^{9 x}-45 e^{8 x} i +120 e^{7 x}+210 e^{6 x} i -252 e^{5 x}-210 e^{4 x} i +120 e^{3 x}+45 e^{2 x} i -10 e^{x}-i \right ) i -9 i x \] Input:
int(1/(sech(x)-I*tanh(x))^5,x)
Output:
- 40*int(e**(8*x)/(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x) *i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e* *(2*x) + 10*e**x*i - 1),x)*i + 240*int(e**(7*x)/(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**( 4*x) - 120*e**(3*x)*i + 45*e**(2*x) + 10*e**x*i - 1),x) + 640*int(e**(6*x) /(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e**(2*x) + 10*e**x*i - 1),x)*i - 1008*int(e**(5*x)/(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3* x)*i + 45*e**(2*x) + 10*e**x*i - 1),x) - 1040*int(e**(4*x)/(e**(10*x) + 10 *e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e**(2*x) + 10*e**x*i - 1),x)*i + 720 *int(e**(3*x)/(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e**(2*x ) + 10*e**x*i - 1),x) + 320*int(e**(2*x)/(e**(10*x) + 10*e**(9*x)*i - 45*e **(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**(5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e**(2*x) + 10*e**x*i - 1),x)*i - 80*int(e**x/(e**(10*x ) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 210*e**(6*x) + 252*e**( 5*x)*i - 210*e**(4*x) - 120*e**(3*x)*i + 45*e**(2*x) + 10*e**x*i - 1),x) - 8*int(1/(e**(10*x) + 10*e**(9*x)*i - 45*e**(8*x) - 120*e**(7*x)*i + 21...