Integrand size = 11, antiderivative size = 40 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=i \log (i+\sinh (x))-\frac {2 i}{(1-i \sinh (x))^2}+\frac {4 i}{1-i \sinh (x)} \] Output:
I*ln(I+sinh(x))-2*I/(1-I*sinh(x))^2+4*I/(1-I*sinh(x))
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.50 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=\arctan (\sinh (x))+i \log (\cosh (x))+i \text {sech}^2(x)-\frac {3}{2} i \text {sech}^4(x)+\text {sech}(x) \tanh (x)-\text {sech}^3(x) \tanh (x)-5 \text {sech}(x) \tanh ^3(x)-\frac {5}{2} i \tanh ^4(x) \] Input:
Integrate[(Sech[x] + I*Tanh[x])^5,x]
Output:
ArcTan[Sinh[x]] + I*Log[Cosh[x]] + I*Sech[x]^2 - ((3*I)/2)*Sech[x]^4 + Sec h[x]*Tanh[x] - Sech[x]^3*Tanh[x] - 5*Sech[x]*Tanh[x]^3 - ((5*I)/2)*Tanh[x] ^4
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int (\text {sech}(x)+i \tanh (x))^5 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\tan (i x)+\sec (i x))^5dx\) |
\(\Big \downarrow \) 4891 |
\(\displaystyle \int (1+i \sinh (x))^5 \text {sech}^5(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(1+\sin (i x))^5}{\cos (i x)^5}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle -i \int \frac {(i \sinh (x)+1)^2}{(1-i \sinh (x))^3}d(i \sinh (x))\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -i \int \left (-\frac {4}{(i \sinh (x)-1)^2}-\frac {4}{(i \sinh (x)-1)^3}+\frac {1}{1-i \sinh (x)}\right )d(i \sinh (x))\) |
\(\Big \downarrow \) 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (34 ) = 68\).
Time = 1.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.95
\[\frac {8 \left (\frac {\operatorname {sech}\left (x \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (x \right )}{8}\right ) \tanh \left (x \right )}{3}+2 \arctan \left ({\mathrm e}^{x}\right )+\frac {5 i}{4 \cosh \left (x \right )^{4}}-\frac {5 \sinh \left (x \right )}{3 \cosh \left (x \right )^{4}}+\frac {5 i \sinh \left (x \right )^{2}}{\cosh \left (x \right )^{4}}-\frac {5 \sinh \left (x \right )^{3}}{\cosh \left (x \right )^{4}}+i \ln \left (\cosh \left (x \right )\right )-\frac {i \tanh \left (x \right )^{2}}{2}-\frac {i \tanh \left (x \right )^{4}}{4}\]
Input:
int((sech(x)+I*tanh(x))^5,x)
Output:
8/3*(1/4*sech(x)^3+3/8*sech(x))*tanh(x)+2*arctan(exp(x))+5/4*I/cosh(x)^4-5 /3*sinh(x)/cosh(x)^4+5*I*sinh(x)^2/cosh(x)^4-5*sinh(x)^3/cosh(x)^4+I*ln(co sh(x))-1/2*I*tanh(x)^2-1/4*I*tanh(x)^4
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 = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.35 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=\frac {-i \, x e^{\left (4 \, x\right )} + 4 \, {\left (x - 2\right )} e^{\left (3 \, x\right )} - 2 \, {\left (-3 i \, x + 4 i\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x - 2\right )} e^{x} - 2 \, {\left (-i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} - 4 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (4 \, x\right )} + 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1} \] Input:
integrate((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)
\[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=\int \left (i \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{5}\, dx \] Input:
integrate((sech(x)+I*tanh(x))**5,x)
Output:
Integral((I*tanh(x) + sech(x))**5, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (28) = 56\).
Time = 0.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 5.88 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=-\frac {5}{2} i \, \tanh \left (x\right )^{4} + i \, x - \frac {5 \, {\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {5 \, {\left (e^{\left (-x\right )} - 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} - e^{\left (-7 \, x\right )}\right )}}{2 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {4 i \, {\left (e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} - \frac {20 i}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} - 2 \, \arctan \left (e^{\left (-x\right )}\right ) + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \] Input:
integrate((sech(x)+I*tanh(x))^5,x, algorithm="maxima")
Output:
-5/2*I*tanh(x)^4 + I*x - 5/4*(5*e^(-x) - 3*e^(-3*x) + 3*e^(-5*x) - 5*e^(-7 *x))/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1) + 1/4*(3*e^(-x) + 11*e^(-3*x) - 11*e^(-5*x) - 3*e^(-7*x))/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^ (-6*x) + e^(-8*x) + 1) - 5/2*(e^(-x) - 7*e^(-3*x) + 7*e^(-5*x) - e^(-7*x)) /(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1) + 4*I*(e^(-2*x) + e ^(-4*x) + e^(-6*x))/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1) - 20*I/(e^(-x) + e^x)^4 - 2*arctan(e^(-x)) + I*log(e^(-2*x) + 1)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=-i \, x - \frac {8 \, {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x}\right )}}{{\left (e^{x} + i\right )}^{4}} + 2 i \, \log \left (e^{x} + i\right ) \] Input:
integrate((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 = 1.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.25 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=-x\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,2{}\mathrm {i}+\frac {16{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {8{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {8}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {16}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}} \] Input:
int((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)
Time = 0.22 (sec) , antiderivative size = 429, normalized size of antiderivative = 10.72 \[ \int (\text {sech}(x)+i \tanh (x))^5 \, dx=\frac {-4 i +4 e^{8 x} \mathrm {log}\left (e^{2 x}+1\right ) i -5 e^{8 x} \mathrm {sech}\left (x \right )^{4} i +10 e^{8 x} \mathrm {sech}\left (x \right )^{2} i -4 e^{8 x} i x +16 e^{6 x} \mathrm {log}\left (e^{2 x}+1\right ) i -20 e^{6 x} \mathrm {sech}\left (x \right )^{4} i +40 e^{6 x} \mathrm {sech}\left (x \right )^{2} i -16 e^{6 x} i x +24 e^{4 x} \mathrm {log}\left (e^{2 x}+1\right ) i -30 e^{4 x} \mathrm {sech}\left (x \right )^{4} i +60 e^{4 x} \mathrm {sech}\left (x \right )^{2} i -24 e^{4 x} i x +16 e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) i -20 e^{2 x} \mathrm {sech}\left (x \right )^{4} i +40 e^{2 x} \mathrm {sech}\left (x \right )^{2} i -16 e^{2 x} i x +10 \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )^{2} i +8 e^{8 x} \mathit {atan} \left (e^{x}\right )+32 e^{6 x} \mathit {atan} \left (e^{x}\right )+48 e^{4 x} \mathit {atan} \left (e^{x}\right )+32 e^{2 x} \mathit {atan} \left (e^{x}\right )-4 e^{8 x} i -8 e^{4 x} i +4 \,\mathrm {log}\left (e^{2 x}+1\right ) i -5 \mathrm {sech}\left (x \right )^{4} i +10 \mathrm {sech}\left (x \right )^{2} i -4 i x +8 \mathit {atan} \left (e^{x}\right )-96 e^{3 x}+96 e^{5 x}-32 e^{7 x}+10 e^{8 x} \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )^{2} i +40 e^{6 x} \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )^{2} i +60 e^{4 x} \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )^{2} i +40 e^{2 x} \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )^{2} i +32 e^{x}}{4 e^{8 x}+16 e^{6 x}+24 e^{4 x}+16 e^{2 x}+4} \] Input:
int((sech(x)+I*tanh(x))^5,x)
Output:
(8*e**(8*x)*atan(e**x) + 32*e**(6*x)*atan(e**x) + 48*e**(4*x)*atan(e**x) + 32*e**(2*x)*atan(e**x) + 8*atan(e**x) + 4*e**(8*x)*log(e**(2*x) + 1)*i - 5*e**(8*x)*sech(x)**4*i + 10*e**(8*x)*sech(x)**2*tanh(x)**2*i + 10*e**(8*x )*sech(x)**2*i - 4*e**(8*x)*i*x - 4*e**(8*x)*i - 32*e**(7*x) + 16*e**(6*x) *log(e**(2*x) + 1)*i - 20*e**(6*x)*sech(x)**4*i + 40*e**(6*x)*sech(x)**2*t anh(x)**2*i + 40*e**(6*x)*sech(x)**2*i - 16*e**(6*x)*i*x + 96*e**(5*x) + 2 4*e**(4*x)*log(e**(2*x) + 1)*i - 30*e**(4*x)*sech(x)**4*i + 60*e**(4*x)*se ch(x)**2*tanh(x)**2*i + 60*e**(4*x)*sech(x)**2*i - 24*e**(4*x)*i*x - 8*e** (4*x)*i - 96*e**(3*x) + 16*e**(2*x)*log(e**(2*x) + 1)*i - 20*e**(2*x)*sech (x)**4*i + 40*e**(2*x)*sech(x)**2*tanh(x)**2*i + 40*e**(2*x)*sech(x)**2*i - 16*e**(2*x)*i*x + 32*e**x + 4*log(e**(2*x) + 1)*i - 5*sech(x)**4*i + 10* sech(x)**2*tanh(x)**2*i + 10*sech(x)**2*i - 4*i*x - 4*i)/(4*(e**(8*x) + 4* e**(6*x) + 6*e**(4*x) + 4*e**(2*x) + 1))