Integrand size = 11, antiderivative size = 38 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=x+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)} \] Output:
x+2/3*I*cosh(x)^3/(1+I*sinh(x))^3-2*I*cosh(x)/(1+I*sinh(x))
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=\frac {3 (8 i+3 x) \cosh \left (\frac {x}{2}\right )-(16 i+3 x) \cosh \left (\frac {3 x}{2}\right )+6 i (4 i+2 x+x \cosh (x)) \sinh \left (\frac {x}{2}\right )}{6 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \] Input:
Integrate[(Sech[x] + I*Tanh[x])^(-4),x]
Output:
(3*(8*I + 3*x)*Cosh[x/2] - (16*I + 3*x)*Cosh[(3*x)/2] + (6*I)*(4*I + 2*x + x*Cosh[x])*Sinh[x/2])/(6*(Cosh[x/2] + I*Sinh[x/2])^3)
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4891, 3042, 3159, 3042, 3159, 24}
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.
\(\displaystyle \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\tan (i x)+\sec (i x))^4}dx\) |
\(\Big \downarrow \) 4891 |
\(\displaystyle \int \frac {\cosh ^4(x)}{(1+i \sinh (x))^4}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)^4}{(1+\sin (i x))^4}dx\) |
\(\Big \downarrow \) 3159 |
\(\displaystyle \frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\int \frac {\cosh ^2(x)}{(i \sinh (x)+1)^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\int \frac {\cos (i x)^2}{(\sin (i x)+1)^2}dx\) |
\(\Big \downarrow \) 3159 |
\(\displaystyle \int 1dx+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle x+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)}\) |
Input:
Int[(Sech[x] + I*Tanh[x])^(-4),x]
Output:
x + (((2*I)/3)*Cosh[x]^3)/(1 + I*Sinh[x])^3 - ((2*I)*Cosh[x])/(1 + I*Sinh[ x])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
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 = 0.95 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08
\[\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {8 i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {16}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}\]
Input:
int(1/(sech(x)+I*tanh(x))^4,x)
Output:
ln(1+tanh(1/2*x))-ln(tanh(1/2*x)-1)+8*I/(tanh(1/2*x)-I)^2-16/3/(tanh(1/2*x )-I)^3
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=\frac {3 \, x e^{\left (3 \, x\right )} - 3 \, {\left (3 i \, x + 8 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (3 \, x + 8\right )} e^{x} + 3 i \, x + 16 i}{3 \, {\left (e^{\left (3 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + i\right )}} \] Input:
integrate(1/(sech(x)+I*tanh(x))^4,x, algorithm="fricas")
Output:
1/3*(3*x*e^(3*x) - 3*(3*I*x + 8*I)*e^(2*x) - 3*(3*x + 8)*e^x + 3*I*x + 16* I)/(e^(3*x) - 3*I*e^(2*x) - 3*e^x + I)
\[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=\int \frac {1}{\left (i \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(sech(x)+I*tanh(x))**4,x)
Output:
Integral((I*tanh(x) + sech(x))**(-4), x)
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=x - \frac {8 \, {\left (3 \, e^{\left (-x\right )} - 3 i \, e^{\left (-2 \, x\right )} + 2 i\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} - 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} + i\right )}} \] Input:
integrate(1/(sech(x)+I*tanh(x))^4,x, algorithm="maxima")
Output:
x - 8/3*(3*e^(-x) - 3*I*e^(-2*x) + 2*I)/(3*e^(-x) - 3*I*e^(-2*x) - e^(-3*x ) + I)
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=x - \frac {8 \, {\left (3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 2 i\right )}}{3 \, {\left (e^{x} - i\right )}^{3}} \] Input:
integrate(1/(sech(x)+I*tanh(x))^4,x, algorithm="giac")
Output:
x - 8/3*(3*I*e^(2*x) + 3*e^x - 2*I)/(e^x - I)^3
Time = 1.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=x+\frac {\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}-\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )} \] Input:
int(1/(tanh(x)*1i + 1/cosh(x))^4,x)
Output:
x + ((exp(2*x)*8i)/3 - 8i/3)/(exp(2*x)*3i - exp(3*x) + 3*exp(x) - 1i) - 8i /(3*(exp(x) - 1i)) + (exp(x)*8i)/(3*(exp(x)*2i - exp(2*x) + 1))
\[ \int \frac {1}{(\text {sech}(x)+i \tanh (x))^4} \, dx=-24 \left (\int \frac {e^{6 x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right )+112 \left (\int \frac {e^{5 x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right ) i +216 \left (\int \frac {e^{4 x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right )-224 \left (\int \frac {e^{3 x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right ) i -136 \left (\int \frac {e^{2 x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right )+48 \left (\int \frac {e^{x}}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right ) i +8 \left (\int \frac {1}{e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1}d x \right )+\mathrm {log}\left (e^{8 x}-8 e^{7 x} i -28 e^{6 x}+56 e^{5 x} i +70 e^{4 x}-56 e^{3 x} i -28 e^{2 x}+8 e^{x} i +1\right )-7 x \] Input:
int(1/(sech(x)+I*tanh(x))^4,x)
Output:
- 24*int(e**(6*x)/(e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x) + 112*int(e **(5*x)/(e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4* x) - 56*e**(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x)*i + 216*int(e**(4*x)/( e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e **(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x) - 224*int(e**(3*x)/(e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x)*i - 136*int(e**(2*x)/(e**(8*x) - 8*e**(7*x )*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3*x)*i - 28*e**(2 *x) + 8*e**x*i + 1),x) + 48*int(e**x/(e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x ) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x)*i + 8*int(1/(e**(8*x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3*x)*i - 28*e**(2*x) + 8*e**x*i + 1),x) + log(e**(8 *x) - 8*e**(7*x)*i - 28*e**(6*x) + 56*e**(5*x)*i + 70*e**(4*x) - 56*e**(3* x)*i - 28*e**(2*x) + 8*e**x*i + 1) - 7*x