Integrand size = 13, antiderivative size = 49 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {i \text {sech}^3(x)}{7 (i+\sinh (x))^2}-\frac {\text {sech}^3(x)}{7 (i+\sinh (x))}-\frac {4 \tanh (x)}{7}+\frac {4 \tanh ^3(x)}{21} \] Output:
-1/7*I*sech(x)^3/(I+sinh(x))^2-sech(x)^3/(7*I+7*sinh(x))-4/7*tanh(x)+4/21* tanh(x)^3
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {\text {sech}^3(x) (8 i \cosh (2 x)+4 i \cosh (4 x)-14 \sinh (x)-3 \sinh (3 x)+\sinh (5 x))}{42 (i+\sinh (x))^2} \] Input:
Integrate[Sech[x]^4/(I + Sinh[x])^2,x]
Output:
-1/42*(Sech[x]^3*((8*I)*Cosh[2*x] + (4*I)*Cosh[4*x] - 14*Sinh[x] - 3*Sinh[ 3*x] + Sinh[5*x]))/(I + Sinh[x])^2
Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 3151, 3042, 3151, 3042, 4254, 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.
\(\displaystyle \int \frac {\text {sech}^4(x)}{(\sinh (x)+i)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i-i \sin (i x))^2 \cos (i x)^4}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle -\frac {5}{7} i \int \frac {\text {sech}^4(x)}{\sinh (x)+i}dx-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5}{7} i \int \frac {1}{\cos (i x)^4 (i-i \sin (i x))}dx-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle -\frac {5}{7} i \left (-\frac {4}{5} i \int \text {sech}^4(x)dx-\frac {i \text {sech}^3(x)}{5 (\sinh (x)+i)}\right )-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5}{7} i \left (-\frac {4}{5} i \int \csc \left (i x+\frac {\pi }{2}\right )^4dx-\frac {i \text {sech}^3(x)}{5 (\sinh (x)+i)}\right )-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {5}{7} i \left (\frac {4}{5} \int \left (1-\tanh ^2(x)\right )d(-i \tanh (x))-\frac {i \text {sech}^3(x)}{5 (\sinh (x)+i)}\right )-\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \text {sech}^3(x)}{7 (\sinh (x)+i)^2}-\frac {5}{7} i \left (\frac {4}{5} \left (\frac {1}{3} i \tanh ^3(x)-i \tanh (x)\right )-\frac {i \text {sech}^3(x)}{5 (\sinh (x)+i)}\right )\) |
Input:
Int[Sech[x]^4/(I + Sinh[x])^2,x]
Output:
((-1/7*I)*Sech[x]^3)/(I + Sinh[x])^2 - ((5*I)/7)*(((-1/5*I)*Sech[x]^3)/(I + Sinh[x]) + (4*((-I)*Tanh[x] + (I/3)*Tanh[x]^3))/5)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (40 ) = 80\).
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.37
\[\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}-\frac {5 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {23 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{7 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{7}}-\frac {4}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}+\frac {55}{12 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {13}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}-\frac {i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {1}{12 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}\]
Input:
int(sech(x)^4/(I+sinh(x))^2,x)
Output:
2*I/(tanh(1/2*x)+I)^6-5*I/(tanh(1/2*x)+I)^4+23/8*I/(tanh(1/2*x)+I)^2+4/7/( tanh(1/2*x)+I)^7-4/(tanh(1/2*x)+I)^5+55/12/(tanh(1/2*x)+I)^3-13/8/(tanh(1/ 2*x)+I)-1/8*I/(tanh(1/2*x)-I)^2+1/12/(tanh(1/2*x)-I)^3-3/8/(tanh(1/2*x)-I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (35) = 70\).
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {16 \, {\left (14 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} - 1\right )}}{21 \, {\left (e^{\left (10 \, x\right )} + 4 i \, e^{\left (9 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 8 i \, e^{\left (7 \, x\right )} - 14 \, e^{\left (6 \, x\right )} - 14 \, e^{\left (4 \, x\right )} - 8 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1\right )}} \] Input:
integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="fricas")
Output:
-16/21*(14*e^(4*x) + 8*I*e^(3*x) + 3*e^(2*x) + 4*I*e^x - 1)/(e^(10*x) + 4* I*e^(9*x) - 3*e^(8*x) + 8*I*e^(7*x) - 14*e^(6*x) - 14*e^(4*x) - 8*I*e^(3*x ) - 3*e^(2*x) - 4*I*e^x + 1)
\[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \] Input:
integrate(sech(x)**4/(I+sinh(x))**2,x)
Output:
Integral(sech(x)**4/(sinh(x) + I)**2, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (35) = 70\).
Time = 0.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 6.47 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {64 i \, e^{\left (-x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac {48 \, e^{\left (-2 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac {128 i \, e^{\left (-3 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} + \frac {224 \, e^{\left (-4 \, x\right )}}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} - \frac {16}{84 i \, e^{\left (-x\right )} - 63 \, e^{\left (-2 \, x\right )} + 168 i \, e^{\left (-3 \, x\right )} - 294 \, e^{\left (-4 \, x\right )} - 294 \, e^{\left (-6 \, x\right )} - 168 i \, e^{\left (-7 \, x\right )} - 63 \, e^{\left (-8 \, x\right )} - 84 i \, e^{\left (-9 \, x\right )} + 21 \, e^{\left (-10 \, x\right )} + 21} \] Input:
integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="maxima")
Output:
-64*I*e^(-x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) + 21) + 48*e^(-2*x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x) - 294*e^( -4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e ^(-10*x) + 21) - 128*I*e^(-3*x)/(84*I*e^(-x) - 63*e^(-2*x) + 168*I*e^(-3*x ) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(- 9*x) + 21*e^(-10*x) + 21) + 224*e^(-4*x)/(84*I*e^(-x) - 63*e^(-2*x) + 168* I*e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 84*I*e^(-9*x) + 21*e^(-10*x) + 21) - 16/(84*I*e^(-x) - 63*e^(-2*x) + 168*I *e^(-3*x) - 294*e^(-4*x) - 294*e^(-6*x) - 168*I*e^(-7*x) - 63*e^(-8*x) - 8 4*I*e^(-9*x) + 21*e^(-10*x) + 21)
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.33 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {6 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 7 i}{24 \, {\left (e^{x} - i\right )}^{3}} - \frac {-42 i \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} + 1015 i \, e^{\left (4 \, x\right )} - 1750 \, e^{\left (3 \, x\right )} - 1344 i \, e^{\left (2 \, x\right )} + 511 \, e^{x} + 79 i}{168 \, {\left (e^{x} + i\right )}^{7}} \] Input:
integrate(sech(x)^4/(I+sinh(x))^2,x, algorithm="giac")
Output:
-1/24*(6*I*e^(2*x) + 15*e^x - 7*I)/(e^x - I)^3 - 1/168*(-42*I*e^(6*x) + 31 5*e^(5*x) + 1015*I*e^(4*x) - 1750*e^(3*x) - 1344*I*e^(2*x) + 511*e^x + 79* I)/(e^x + I)^7
Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.84 \[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\frac {\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{7}+\frac {32\,{\mathrm {e}}^{4\,x}}{3}-\frac {16}{21}\right )\,1{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{7}+\frac {32\,{\mathrm {e}}^{4\,x}}{3}-\frac {16}{21}\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,\left (\frac {128\,{\mathrm {e}}^{3\,x}}{21}+\frac {64\,{\mathrm {e}}^x}{21}\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7}-\frac {\left (\frac {128\,{\mathrm {e}}^{3\,x}}{21}+\frac {64\,{\mathrm {e}}^x}{21}\right )\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )\,1{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^7} \] Input:
int(1/(cosh(x)^4*(sinh(x) + 1i)^2),x)
Output:
((4*exp(3*x) - 4*exp(x))*((16*exp(2*x))/7 + (32*exp(4*x))/3 - 16/21)*1i)/( exp(2*x) + 1)^7 - ((exp(4*x) - 6*exp(2*x) + 1)*((16*exp(2*x))/7 + (32*exp( 4*x))/3 - 16/21))/(exp(2*x) + 1)^7 - ((4*exp(3*x) - 4*exp(x))*((128*exp(3* x))/21 + (64*exp(x))/21))/(exp(2*x) + 1)^7 - (((128*exp(3*x))/21 + (64*exp (x))/21)*(exp(4*x) - 6*exp(2*x) + 1)*1i)/(exp(2*x) + 1)^7
\[ \int \frac {\text {sech}^4(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\mathrm {sech}\left (x \right )^{4}}{\sinh \left (x \right )^{2}+2 \sinh \left (x \right ) i -1}d x \] Input:
int(sech(x)^4/(I+sinh(x))^2,x)
Output:
int(sech(x)**4/(sinh(x)**2 + 2*sinh(x)*i - 1),x)