Integrand size = 13, antiderivative size = 60 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{4} \arctan (\sinh (x))+\frac {1}{16 (i-\sinh (x))}+\frac {1}{12 (i+\sinh (x))^3}-\frac {i}{8 (i+\sinh (x))^2}-\frac {3}{16 (i+\sinh (x))} \] Output:
-1/4*arctan(sinh(x))+1/(16*I-16*sinh(x))+1/12/(I+sinh(x))^3-1/8*I/(I+sinh( x))^2-3/(16*I+16*sinh(x))
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {\text {sech}^2(x) \left (4 i-3 \arctan (\sinh (x))+(-1+6 i \arctan (\sinh (x))) \sinh (x)+6 i \sinh ^2(x)+(3+6 i \arctan (\sinh (x))) \sinh ^3(x)+3 \arctan (\sinh (x)) \sinh ^4(x)\right )}{12 (i+\sinh (x))^2} \] Input:
Integrate[Sech[x]^3/(I + Sinh[x])^2,x]
Output:
-1/12*(Sech[x]^2*(4*I - 3*ArcTan[Sinh[x]] + (-1 + (6*I)*ArcTan[Sinh[x]])*S inh[x] + (6*I)*Sinh[x]^2 + (3 + (6*I)*ArcTan[Sinh[x]])*Sinh[x]^3 + 3*ArcTa n[Sinh[x]]*Sinh[x]^4))/(I + Sinh[x])^2
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3146, 54, 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.
\(\displaystyle \int \frac {\text {sech}^3(x)}{(\sinh (x)+i)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i-i \sin (i x))^2 \cos (i x)^3}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \int \frac {1}{(-\sinh (x)+i)^2 (\sinh (x)+i)^4}d\sinh (x)\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {1}{4 \left (\sinh ^2(x)+1\right )}+\frac {1}{16 (\sinh (x)-i)^2}+\frac {3}{16 (\sinh (x)+i)^2}+\frac {i}{4 (\sinh (x)+i)^3}-\frac {1}{4 (\sinh (x)+i)^4}\right )d\sinh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{4} \arctan (\sinh (x))+\frac {1}{16 (-\sinh (x)+i)}-\frac {3}{16 (\sinh (x)+i)}-\frac {i}{8 (\sinh (x)+i)^2}+\frac {1}{12 (\sinh (x)+i)^3}\) |
Input:
Int[Sech[x]^3/(I + Sinh[x])^2,x]
Output:
-1/4*ArcTan[Sinh[x]] + 1/(16*(I - Sinh[x])) + 1/(12*(I + Sinh[x])^3) - (I/ 8)/(I + Sinh[x])^2 - 3/(16*(I + Sinh[x]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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])
Time = 169.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.33
method | result | size |
risch | \(-\frac {-13 \,{\mathrm e}^{5 x}+8 i {\mathrm e}^{4 x}+13 \,{\mathrm e}^{3 x}+12 i {\mathrm e}^{2 x}+12 i {\mathrm e}^{6 x}+3 \,{\mathrm e}^{7 x}-3 \,{\mathrm e}^{x}}{6 \left ({\mathrm e}^{x}-i\right )^{2} \left ({\mathrm e}^{x}+i\right )^{6}}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}\) | \(80\) |
default | \(\frac {7 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{4}-\frac {23 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {11}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {11}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{4}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )-8 i}\) | \(116\) |
Input:
int(sech(x)^3/(I+sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-1/6*(-13*exp(x)^5+8*I*exp(x)^4+13*exp(x)^3+12*I*exp(x)^2+12*I*exp(x)^6+3* exp(x)^7-3*exp(x))/(exp(x)-I)^2/(exp(x)+I)^6-1/4*I*ln(exp(x)+I)+1/4*I*ln(e xp(x)-I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (38) = 76\).
Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.35 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {3 \, {\left (i \, e^{\left (8 \, x\right )} - 4 \, e^{\left (7 \, x\right )} - 4 i \, e^{\left (6 \, x\right )} - 4 \, e^{\left (5 \, x\right )} - 10 i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - 4 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} + i\right )} \log \left (e^{x} + i\right ) + 3 \, {\left (-i \, e^{\left (8 \, x\right )} + 4 \, e^{\left (7 \, x\right )} + 4 i \, e^{\left (6 \, x\right )} + 4 \, e^{\left (5 \, x\right )} + 10 i \, e^{\left (4 \, x\right )} - 4 \, e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} - 4 \, e^{x} - i\right )} \log \left (e^{x} - i\right ) + 6 \, e^{\left (7 \, x\right )} + 24 i \, e^{\left (6 \, x\right )} - 26 \, e^{\left (5 \, x\right )} + 16 i \, e^{\left (4 \, x\right )} + 26 \, e^{\left (3 \, x\right )} + 24 i \, e^{\left (2 \, x\right )} - 6 \, e^{x}}{12 \, {\left (e^{\left (8 \, x\right )} + 4 i \, e^{\left (7 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 4 i \, e^{\left (5 \, x\right )} - 10 \, e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1\right )}} \] Input:
integrate(sech(x)^3/(I+sinh(x))^2,x, algorithm="fricas")
Output:
-1/12*(3*(I*e^(8*x) - 4*e^(7*x) - 4*I*e^(6*x) - 4*e^(5*x) - 10*I*e^(4*x) + 4*e^(3*x) - 4*I*e^(2*x) + 4*e^x + I)*log(e^x + I) + 3*(-I*e^(8*x) + 4*e^( 7*x) + 4*I*e^(6*x) + 4*e^(5*x) + 10*I*e^(4*x) - 4*e^(3*x) + 4*I*e^(2*x) - 4*e^x - I)*log(e^x - I) + 6*e^(7*x) + 24*I*e^(6*x) - 26*e^(5*x) + 16*I*e^( 4*x) + 26*e^(3*x) + 24*I*e^(2*x) - 6*e^x)/(e^(8*x) + 4*I*e^(7*x) - 4*e^(6* x) + 4*I*e^(5*x) - 10*e^(4*x) - 4*I*e^(3*x) - 4*e^(2*x) - 4*I*e^x + 1)
\[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \] Input:
integrate(sech(x)**3/(I+sinh(x))**2,x)
Output:
Integral(sech(x)**3/(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. 120 vs. \(2 (38) = 76\).
Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {8 \, {\left (3 \, e^{\left (-x\right )} + 12 i \, e^{\left (-2 \, x\right )} - 13 \, e^{\left (-3 \, x\right )} + 8 i \, e^{\left (-4 \, x\right )} + 13 \, e^{\left (-5 \, x\right )} + 12 i \, e^{\left (-6 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}\right )}}{192 i \, e^{\left (-x\right )} - 192 \, e^{\left (-2 \, x\right )} + 192 i \, e^{\left (-3 \, x\right )} - 480 \, e^{\left (-4 \, x\right )} - 192 i \, e^{\left (-5 \, x\right )} - 192 \, e^{\left (-6 \, x\right )} - 192 i \, e^{\left (-7 \, x\right )} + 48 \, e^{\left (-8 \, x\right )} + 48} - \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) + \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \] Input:
integrate(sech(x)^3/(I+sinh(x))^2,x, algorithm="maxima")
Output:
-8*(3*e^(-x) + 12*I*e^(-2*x) - 13*e^(-3*x) + 8*I*e^(-4*x) + 13*e^(-5*x) + 12*I*e^(-6*x) - 3*e^(-7*x))/(192*I*e^(-x) - 192*e^(-2*x) + 192*I*e^(-3*x) - 480*e^(-4*x) - 192*I*e^(-5*x) - 192*e^(-6*x) - 192*I*e^(-7*x) + 48*e^(-8 *x) + 48) - 1/4*I*log(I*e^(-x) + 1) + 1/4*I*log(I*e^(-x) - 1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (38) = 76\).
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=\frac {-i \, e^{\left (-x\right )} + i \, e^{x} + 3}{8 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}} + \frac {11 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 84 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 228 i \, e^{\left (-x\right )} + 228 i \, e^{x} - 240}{48 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}^{3}} - \frac {1}{8} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) + \frac {1}{8} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \] Input:
integrate(sech(x)^3/(I+sinh(x))^2,x, algorithm="giac")
Output:
1/8*(-I*e^(-x) + I*e^x + 3)/(e^(-x) - e^x + 2*I) + 1/48*(11*I*(e^(-x) - e^ x)^3 + 84*(e^(-x) - e^x)^2 - 228*I*e^(-x) + 228*I*e^x - 240)/(e^(-x) - e^x - 2*I)^3 - 1/8*I*log(-e^(-x) + e^x + 2*I) + 1/8*I*log(-e^(-x) + e^x - 2*I )
Time = 2.06 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.30 \[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}-\frac {2}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {1{}\mathrm {i}}{8\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {3{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{8\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {1}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {3}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}+\frac {2{}\mathrm {i}}{3\,\left (15\,{\mathrm {e}}^{2\,x}-15\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1-{\mathrm {e}}^{3\,x}\,20{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,6{}\mathrm {i}+{\mathrm {e}}^x\,6{}\mathrm {i}\right )}-\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )} \] Input:
int(1/(cosh(x)^3*(sinh(x) + 1i)^2),x)
Output:
1i/(8*(exp(x)*2i - exp(2*x) + 1)) - 2/(exp(4*x)*5i - 10*exp(3*x) - exp(2*x )*10i + exp(5*x) + 5*exp(x) + 1i) - 1i/(8*(exp(2*x) + exp(x)*2i - 1)) - 3i /(2*(exp(3*x)*4i - 6*exp(2*x) + exp(4*x) - exp(x)*4i + 1)) - atan(exp(x))/ 2 - 1/(8*(exp(x) - 1i)) - 3/(8*(exp(x) + 1i)) + 2i/(3*(15*exp(2*x) - exp(3 *x)*20i - 15*exp(4*x) + exp(5*x)*6i + exp(6*x) + exp(x)*6i - 1)) - 1/(3*(e xp(2*x)*3i + exp(3*x) - 3*exp(x) - 1i))
\[ \int \frac {\text {sech}^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\mathrm {sech}\left (x \right )^{3}}{\sinh \left (x \right )^{2}+2 \sinh \left (x \right ) i -1}d x \] Input:
int(sech(x)^3/(I+sinh(x))^2,x)
Output:
int(sech(x)**3/(sinh(x)**2 + 2*sinh(x)*i - 1),x)