Integrand size = 13, antiderivative size = 44 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i x+\frac {4 \cosh (x)}{3}-\frac {\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}+\frac {2 i \cosh (x)}{i+\sinh (x)} \] Output:
-2*I*x+4/3*cosh(x)-1/3*cosh(x)*sinh(x)^2/(I+sinh(x))^2+2*I*cosh(x)/(I+sinh (x))
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{3} \cosh (x) \left (-\frac {6 i \text {arcsinh}(\sinh (x))}{\sqrt {\cosh ^2(x)}}+\frac {-10+14 i \sinh (x)+3 \sinh ^2(x)}{(i+\sinh (x))^2}\right ) \] Input:
Integrate[Sinh[x]^3/(I + Sinh[x])^2,x]
Output:
(Cosh[x]*(((-6*I)*ArcSinh[Sinh[x]])/Sqrt[Cosh[x]^2] + (-10 + (14*I)*Sinh[x ] + 3*Sinh[x]^2)/(I + Sinh[x])^2))/3
Time = 0.55 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.231, Rules used = {3042, 26, 25, 3244, 27, 3042, 26, 3447, 3042, 3502, 27, 3042, 26, 3214, 3042, 3127}
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.
\(\displaystyle \int \frac {\sinh ^3(x)}{(\sinh (x)+i)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sin (i x)^3}{(i-i \sin (i x))^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int -\frac {\sin (i x)^3}{(1-\sin (i x))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -i \int \frac {\sin (i x)^3}{(1-\sin (i x))^2}dx\) |
\(\Big \downarrow \) 3244 |
\(\displaystyle -i \left (\frac {1}{3} \int -\frac {2 i (2 i \sinh (x)+1) \sinh (x)}{1-i \sinh (x)}dx+\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} i \int \frac {(2 i \sinh (x)+1) \sinh (x)}{1-i \sinh (x)}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} i \int -\frac {i \sin (i x) (2 \sin (i x)+1)}{1-\sin (i x)}dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \int \frac {\sin (i x) (2 \sin (i x)+1)}{1-\sin (i x)}dx\right )\) |
\(\Big \downarrow \) 3447 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \int \frac {i \sinh (x)-2 \sinh ^2(x)}{1-i \sinh (x)}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \int \frac {2 \sin (i x)^2+\sin (i x)}{1-\sin (i x)}dx\right )\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (-\int -\frac {3 i \sinh (x)}{1-i \sinh (x)}dx-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 i \int \frac {\sinh (x)}{1-i \sinh (x)}dx-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 i \int -\frac {i \sin (i x)}{1-\sin (i x)}dx-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 \int \frac {\sin (i x)}{1-\sin (i x)}dx-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 \left (-x+\int \frac {1}{1-i \sinh (x)}dx\right )-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 \left (-x+\int \frac {1}{1-\sin (i x)}dx\right )-2 i \cosh (x)\right )\right )\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -i \left (\frac {i \sinh ^2(x) \cosh (x)}{3 (1-i \sinh (x))^2}-\frac {2}{3} \left (3 \left (-x-\frac {i \cosh (x)}{1-i \sinh (x)}\right )-2 i \cosh (x)\right )\right )\) |
Input:
Int[Sinh[x]^3/(I + Sinh[x])^2,x]
Output:
(-I)*((-2*((-2*I)*Cosh[x] + 3*(-x - (I*Cosh[x])/(1 - I*Sinh[x]))))/3 + ((I /3)*Cosh[x]*Sinh[x]^2)/(1 - I*Sinh[x])^2)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-2 i x +\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {10 i {\mathrm e}^{x}+6 \,{\mathrm e}^{2 x}-\frac {16}{3}}{\left ({\mathrm e}^{x}+i\right )^{3}}\) | \(38\) |
default | \(\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {4 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-2 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+2 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}\) | \(75\) |
parallelrisch | \(\frac {\left (-36 i \cosh \left (\frac {x}{2}\right )+12 i \cosh \left (\frac {3 x}{2}\right )-12 \sinh \left (\frac {3 x}{2}\right )-36 \sinh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\left (36 i \cosh \left (\frac {x}{2}\right )-12 i \cosh \left (\frac {3 x}{2}\right )+36 \sinh \left (\frac {x}{2}\right )+12 \sinh \left (\frac {3 x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-24 i \sinh \left (\frac {x}{2}\right )+23 i \sinh \left (\frac {3 x}{2}\right )+3 i \sinh \left (\frac {5 x}{2}\right )-12 \cosh \left (\frac {x}{2}\right )-27 \cosh \left (\frac {3 x}{2}\right )+3 \cosh \left (\frac {5 x}{2}\right )}{6 i \sinh \left (\frac {3 x}{2}\right )+18 i \sinh \left (\frac {x}{2}\right )+6 \cosh \left (\frac {3 x}{2}\right )-18 \cosh \left (\frac {x}{2}\right )}\) | \(141\) |
Input:
int(sinh(x)^3/(I+sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-2*I*x+1/2*exp(x)+1/2*exp(-x)+2/3*(15*I*exp(x)+9*exp(2*x)-8)/(exp(x)+I)^3
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.68 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {3 \, {\left (4 i \, x - 3 i\right )} e^{\left (4 \, x\right )} - 6 \, {\left (6 \, x + 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (-6 i \, x - 11 i\right )} e^{\left (2 \, x\right )} + {\left (12 \, x + 41\right )} e^{x} - 3 \, e^{\left (5 \, x\right )} + 3 i}{6 \, {\left (e^{\left (4 \, x\right )} + 3 i \, e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - i \, e^{x}\right )}} \] Input:
integrate(sinh(x)^3/(I+sinh(x))^2,x, algorithm="fricas")
Output:
-1/6*(3*(4*I*x - 3*I)*e^(4*x) - 6*(6*x + 5)*e^(3*x) + 6*(-6*I*x - 11*I)*e^ (2*x) + (12*x + 41)*e^x - 3*e^(5*x) + 3*I)/(e^(4*x) + 3*I*e^(3*x) - 3*e^(2 *x) - I*e^x)
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=- 2 i x + \frac {18 e^{2 x} + 30 i e^{x} - 16}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} + \frac {e^{x}}{2} + \frac {e^{- x}}{2} \] Input:
integrate(sinh(x)**3/(I+sinh(x))**2,x)
Output:
-2*I*x + (18*exp(2*x) + 30*I*exp(x) - 16)/(3*exp(3*x) + 9*I*exp(2*x) - 9*e xp(x) - 3*I) + exp(x)/2 + exp(-x)/2
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i \, x - \frac {41 \, e^{\left (-x\right )} + 69 i \, e^{\left (-2 \, x\right )} - 39 \, e^{\left (-3 \, x\right )} - 3 i}{2 \, {\left (3 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} - 9 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac {1}{2} \, e^{\left (-x\right )} \] Input:
integrate(sinh(x)^3/(I+sinh(x))^2,x, algorithm="maxima")
Output:
-2*I*x - 1/2*(41*e^(-x) + 69*I*e^(-2*x) - 39*e^(-3*x) - 3*I)/(3*I*e^(-x) - 9*e^(-2*x) - 9*I*e^(-3*x) + 3*e^(-4*x)) + 1/2*e^(-x)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i \, x + \frac {{\left (39 \, e^{\left (3 \, x\right )} + 69 i \, e^{\left (2 \, x\right )} - 41 \, e^{x} - 3 i\right )} e^{\left (-x\right )}}{6 \, {\left (e^{x} + i\right )}^{3}} + \frac {1}{2} \, e^{x} \] Input:
integrate(sinh(x)^3/(I+sinh(x))^2,x, algorithm="giac")
Output:
-2*I*x + 1/6*(39*e^(3*x) + 69*I*e^(2*x) - 41*e^x - 3*I)*e^(-x)/(e^x + I)^3 + 1/2*e^x
Time = 1.76 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.80 \[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{-x}}{2}-x\,2{}\mathrm {i}+\frac {{\mathrm {e}}^x}{2}+\frac {2\,{\mathrm {e}}^x+\frac {4}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {2\,{\mathrm {e}}^{2\,x}-2+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \] Input:
int(sinh(x)^3/(sinh(x) + 1i)^2,x)
Output:
exp(-x)/2 - x*2i + exp(x)/2 + (2*exp(x) + 4i/3)/(exp(2*x) + exp(x)*2i - 1) + (2*exp(2*x) + (exp(x)*8i)/3 - 2)/(exp(2*x)*3i + exp(3*x) - 3*exp(x) - 1 i) + 2/(exp(x) + 1i)
\[ \int \frac {\sinh ^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\sinh \left (x \right )^{3}}{\sinh \left (x \right )^{2}+2 \sinh \left (x \right ) i -1}d x \] Input:
int(sinh(x)^3/(I+sinh(x))^2,x)
Output:
int(sinh(x)**3/(sinh(x)**2 + 2*sinh(x)*i - 1),x)