Integrand size = 24, antiderivative size = 87 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 \text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {2 i \coth (c+d x)}{a d}-\frac {3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))} \] Output:
3/2*arctanh(cosh(d*x+c))/a/d+2*I*coth(d*x+c)/a/d-3/2*coth(d*x+c)*csch(d*x+ c)/a/d+coth(d*x+c)*csch(d*x+c)/d/(a+I*a*sinh(d*x+c))
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {4 i \text {csch}(2 (c+d x))-3 \text {sech}(c+d x)+3 \text {arctanh}\left (\sqrt {\cosh ^2(c+d x)}\right ) \sqrt {\cosh ^2(c+d x)} \text {sech}(c+d x)-\text {csch}^2(c+d x) \text {sech}(c+d x)+4 i \tanh (c+d x)}{2 a d} \] Input:
Integrate[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]
Output:
((4*I)*Csch[2*(c + d*x)] - 3*Sech[c + d*x] + 3*ArcTanh[Sqrt[Cosh[c + d*x]^ 2]]*Sqrt[Cosh[c + d*x]^2]*Sech[c + d*x] - Csch[c + d*x]^2*Sech[c + d*x] + (4*I)*Tanh[c + d*x])/(2*a*d)
Time = 0.63 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {3042, 26, 3247, 26, 3042, 26, 3227, 25, 26, 3042, 25, 26, 4254, 24, 4255, 26, 3042, 26, 4257}
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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 (a+a \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 (\sin (i c+i d x) a+a)}dx\) |
| \(\Big \downarrow \) 3247 |
| \(\displaystyle -i \left (\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}-\frac {\int -i \text {csch}^3(c+d x) (3 a-2 i a \sinh (c+d x))dx}{a^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \int \text {csch}^3(c+d x) (3 a-2 i a \sinh (c+d x))dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \int -\frac {i (3 a-2 a \sin (i c+i d x))}{\sin (i c+i d x)^3}dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\int \frac {3 a-2 a \sin (i c+i d x)}{\sin (i c+i d x)^3}dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -i \left (\frac {-2 a \int -\text {csch}^2(c+d x)dx+3 a \int i \text {csch}^3(c+d x)dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {2 a \int \text {csch}^2(c+d x)dx+3 a \int i \text {csch}^3(c+d x)dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {2 a \int \text {csch}^2(c+d x)dx+3 i a \int \text {csch}^3(c+d x)dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {2 a \int -\csc (i c+i d x)^2dx+3 i a \int -i \csc (i c+i d x)^3dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {3 i a \int -i \csc (i c+i d x)^3dx-2 a \int \csc (i c+i d x)^2dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {3 a \int \csc (i c+i d x)^3dx-2 a \int \csc (i c+i d x)^2dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -i \left (\frac {3 a \int \csc (i c+i d x)^3dx-\frac {2 i a \int 1d(-i \coth (c+d x))}{d}}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \int \csc (i c+i d x)^3dx}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \left (\frac {1}{2} \int -i \text {csch}(c+d x)dx-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \left (-\frac {1}{2} i \int \text {csch}(c+d x)dx-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \left (-\frac {1}{2} i \int i \csc (i c+i d x)dx-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \left (\frac {1}{2} \int \csc (i c+i d x)dx-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -i \left (\frac {-\frac {2 a \coth (c+d x)}{d}+3 a \left (\frac {i \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {i \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a^2}+\frac {i \coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}\right )\) |
Input:
Int[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]
Output:
(-I)*(((-2*a*Coth[c + d*x])/d + 3*a*(((I/2)*ArcTanh[Cosh[c + d*x]])/d - (( I/2)*Coth[c + d*x]*Csch[c + d*x])/d))/a^2 + (I*Coth[c + d*x]*Csch[c + d*x] )/(d*(a + I*a*Sinh[c + d*x])))
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{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[n, 0] && (IntegerQ[2*n] || EqQ[c, 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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {2 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 i}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{4 d a}\) | \(91\) |
| default | \(\frac {2 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 i}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{4 d a}\) | \(91\) |
| parallelrisch | \(\frac {12 \left (i-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+i \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+3 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(111\) |
| risch | \(-\frac {-3 i {\mathrm e}^{3 d x +3 c}+i {\mathrm e}^{d x +c}+3 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+4}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{d x +c}-i\right ) d a}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}\) | \(113\) |
Input:
int(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/4/d/a*(2*I*tanh(1/2*d*x+1/2*c)+1/2*tanh(1/2*d*x+1/2*c)^2-1/2/tanh(1/2*d* x+1/2*c)^2+2*I/tanh(1/2*d*x+1/2*c)-6*ln(tanh(1/2*d*x+1/2*c))+8*I/(-I+tanh( 1/2*d*x+1/2*c)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (79) = 158\).
Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.69 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 \, {\left (e^{\left (5 \, d x + 5 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, {\left (e^{\left (5 \, d x + 5 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 6 \, e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, e^{\left (3 \, d x + 3 \, c\right )} + 10 \, e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, e^{\left (d x + c\right )} - 8}{2 \, {\left (a d e^{\left (5 \, d x + 5 \, c\right )} - i \, a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d e^{\left (d x + c\right )} - i \, a d\right )}} \] Input:
integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
Output:
1/2*(3*(e^(5*d*x + 5*c) - I*e^(4*d*x + 4*c) - 2*e^(3*d*x + 3*c) + 2*I*e^(2 *d*x + 2*c) + e^(d*x + c) - I)*log(e^(d*x + c) + 1) - 3*(e^(5*d*x + 5*c) - I*e^(4*d*x + 4*c) - 2*e^(3*d*x + 3*c) + 2*I*e^(2*d*x + 2*c) + e^(d*x + c) - I)*log(e^(d*x + c) - 1) - 6*e^(4*d*x + 4*c) + 6*I*e^(3*d*x + 3*c) + 10* e^(2*d*x + 2*c) - 2*I*e^(d*x + c) - 8)/(a*d*e^(5*d*x + 5*c) - I*a*d*e^(4*d *x + 4*c) - 2*a*d*e^(3*d*x + 3*c) + 2*I*a*d*e^(2*d*x + 2*c) + a*d*e^(d*x + c) - I*a*d)
\[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx}{a} \] Input:
integrate(csch(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)
Output:
-I*Integral(csch(c + d*x)**3/(sinh(c + d*x) - I), x)/a
Time = 0.05 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.79 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4}{{\left (a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a e^{\left (-4 \, d x - 4 \, c\right )} + a e^{\left (-5 \, d x - 5 \, c\right )} + i \, a\right )} d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a d} \] Input:
integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
Output:
-(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d* x - 4*c) + 4)/((a*e^(-d*x - c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c) + I*a*e^(-4*d*x - 4*c) + a*e^(-5*d*x - 5*c) + I*a)*d) + 3/2*log(e^(-d *x - c) + 1)/(a*d) - 3/2*log(e^(-d*x - c) - 1)/(a*d)
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\frac {3 \, \log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {3 \, \log \left (e^{\left (d x + c\right )} - 1\right )}{a} - \frac {2 \, {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} + 2 i\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} - \frac {4 i}{a {\left (i \, e^{\left (d x + c\right )} + 1\right )}}}{2 \, d} \] Input:
integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
1/2*(3*log(e^(d*x + c) + 1)/a - 3*log(e^(d*x + c) - 1)/a - 2*(e^(3*d*x + 3 *c) - 2*I*e^(2*d*x + 2*c) + e^(d*x + c) + 2*I)/(a*(e^(2*d*x + 2*c) - 1)^2) - 4*I/(a*(I*e^(d*x + c) + 1)))/d
Time = 1.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^2\,d^2}}{a\,d}\right )}{\sqrt {-a^2\,d^2}}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}^2}+\frac {2{}\mathrm {i}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \] Input:
int(1/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)
Output:
(3*atan((exp(d*x)*exp(c)*(-a^2*d^2)^(1/2))/(a*d)))/(-a^2*d^2)^(1/2) - 2/(a *d*(exp(c + d*x) - 1i)) + 2i/(a*d*(exp(2*c + 2*d*x) - 1)) - exp(c + d*x)/( a*d*(exp(2*c + 2*d*x) - 1)) - (2*exp(c + d*x))/(a*d*(exp(2*c + 2*d*x) - 1) ^2)
\[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\int \frac {\mathrm {csch}\left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x}{a} \] Input:
int(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
int(csch(c + d*x)**3/(sinh(c + d*x)*i + 1),x)/a