\(\int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [220]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 \text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {2 i \coth (c+d x)}{a d}-\frac {3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))} \]

[Out]

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))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2847, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 \text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {2 i \coth (c+d x)}{a d}-\frac {3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))} \]

[In]

Int[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

(3*ArcTanh[Cosh[c + d*x]])/(2*a*d) + ((2*I)*Coth[c + d*x])/(a*d) - (3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) + (
Coth[c + d*x]*Csch[c + d*x])/(d*(a + I*a*Sinh[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2847

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] + Dist[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] /; FreeQ[{a, b, c, d, e, f}, x] && Ne
Q[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])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

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] + Dist[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]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}-\frac {\int \text {csch}^3(c+d x) (-3 a+2 i a \sinh (c+d x)) \, dx}{a^2} \\ & = \frac {\coth (c+d x) \text {csch}(c+d x)}{d (a+i a \sinh (c+d x))}-\frac {(2 i) \int \text {csch}^2(c+d x) \, dx}{a}+\frac {3 \int \text {csch}^3(c+d x) \, dx}{a} \\ & = -\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))}-\frac {3 \int \text {csch}(c+d x) \, dx}{2 a}-\frac {2 \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{a d} \\ & = \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))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {4 i \text {csch}(2 (c+d x))-3 \text {sech}(c+d x)+3 \text {arctanh}\left (\sqrt {\cosh ^2(c+d x)}\right ) \sqrt {\cosh ^2(c+d x)} \text {sech}(c+d x)-\text {csch}^2(c+d x) \text {sech}(c+d x)+4 i \tanh (c+d x)}{2 a d} \]

[In]

Integrate[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

((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)

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {2 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {8 i}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\)	\(91\)
default	\(\frac {2 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {8 i}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\)	\(91\)
risch	\(-\frac {i {\mathrm e}^{d x +c}-3 i {\mathrm e}^{3 d x +3 c}+3 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+4}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{d x +c}-i\right ) a d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}\)	\(113\)
parallelrisch	\(\frac {12 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-i \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a \left (i-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(113\)

[In]

int(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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+8*I/(-I+tanh(1/2*d*x+1/2*c))-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)))

Fricas [B] (verification not implemented)

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 = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.69 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 \, {\left (e^{\left (5 \, d x + 5 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, {\left (e^{\left (5 \, d x + 5 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 6 \, e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, e^{\left (3 \, d x + 3 \, c\right )} + 10 \, e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, e^{\left (d x + c\right )} - 8}{2 \, {\left (a d e^{\left (5 \, d x + 5 \, c\right )} - i \, a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d e^{\left (d x + c\right )} - i \, a d\right )}} \]

[In]

integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx}{a} \]

[In]

integrate(csch(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*Integral(csch(c + d*x)**3/(sinh(c + d*x) - I), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.79 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4}{{\left (a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a e^{\left (-4 \, d x - 4 \, c\right )} + a e^{\left (-5 \, d x - 5 \, c\right )} + i \, a\right )} d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a d} \]

[In]

integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-(-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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\frac {3 \, \log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {3 \, \log \left (e^{\left (d x + c\right )} - 1\right )}{a} - \frac {2 \, {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} + 2 i\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} - \frac {4 i}{a {\left (i \, e^{\left (d x + c\right )} + 1\right )}}}{2 \, d} \]

[In]

integrate(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^2\,d^2}}{a\,d}\right )}{\sqrt {-a^2\,d^2}}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}^2}+\frac {2{}\mathrm {i}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]

[In]

int(1/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

(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*(e
xp(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)