\(\int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx\) [105]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=-i x+\frac {1}{3} (3 i-2 \text {csch}(x)) \tanh (x)+\frac {1}{3} (i-\text {csch}(x)) \tanh ^3(x) \]

[Out]

-I*x+1/3*(3*I-2*csch(x))*tanh(x)+1/3*(I-csch(x))*tanh(x)^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3967, 8} \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=-i x+\frac {1}{3} \tanh ^3(x) (-\text {csch}(x)+i)+\frac {1}{3} \tanh (x) (-2 \text {csch}(x)+3 i) \]

[In]

Int[Tanh[x]^2/(I + Csch[x]),x]

[Out]

(-I)*x + ((3*I - 2*Csch[x])*Tanh[x])/3 + ((I - Csch[x])*Tanh[x]^3)/3

Rule 8

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

Rule 3967

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-(e*Cot[c
+ d*x])^(m + 1))*((a + b*Csc[c + d*x])/(d*e*(m + 1))), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)
*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 3973

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int (-i+\text {csch}(x)) \tanh ^4(x) \, dx \\ & = \frac {1}{3} (i-\text {csch}(x)) \tanh ^3(x)-\frac {1}{3} \int (3 i-2 \text {csch}(x)) \tanh ^2(x) \, dx \\ & = \frac {1}{3} (3 i-2 \text {csch}(x)) \tanh (x)+\frac {1}{3} (i-\text {csch}(x)) \tanh ^3(x)+\frac {1}{3} \int -3 i \, dx \\ & = -i x+\frac {1}{3} (3 i-2 \text {csch}(x)) \tanh (x)+\frac {1}{3} (i-\text {csch}(x)) \tanh ^3(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.97 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {-4 \cosh (2 x)+2 i \sinh (x)+(5 i+6 x) \cosh (x) (-i+\sinh (x))}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]

[In]

Integrate[Tanh[x]^2/(I + Csch[x]),x]

[Out]

(-4*Cosh[2*x] + (2*I)*Sinh[x] + (5*I + 6*x)*Cosh[x]*(-I + Sinh[x]))/(6*(Cosh[x/2] - I*Sinh[x/2])*(Cosh[x/2] +
I*Sinh[x/2])^3)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97

method	result	size
risch	\(-i x -\frac {2 \left (-4 i+5 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{3 x}\right )}{3 \left ({\mathrm e}^{x}+i\right ) \left ({\mathrm e}^{x}-i\right )^{3}}\)	\(35\)
default	\(\frac {3 i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {i}{2 \tanh \left (\frac {x}{2}\right )+2 i}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\)	\(67\)

[In]

int(tanh(x)^2/(I+csch(x)),x,method=_RETURNVERBOSE)

[Out]

-I*x-2/3*(-4*I+5*exp(x)+3*exp(3*x))/(exp(x)+I)/(exp(x)-I)^3

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. 50 vs. \(2 (24) = 48\).

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=\frac {-3 i \, x e^{\left (4 \, x\right )} - 6 \, {\left (x + 1\right )} e^{\left (3 \, x\right )} - 2 \, {\left (3 \, x + 5\right )} e^{x} + 3 i \, x + 8 i}{3 \, {\left (e^{\left (4 \, x\right )} - 2 i \, e^{\left (3 \, x\right )} - 2 i \, e^{x} - 1\right )}} \]

[In]

integrate(tanh(x)^2/(I+csch(x)),x, algorithm="fricas")

[Out]

1/3*(-3*I*x*e^(4*x) - 6*(x + 1)*e^(3*x) - 2*(3*x + 5)*e^x + 3*I*x + 8*I)/(e^(4*x) - 2*I*e^(3*x) - 2*I*e^x - 1)

Sympy [F]

\[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(tanh(x)**2/(I+csch(x)),x)

[Out]

Integral(tanh(x)**2/(csch(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=-i \, x - \frac {2 \, {\left (5 \, e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )} + 4 i\right )}}{6 i \, e^{\left (-x\right )} + 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \]

[In]

integrate(tanh(x)^2/(I+csch(x)),x, algorithm="maxima")

[Out]

-I*x - 2*(5*e^(-x) + 3*e^(-3*x) + 4*I)/(6*I*e^(-x) + 6*I*e^(-3*x) + 3*e^(-4*x) - 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {i}{2 \, {\left (i \, e^{x} - 1\right )}} - \frac {15 \, e^{\left (2 \, x\right )} - 24 i \, e^{x} - 13}{6 \, {\left (e^{x} - i\right )}^{3}} \]

[In]

integrate(tanh(x)^2/(I+csch(x)),x, algorithm="giac")

[Out]

-I*x + 1/2*I/(I*e^x - 1) - 1/6*(15*e^(2*x) - 24*I*e^x - 13)/(e^x - I)^3

Mupad [B] (verification not implemented)

Time = 2.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.36 \[ \int \frac {\tanh ^2(x)}{i+\text {csch}(x)} \, dx=-x\,1{}\mathrm {i}+\frac {\frac {5\,{\mathrm {e}}^x}{6}-\frac {1}{2}{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {5}{6\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {1}{2\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {\frac {5}{6}-\frac {5\,{\mathrm {e}}^{2\,x}}{6}+{\mathrm {e}}^x\,1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}} \]

[In]

int(tanh(x)^2/(1/sinh(x) + 1i),x)

[Out]

((5*exp(x))/6 - 1i/2)/(exp(x)*2i - exp(2*x) + 1) - x*1i - 5/(6*(exp(x) - 1i)) + 1/(2*(exp(x) + 1i)) - (exp(x)*
1i - (5*exp(2*x))/6 + 5/6)/(exp(2*x)*3i - exp(3*x) + 3*exp(x) - 1i)