### 3.77 $$\int \frac{\text{csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$$

Optimal. Leaf size=30 $i \text{Unintegrable}\left (-\frac{i \text{csch}(c+d x)}{a+b \tanh ^3(c+d x)},x\right )$

[Out]

I*Unintegrable[((-I)*Csch[c + d*x])/(a + b*Tanh[c + d*x]^3), x]

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Rubi [A]  time = 0.0408265, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\text{csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

I*Defer[Int][((-I)*Csch[c + d*x])/(a + b*Tanh[c + d*x]^3), x]

Rubi steps

\begin{align*} \int \frac{\text{csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=i \int -\frac{i \text{csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.158439, size = 319, normalized size = 10.63 $\frac{6 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-b \text{RootSum}\left [\text{\#1}^6 a+3 \text{\#1}^4 a+3 \text{\#1}^2 a+\text{\#1}^6 b-3 \text{\#1}^4 b+3 \text{\#1}^2 b+a-b\& ,\frac{2 \text{\#1}^4 \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-4 \text{\#1}^2 \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{\#1}^4 c-2 \text{\#1}^2 c+\text{\#1}^4 d x-2 \text{\#1}^2 d x+2 \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+c+d x}{\text{\#1}^5 a+2 \text{\#1}^3 a+\text{\#1}^5 b-2 \text{\#1}^3 b+\text{\#1} a+\text{\#1} b}\& \right ]}{6 a d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & ,
(c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#
1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]
*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*
x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ])/(6*a*d)

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Maple [A]  time = 0.105, size = 98, normalized size = 3.3 \begin{align*}{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{4\,b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)

[Out]

1/d/a*ln(tanh(1/2*d*x+1/2*c))-4/3/d/a*b*sum(_R^2/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R
=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a d} - 2 \, \int \frac{b e^{\left (5 \, d x + 5 \, c\right )} - 2 \, b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}}{a^{2} - a b +{\left (a^{2} e^{\left (6 \, c\right )} + a b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \,{\left (a^{2} e^{\left (4 \, c\right )} - a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \,{\left (a^{2} e^{\left (2 \, c\right )} + a b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(5*d*x + 5*c) -
2*b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^2 - a*b + (a^2*e^(6*c) + a*b*e^(6*c))*e^(6*d*x) + 3*(a^2*e^(4*c) - a*b
*e^(4*c))*e^(4*d*x) + 3*(a^2*e^(2*c) + a*b*e^(2*c))*e^(2*d*x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a+b*tanh(d*x+c)**3),x)

[Out]

Integral(csch(c + d*x)/(a + b*tanh(c + d*x)**3), x)

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Giac [A]  time = 1.45323, size = 207, normalized size = 6.9 \begin{align*} -\frac{\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a}}{d} - \frac{\frac{6 \, b d x e^{c}}{a d - b d} - \frac{b e^{c} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a d - b d}}{3 \, a d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(e^(d*x + c) + 1)/a - log(abs(e^(d*x + c) - 1))/a)/d - 1/3*(6*b*d*x*e^c/(a*d - b*d) - b*e^c*log(abs(a*e^(
6*d*x + 6*c) + b*e^(6*d*x + 6*c) + 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*
d*x + 2*c) + a - b))/(a*d - b*d))/(a*d)