3.79 $$\int \frac{\text{csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$$

Optimal. Leaf size=32 $-i \text{Unintegrable}\left (\frac{i \text{csch}^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right )$

[Out]

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

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Rubi [A]  time = 0.0471116, 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 = {} $\int \frac{\text{csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.361888, size = 201, normalized size = 6.28 $-\frac{16 b \text{RootSum}\left [\text{\#1}^6 a+3 \text{\#1}^4 a+3 \text{\#1}^2 a+\text{\#1}^6 b-3 \text{\#1}^4 b+3 \text{\#1}^2 b+a-b\& ,\frac{2 \text{\#1} \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{\#1} c+\text{\#1} d x}{\text{\#1}^4 a+2 \text{\#1}^2 a+\text{\#1}^4 b-2 \text{\#1}^2 b+a+b}\& \right ]+3 \left (\text{csch}^2\left (\frac{1}{2} (c+d x)\right )+\text{sech}^2\left (\frac{1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 a d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(16*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*#1 + d*x*#1 + 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)/(a + b + 2*a*#1^2 -
2*b*#1^2 + a*#1^4 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(2
4*a*d)

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Maple [A]  time = 0.125, size = 144, normalized size = 4.5 \begin{align*}{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{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}}^{4}-2\,{{\it \_R}}^{2}+1}{{{\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)^3/(a+b*tanh(d*x+c)^3),x)

[Out]

1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/8/d/a/tanh(1/2*d*x+1/2*c)^2-1/2/d/a*ln(tanh(1/2*d*x+1/2*c))-1/3/d/a*b*sum((_R^
4-2*_R^2+1)/(_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*} -8 \, b \int \frac{e^{\left (3 \, d x + 3 \, 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} - \frac{e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} - \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*b*integrate(e^(3*d*x + 3*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) - (e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x +
4*c) - 2*a*d*e^(2*d*x + 2*c) + a*d) + 1/2*log((e^(d*x + c) + 1)*e^(-c))/(a*d) - 1/2*log((e^(d*x + c) - 1)*e^(
-c))/(a*d)

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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)^3/(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}^{3}{\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)**3/(a+b*tanh(d*x+c)**3),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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