### 3.76 $$\int \frac{\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0276825, 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 = {} $\int \frac{\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.226157, size = 409, normalized size = 13.63 $\frac{b \text{RootSum}\left [\text{\#1}^6 a+3 \text{\#1}^4 a+3 \text{\#1}^2 a+\text{\#1}^6 b-3 \text{\#1}^4 b+3 \text{\#1}^2 b+a-b\& ,\frac{4 \text{\#1}^4 a \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 \text{\#1}^4 a c+2 \text{\#1}^4 a d x-2 \text{\#1}^4 b \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\text{\#1}^4 b c-\text{\#1}^4 b d x+4 a \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 b \log \left (-\text{\#1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{\#1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 a c+2 a d x+b c+b d x}{\text{\#1}^5 a+2 \text{\#1}^3 a+\text{\#1}^5 b-2 \text{\#1}^3 b+\text{\#1} a+\text{\#1} b}\& \right ]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.108, size = 164, normalized size = 5.5 \begin{align*} -4\,{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+{\frac{b}{3\,d \left ( a-b \right ) \left ( a+b \right ) }\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}a-2\,{{\it \_R}}^{3}b+6\,{{\it \_R}}^{2}a-2\,{\it \_R}\,b+a}{{{\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 ) }}+4\,{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/d/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)+1/3/d*b/(a-b)/(a+b)*sum((_R^4*a-2*_R^3*b+6*_R^2*a-2*_R*b+a)/(_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))+4/d/(4*a-4*b)/
(tanh(1/2*d*x+1/2*c)+1)

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

[Out]

1/2*((a*e^(2*c) - b*e^(2*c))*e^(2*d*x) + a + b)*e^(-d*x)/(a^2*d*e^c - b^2*d*e^c) + 1/2*integrate(4*((2*a*b*e^(
5*c) - b^2*e^(5*c))*e^(5*d*x) + (2*a*b*e^c + b^2*e^c)*e^(d*x))/(a^3 - a^2*b - a*b^2 + b^3 + (a^3*e^(6*c) + a^2
*b*e^(6*c) - a*b^2*e^(6*c) - b^3*e^(6*c))*e^(6*d*x) + 3*(a^3*e^(4*c) - a^2*b*e^(4*c) - a*b^2*e^(4*c) + b^3*e^(
4*c))*e^(4*d*x) + 3*(a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*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(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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(sinh(d*x+c)/(a+b*tanh(d*x+c)**3),x)

[Out]

Timed out

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Giac [A]  time = 1.62107, size = 263, normalized size = 8.77 \begin{align*} \frac{\frac{e^{\left (d x + 8 \, c\right )}}{a e^{\left (7 \, c\right )} + b e^{\left (7 \, c\right )}} + \frac{e^{\left (-d x\right )}}{a e^{c} - b e^{c}}}{2 \, d} + \frac{\frac{6 \,{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} d x}{a d - b d} - \frac{{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} \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 \,{\left (a^{2} - b^{2}\right )} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e^(d*x + 8*c)/(a*e^(7*c) + b*e^(7*c)) + e^(-d*x)/(a*e^c - b*e^c))/d + 1/3*(6*(2*a*b*e^c + b^2*e^c)*d*x/(a
*d - b*d) - (2*a*b*e^c + b^2*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^2 - b^2)*d)