### 3.197 $$\int \frac{\coth (c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=138 $-\frac{b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^3}+\frac{b (2 a+b)}{2 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log (\tanh (c+d x))}{a^3 d}+\frac{b}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}$

[Out]

Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[Tanh[c + d*x]]/(a^3*d) - (b*(3*a^2 + 3*a*b + b^2)*Log[a + b*Tanh[c + d*
x]^2])/(2*a^3*(a + b)^3*d) + b/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(2*a + b))/(2*a^2*(a + b)^2*d*(a
+ b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.204705, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {3670, 446, 72} $-\frac{b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^3}+\frac{b (2 a+b)}{2 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log (\tanh (c+d x))}{a^3 d}+\frac{b}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[Tanh[c + d*x]]/(a^3*d) - (b*(3*a^2 + 3*a*b + b^2)*Log[a + b*Tanh[c + d*
x]^2])/(2*a^3*(a + b)^3*d) + b/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(2*a + b))/(2*a^2*(a + b)^2*d*(a
+ b*Tanh[c + d*x]^2))

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) x (a+b x)^3} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b)^3 (-1+x)}+\frac{1}{a^3 x}-\frac{b^2}{a (a+b) (a+b x)^3}-\frac{b^2 (2 a+b)}{a^2 (a+b)^2 (a+b x)^2}-\frac{b^2 \left (3 a^2+3 a b+b^2\right )}{a^3 (a+b)^3 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\log (\cosh (c+d x))}{(a+b)^3 d}+\frac{\log (\tanh (c+d x))}{a^3 d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac{b}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b (2 a+b)}{2 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.56875, size = 117, normalized size = 0.85 $\frac{\frac{\frac{b \left (\frac{a (a+b) \left (2 b (2 a+b) \tanh ^2(c+d x)+a (5 a+3 b)\right )}{\left (a+b \tanh ^2(c+d x)\right )^2}-2 \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )\right )}{(a+b)^3}+4 \log (\tanh (c+d x))}{a^3}+\frac{4 \log (\cosh (c+d x))}{(a+b)^3}}{4 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

((4*Log[Cosh[c + d*x]])/(a + b)^3 + (4*Log[Tanh[c + d*x]] + (b*(-2*(3*a^2 + 3*a*b + b^2)*Log[a + b*Tanh[c + d*
x]^2] + (a*(a + b)*(a*(5*a + 3*b) + 2*b*(2*a + b)*Tanh[c + d*x]^2))/(a + b*Tanh[c + d*x]^2)^2))/(a + b)^3)/a^3
)/(4*d)

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Maple [B]  time = 0.117, size = 952, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x)

[Out]

-1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)+1)-6/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*ta
nh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^6-10/d*b^3/(a+b)^3/a/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/
2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^6-4/d*b^4/(a+b)^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*t
anh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^6-12/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2
*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^4-40/d*b^3/(a+b)^3/a/(tan
h(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^4-40/d*b^4/(
a+b)^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*
c)^4-12/d*b^5/(a+b)^3/a^3/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*ta
nh(1/2*d*x+1/2*c)^4-6/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2
*b+a)^2*tanh(1/2*d*x+1/2*c)^2-10/d*b^3/(a+b)^3/a/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2
*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^2-4/d*b^4/(a+b)^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^
2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^2-3/2/d*b/(a+b)^3/a*ln(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1
/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)-3/2/d*b^2/(a+b)^3/a^2*ln(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x
+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)-1/2/d*b^3/(a+b)^3/a^3*ln(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c
)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)+1/d/a^3*ln(tanh(1/2*d*x+1/2*c))-1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)-1)

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Maxima [B]  time = 1.30807, size = 672, normalized size = 4.87 \begin{align*} -\frac{{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (2 \,{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \,{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} d} + \frac{d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac{2 \,{\left ({\left (3 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \,{\left (3 \, a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (3 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + 4 \,{\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \,{\left (3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \,{\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} +{\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/2*(3*a^2*b + 3*a*b^2 + b^3)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^6 + 3*a^
5*b + 3*a^4*b^2 + a^3*b^3)*d) + (d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 2*((3*a^2*b^2 + 4*a*b^3 + b^4)
*e^(-2*d*x - 2*c) + 2*(3*a^2*b^2 - a*b^3 - b^4)*e^(-4*d*x - 4*c) + (3*a^2*b^2 + 4*a*b^3 + b^4)*e^(-6*d*x - 6*c
))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3
- 3*a^3*b^4 - a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*
e^(-4*d*x - 4*c) + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-6*d*x - 6*c) + (a^7 + 5
*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c))*d) + log(e^(-d*x - c) + 1)/(a^3*d) +
log(e^(-d*x - c) - 1)/(a^3*d)

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Fricas [B]  time = 5.28184, size = 10714, normalized size = 77.64 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/2*(2*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^8 + 16*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)*sinh(d*
x + c)^7 + 2*(a^5 + 2*a^4*b + a^3*b^2)*d*x*sinh(d*x + c)^8 - 4*(3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b
^2)*d*x)*cosh(d*x + c)^6 - 4*(3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 14*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^2
- 2*(a^5 - a^3*b^2)*d*x)*sinh(d*x + c)^6 + 8*(14*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^3 - 3*(3*a^3*b^2
+ 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(6*a^3*b^2 - 2*a^2*b^3 - 2*a*
b^4 - (3*a^5 - 2*a^4*b + 3*a^3*b^2)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^4
- 6*a^3*b^2 + 2*a^2*b^3 + 2*a*b^4 + (3*a^5 - 2*a^4*b + 3*a^3*b^2)*d*x - 15*(3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2
*(a^5 - a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(7*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^5 -
5*(3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*cosh(d*x + c)^3 - (6*a^3*b^2 - 2*a^2*b^3 - 2*a*b^4
- (3*a^5 - 2*a^4*b + 3*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^5 + 2*a^4*b + a^3*b^2)*d*x - 4*(3*a
^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*cosh(d*x + c)^2 + 4*(14*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh
(d*x + c)^6 - 3*a^3*b^2 - 4*a^2*b^3 - a*b^4 - 15*(3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*cosh(
d*x + c)^4 + 2*(a^5 - a^3*b^2)*d*x - 6*(6*a^3*b^2 - 2*a^2*b^3 - 2*a*b^4 - (3*a^5 - 2*a^4*b + 3*a^3*b^2)*d*x)*c
osh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^8 + 8*(3*a
^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^4*b + 9*a^3*b^2 + 10*a^2*b
^3 + 5*a*b^4 + b^5)*sinh(d*x + c)^8 + 4*(3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^6 + 4*
(3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 7*(3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d
*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^3 + 3*(3*a^
4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3
+ 5*a*b^4 + b^5 + 2*(9*a^4*b + 3*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c)^4 + 2*(9*a^4*b + 3*a^3*
b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5 + 35*(3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^4 + 30
*(3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(3*a^4*b + 9*a^3*b^
2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^5 + 10*(3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*
x + c)^3 + (9*a^4*b + 3*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4*b + 3
*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^2 + 4*(7*(3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5
)*cosh(d*x + c)^6 + 3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 15*(3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*
a*b^4 - b^5)*cosh(d*x + c)^4 + 3*(9*a^4*b + 3*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((3*a^4*b + 9*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^7 + 3*(3*a^4*b + 3*a^3*b^2 - 2*a
^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^5 + (9*a^4*b + 3*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c)^3
+ (3*a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c))*log(2*((a + b)*cosh(d*x + c)
^2 + (a + b)*sinh(d*x + c)^2 + a - b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2
*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^8 + 8*(a^5 + 5*a^4*b + 10*a^3*b^2 +
10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4
+ b^5)*sinh(d*x + c)^8 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^6 + 4*(a^5 +
3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)
*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x +
c)^3 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + a^5 + 5*a^4*
b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*co
sh(d*x + c)^4 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5 + 35*(a^5 + 5*a^4*b + 10*a^3*b^2
+ 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^4 + 30*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cos
h(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^5
+ 10*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^3 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 +
6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*
b^4 - b^5)*cosh(d*x + c)^2 + 4*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^6 +
a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 15*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 -
b^5)*cosh(d*x + c)^4 + 3*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(d*x + c)^7 + 3*(a^5 + 3*a^4*b + 2*
a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c)^5 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*
b^5)*cosh(d*x + c)^3 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c))*l
og(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(2*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^7 - 3*(
3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*cosh(d*x + c)^5 - 2*(6*a^3*b^2 - 2*a^2*b^3 - 2*a*b^4 -
(3*a^5 - 2*a^4*b + 3*a^3*b^2)*d*x)*cosh(d*x + c)^3 - (3*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*(a^5 - a^3*b^2)*d*x)*c
osh(d*x + c))*sinh(d*x + c))/((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^
8 + 8*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^8 +
5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2
*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4
+ a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^
6 + 2*(3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^8 + 5*a^7*
b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^
3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5
*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cos
h(d*x + c)^2 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^8 +
3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 +
10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 -
a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c
))*sinh(d*x + c)^3 + 4*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 +
15*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^8 + 7*a^7*b + 6*a^
6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*
b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d + 8*((a^
8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^8 + 3*a^7*b + 2*a^6*b^2
- 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 +
3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*
sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral(coth(c + d*x)/(a + b*tanh(c + d*x)**2)**3, x)

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Giac [B]  time = 1.28943, size = 417, normalized size = 3.02 \begin{align*} -\frac{{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{2 \,{\left (a^{6} d + 3 \, a^{5} b d + 3 \, a^{4} b^{2} d + a^{3} b^{3} d\right )}} - \frac{d x + c}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + \frac{\log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{3} d} + \frac{2 \,{\left ({\left (3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} +{\left (3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac{2 \,{\left (3 \, a^{3} b^{2} - a^{2} b^{3} - a b^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{a + b}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}{\left (a + b\right )}^{2} a^{3} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

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