### 3.188 $$\int \frac{\coth ^3(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=124 $-\frac{b^2}{2 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^2}+\frac{(a-2 b) \log (\tanh (c+d x))}{a^3 d}-\frac{\coth ^2(c+d x)}{2 a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}$

[Out]

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

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Rubi [A]  time = 0.188638, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {3670, 446, 88} $-\frac{b^2}{2 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^2}+\frac{(a-2 b) \log (\tanh (c+d x))}{a^3 d}-\frac{\coth ^2(c+d x)}{2 a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Coth[c + d*x]^2/(2*a^2*d) + Log[Cosh[c + d*x]]/((a + b)^2*d) + ((a - 2*b)*Log[Tanh[c + d*x]])/(a^3*d) + (b^2*
(3*a + 2*b)*Log[a + b*Tanh[c + d*x]^2])/(2*a^3*(a + b)^2*d) - b^2/(2*a^2*(a + b)*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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.799968, size = 93, normalized size = 0.75 $\frac{\frac{b^3}{a^3 (a+b) \left (a \coth ^2(c+d x)+b\right )}+\frac{b^2 (3 a+2 b) \log \left (a \coth ^2(c+d x)+b\right )}{a^3 (a+b)^2}-\frac{\coth ^2(c+d x)}{a^2}+\frac{2 \log (\sinh (c+d x))}{(a+b)^2}}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(Coth[c + d*x]^2/a^2) + b^3/(a^3*(a + b)*(b + a*Coth[c + d*x]^2)) + (b^2*(3*a + 2*b)*Log[b + a*Coth[c + d*x]
^2])/(a^3*(a + b)^2) + (2*Log[Sinh[c + d*x]])/(a + b)^2)/(2*d)

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Maple [B]  time = 0.113, size = 383, normalized size = 3.1 \begin{align*} -{\frac{1}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+2\,{\frac{{b}^{3} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d \left ( a+b \right ) ^{2}{a}^{2} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }}+2\,{\frac{{b}^{4} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{3} \left ( a+b \right ) ^{2} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }}+{\frac{3\,{b}^{2}}{2\,d \left ( a+b \right ) ^{2}{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }+{\frac{{b}^{3}}{d{a}^{3} \left ( a+b \right ) ^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }-{\frac{1}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{1}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{\ln \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) b}{d{a}^{3}}}-{\frac{1}{d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d*tanh(1/2*d*x+1/2*c)^2/a^2-1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)+1)+2/d*b^3/(a+b)^2/a^2*tanh(1/2*d*x+1/2*c)
^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/d*b^4/a^3/(a+b)^2*tanh(1/
2*d*x+1/2*c)^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)+3/2/d*b^2/(a+b)
^2/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/d*b^3/a^3/(a+b)^2*l
n(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)-1/8/d/a^2/tanh(1/2*d*x+1/2*c)
^2+1/d/a^2*ln(tanh(1/2*d*x+1/2*c))-2/d/a^3*ln(tanh(1/2*d*x+1/2*c))*b-1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)-1)

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Maxima [B]  time = 1.16502, size = 543, normalized size = 4.38 \begin{align*} \frac{{\left (3 \, a b^{2} + 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^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d} + \frac{d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac{2 \,{\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \,{\left (a^{3} + a^{2} b - a b^{2} - 2 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} - 4 \,{\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \,{\left (a^{5} - a^{4} b - 5 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \,{\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} +{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac{{\left (a - 2 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac{{\left (a - 2 \, b\right )} \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)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 4.11124, size = 7919, normalized size = 63.86 \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)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/2*(2*(a^4 + a^3*b)*d*x*cosh(d*x + c)^8 + 16*(a^4 + a^3*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*(a^4 + a^3*
b)*d*x*sinh(d*x + c)^8 - 4*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 4*(2*a^3*b*d*
x - 14*(a^4 + a^3*b)*d*x*cosh(d*x + c)^2 - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*sinh(d*x + c)^6 + 8*(14*(a^4 +
a^3*b)*d*x*cosh(d*x + c)^3 - 3*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x +
c)^5 + 4*(2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^4 + a^3*b)*d
*x*cosh(d*x + c)^4 + 2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x - 15*(2*a^3*b*d*x - a^4 - 3*a
^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(7*(a^4 + a^3*b)*d*x*cosh(d*x + c)^5 - 5*(2*
a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a
^4 - 3*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^4 + a^3*b)*d*x - 4*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a
^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2 + 4*(14*(a^4 + a^3*b)*d*x*cosh(d*x + c)^6 - 2*a^3*b*d*x - 15*(2*a^3*b*d*x -
a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 + a^4 + 3*a^3*b + 3*a^2*b^2 + 2*a*b^3 + 6*(2*a^4 + 2*a^3*
b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^2*b^2 + 5*a*b^3 + 2*b^
4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b^2 + 5*a*b^3 + 2*
b^4)*sinh(d*x + c)^8 - 4*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^6 - 4*(3*a*b^3 + 2*b^4 - 7*(3*a^2*b^2 + 5*a*b^3 + 2*b
^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^3 - 3*(3*a*b^3 + 2*b^4
)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 5*a*b^
3 + 2*b^4)*cosh(d*x + c)^4 - 3*a^2*b^2 + 7*a*b^3 + 6*b^4 - 30*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + 3*a^2*b^2 + 5*a*b^3 + 2*b^4 + 8*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^5 - 10*(3*a*b^3 + 2*b^4)*c
osh(d*x + c)^3 - (3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a*b^3 + 2*b^4)*cosh(d*x +
c)^2 + 4*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^6 - 15*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^4 - 3*a*b^3 -
2*b^4 - 3*(3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^2*b^2 + 5*a*b^3 + 2*b^4)*co
sh(d*x + c)^7 - 3*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^5 - (3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^3 - (3*a*b^3
+ 2*b^4)*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)/(cos
h(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*((a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b
^4)*cosh(d*x + c)^8 + 8*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + a^3
*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*sinh(d*x + c)^8 - 4*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^6 - 4*(a^3*b - 3
*a*b^3 - 2*b^4 - 7*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 +
a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^3 - 3*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c
)^5 - 2*(a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b
^3 - 2*b^4)*cosh(d*x + c)^4 - a^4 + 3*a^3*b + 3*a^2*b^2 - 7*a*b^3 - 6*b^4 - 30*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(
d*x + c)^2)*sinh(d*x + c)^4 + a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4 + 8*(7*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*
b^3 - 2*b^4)*cosh(d*x + c)^5 - 10*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^3 - (a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a
*b^3 + 6*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^2 + 4*(7*(a^4 + a^3*b
- 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^6 - 15*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^4 - a^3*b + 3*a*b
^3 + 2*b^4 - 3*(a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + a^3*
b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^7 - 3*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^5 - (a^4 - 3*a^3*
b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^3 - (a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c))*log
(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(2*(a^4 + a^3*b)*d*x*cosh(d*x + c)^7 - 3*(2*a^3*b*d*x -
a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + 2*(2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3
*b)*d*x)*cosh(d*x + c)^3 - (2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/(
(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^8 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^8 - 4*(a^5*b + 2*a^4*b^2 + a^3*b^3
)*d*cosh(d*x + c)^6 + 4*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 - (a^5*b + 2*a^4*b^2 + a^3*
b^3)*d)*sinh(d*x + c)^6 - 2*(a^6 - a^5*b - 5*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a^6 + 3*a^5*b + 3*
a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 - 3*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3
5*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^4 - 30*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2
- (a^6 - a^5*b - 5*a^4*b^2 - 3*a^3*b^3)*d)*sinh(d*x + c)^4 - 4*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^
2 + 8*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^5 - 10*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x
+ c)^3 - (a^6 - a^5*b - 5*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6 + 3*a^5*b + 3*a^4
*b^2 + a^3*b^3)*d*cosh(d*x + c)^6 - 15*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^4 - 3*(a^6 - a^5*b - 5*a^
4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^2 - (a^5*b + 2*a^4*b^2 + a^3*b^3)*d)*sinh(d*x + c)^2 + (a^6 + 3*a^5*b + 3*a
^4*b^2 + a^3*b^3)*d + 8*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^7 - 3*(a^5*b + 2*a^4*b^2 + a^3*
b^3)*d*cosh(d*x + c)^5 - (a^6 - a^5*b - 5*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^3 - (a^5*b + 2*a^4*b^2 + a^3*b^
3)*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 ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.28295, size = 450, normalized size = 3.63 \begin{align*} \frac{{\left (3 \, a b^{2} + 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^{5} d + 2 \, a^{4} b d + a^{3} b^{2} d\right )}} - \frac{d x + c}{a^{2} d + 2 \, a b d + b^{2} d} + \frac{{\left (a - 2 \, b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{3} d} - \frac{2 \,{\left (\frac{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{a + b} + \frac{2 \,{\left (a^{4} + a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{a + b} + \frac{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (2 \, d x + 2 \, 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 )}{\left (a + b\right )} a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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