### 3.165 $$\int \coth ^5(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx$$

Optimal. Leaf size=83 $\frac{a \left (a^2+3 a b+3 b^2\right ) \log (\tanh (c+d x))}{d}-\frac{a^2 (a+3 b) \coth ^2(c+d x)}{2 d}-\frac{a^3 \coth ^4(c+d x)}{4 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}$

[Out]

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

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Rubi [A]  time = 0.117336, antiderivative size = 83, 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} $\frac{a \left (a^2+3 a b+3 b^2\right ) \log (\tanh (c+d x))}{d}-\frac{a^2 (a+3 b) \coth ^2(c+d x)}{2 d}-\frac{a^3 \coth ^4(c+d x)}{4 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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 \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{x^5 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{(1-x) x^3} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^3}{-1+x}+\frac{a^3}{x^3}+\frac{a^2 (a+3 b)}{x^2}+\frac{a \left (a^2+3 a b+3 b^2\right )}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac{a^2 (a+3 b) \coth ^2(c+d x)}{2 d}-\frac{a^3 \coth ^4(c+d x)}{4 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}+\frac{a \left (a^2+3 a b+3 b^2\right ) \log (\tanh (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.555253, size = 67, normalized size = 0.81 $\frac{-a^2 (a+3 b) \coth ^2(c+d x)-\frac{1}{2} a^3 \coth ^4(c+d x)+2 (a+b)^3 \log (\sinh (c+d x))-2 b^3 \log (\tanh (c+d x))}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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Maple [A]  time = 0.062, size = 111, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}+3\,{\frac{{a}^{2}b\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,{a}^{2}b \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}+3\,{\frac{a{b}^{2}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{3}\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

1/d*a^3*ln(sinh(d*x+c))-1/2*a^3*coth(d*x+c)^2/d-1/4*a^3*coth(d*x+c)^4/d+3/d*a^2*b*ln(sinh(d*x+c))-3/2/d*a^2*b*
coth(d*x+c)^2+3/d*a*b^2*ln(sinh(d*x+c))+1/d*b^3*ln(cosh(d*x+c))

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Maxima [B]  time = 1.07568, size = 356, normalized size = 4.29 \begin{align*} a^{3}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{4 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 3 \, a^{2} b{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac{b^{3} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac{3 \, a b^{2} \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.36982, size = 5854, normalized size = 70.53 \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)^5*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

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

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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(coth(d*x+c)**5*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.50226, size = 366, normalized size = 4.41 \begin{align*} \frac{b^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right )}{2 \, d} + \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right )}{2 \, d} - \frac{3 \, a^{3}{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 9 \, a^{2} b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 9 \, a b^{2}{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 4 \, a^{3}{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 12 \, a^{2} b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 36 \, a b^{2}{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 4 \, a^{3} - 12 \, a^{2} b + 36 \, a b^{2}}{4 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/2*b^3*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2)/d + 1/2*(a^3 + 3*a^2*b + 3*a*b^2)*log(e^(2*d*x + 2*c) + e^
(-2*d*x - 2*c) - 2)/d - 1/4*(3*a^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 9*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d
*x - 2*c))^2 + 9*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 4*a^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 1
2*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 36*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 4*a^3 - 12*a^2*
b + 36*a*b^2)/(d*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2)^2)