3.131 \(\int \coth ^3(c+d x) (a+b \text{sech}^2(c+d x))^3 \, dx\)

Optimal. Leaf size=81 \[ \frac{b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^2(c+d x)}{2 d}+\frac{(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b^3 \text{sech}^2(c+d x)}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.116445, antiderivative size = 81, 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 = {4138, 446, 88} \[ \frac{b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^2(c+d x)}{2 d}+\frac{(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b^3 \text{sech}^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4138

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

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

Mathematica [A]  time = 1.22554, size = 110, normalized size = 1.36 \[ -\frac{4 \cosh ^6(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \left (-2 b^2 (3 a+2 b) \log (\cosh (c+d x))+(a+b)^3 \text{csch}^2(c+d x)-2 (a-2 b) (a+b)^2 \log (\sinh (c+d x))+b^3 \text{sech}^2(c+d x)\right )}{d (a \cosh (2 c+2 d x)+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*((a + b)^3*Csch[c + d*x]^2 - 2*b^2*(3*a + 2*b)*Log[Cosh[c + d*x]
] - 2*(a - 2*b)*(a + b)^2*Log[Sinh[c + d*x]] + b^3*Sech[c + d*x]^2))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 145, normalized size = 1.8 \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{3\,{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a{b}^{2}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{a{b}^{2}\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{3}}{2\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{{b}^{3}\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Maxima [B]  time = 1.78659, size = 424, normalized size = 5.23 \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{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 )} - 2 \, b^{3}{\left (\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{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac{2 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, a b^{2}{\left (\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{\log \left (e^{\left (-2 \, d x - 2 \, 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{6 \, a^{2} b}{d{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [B]  time = 2.72197, size = 4127, normalized size = 50.95 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-(a^3*d*x*cosh(d*x + c)^8 + 8*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*d*x*sinh(d*x + c)^8 + 2*(a^3 + 3*a^2
*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*sinh(
d*x + c)^6 + 4*(14*a^3*d*x*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^
5 + a^3*d*x - 2*(a^3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 - a^3*d*
x + 2*a^3 + 6*a^2*b + 6*a*b^2 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a
^3*d*x*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^3 - (a^3*d*x - 2*a^3 - 6*a^2*b - 6*
a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^2 + 2*(14*a^3*d*x*co
sh(d*x + c)^6 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3 - 6*(a^
3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a*b^2 + 2*b^3)*cosh(d*x + c)^8 + 56*
(3*a*b^2 + 2*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(
3*a*b^2 + 2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a*b^2 + 2*b^3)*sinh(d*x + c)^8 - 2*(3*a*b^2 + 2*b^3)*cosh(
d*x + c)^4 + 2*(35*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 - 3*a*b^2 - 2*b^3)*sinh(d*x + c)^4 + 8*(7*(3*a*b^2 + 2*b^
3)*cosh(d*x + c)^5 - (3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a*b^2 + 2*b^3 + 4*(7*(3*a*b^2 + 2*b^
3)*cosh(d*x + c)^6 - 3*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a*b^2 + 2*b^3)*cosh(d*x + c)
^7 - (3*a*b^2 + 2*b^3)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) -
((a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^8 + 56*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(a^
3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^3 - 3*a*b^2 - 2*b^3)*sinh(d*x + c)^8 - 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^4 + 2*(35*(a^3 - 3*a*b^2
- 2*b^3)*cosh(d*x + c)^4 - a^3 + 3*a*b^2 + 2*b^3)*sinh(d*x + c)^4 + 8*(7*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)
^5 - (a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 3*a*b^2 - 2*b^3 + 4*(7*(a^3 - 3*a*b^2 - 2*
b^3)*cosh(d*x + c)^6 - 3*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 - 3*a*b^2 - 2*b^3)
*cosh(d*x + c)^7 - (a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c)
- sinh(d*x + c))) + 4*(2*a^3*d*x*cosh(d*x + c)^7 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^5 - 2*(a^
3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x
+ c))/(d*cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*
cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 2*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - d)*sinh(d*
x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d*x + c)^6 - 3*d*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + d)

________________________________________________________________________________________

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)**3*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.40154, size = 394, normalized size = 4.86 \begin{align*} -\frac{4 \, a^{3} d x - 4 \,{\left (3 \, a b^{2} e^{\left (2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 4 \,{\left (a^{3} e^{\left (2 \, c\right )} - 3 \, a b^{2} e^{\left (2 \, c\right )} - 2 \, b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac{3 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3}}{{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{2}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a^3*d*x - 4*(3*a*b^2*e^(2*c) + 2*b^3*e^(2*c))*e^(-2*c)*log(e^(2*d*x + 2*c) + 1) - 4*(a^3*e^(2*c) - 3*a
*b^2*e^(2*c) - 2*b^3*e^(2*c))*e^(-2*c)*log(abs(e^(2*d*x + 2*c) - 1)) + (3*a^3*e^(8*d*x + 8*c) + 8*a^3*e^(6*d*x
+ 6*c) + 24*a^2*b*e^(6*d*x + 6*c) + 24*a*b^2*e^(6*d*x + 6*c) + 16*b^3*e^(6*d*x + 6*c) + 10*a^3*e^(4*d*x + 4*c
) + 48*a^2*b*e^(4*d*x + 4*c) + 48*a*b^2*e^(4*d*x + 4*c) + 8*a^3*e^(2*d*x + 2*c) + 24*a^2*b*e^(2*d*x + 2*c) + 2
4*a*b^2*e^(2*d*x + 2*c) + 16*b^3*e^(2*d*x + 2*c) + 3*a^3)/(e^(4*d*x + 4*c) - 1)^2)/d