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

Optimal. Leaf size=130 \[ -\frac {b^3}{4 a^3 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^3}+\frac {\log (\sinh (c+d x))}{d (a+b)^3} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 130, normalized size of antiderivative = 1.00, 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 = {4138, 446, 88} \[ -\frac {b^3}{4 a^3 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^3}+\frac {\log (\sinh (c+d x))}{d (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

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 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]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^7}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^3}{(1-x) (b+a x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{(a+b)^3 (-1+x)}-\frac {b^3}{a^2 (a+b) (b+a x)^3}+\frac {b^2 (3 a+2 b)}{a^2 (a+b)^2 (b+a x)^2}-\frac {b \left (3 a^2+3 a b+b^2\right )}{a^2 (a+b)^3 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b^3}{4 a^3 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac {\log (\sinh (c+d x))}{(a+b)^3 d}\\ \end {align*}

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Mathematica [A]  time = 1.06, size = 155, normalized size = 1.19 \[ \frac {\text {sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left (-\frac {b^3 (a+b)^2}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac {2 b^2 (a+b) (3 a+2 b)}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {2 b \left (3 a^2+3 a b+b^2\right ) \log \left (a \sinh ^2(c+d x)+a+b\right )}{a^3}+4 \log (\sinh (c+d x))\right )}{32 d (a+b)^3 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(4*Log[Sinh[c + d*x]] + (2*b*(3*a^2 + 3*a*b + b^2)*Log[a +
b + a*Sinh[c + d*x]^2])/a^3 - (b^3*(a + b)^2)/(a^3*(a + b + a*Sinh[c + d*x]^2)^2) + (2*b^2*(a + b)*(3*a + 2*b)
)/(a^3*(a + b + a*Sinh[c + d*x]^2))))/(32*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [B]  time = 0.85, size = 4132, normalized size = 31.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 16*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d
*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 - 4*(3*a^3*b^2
+ 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 - 4*(3*a^3*b^
2 + 5*a^2*b^3 + 2*a*b^4 - 14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 - 2*(a^5 + 5*a^4*b + 9*
a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6 + 8*(14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x
+ c)^3 - 3*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d
*x + c))*sinh(d*x + c)^5 - 4*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*
a^2*b^3 + 32*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x +
c)^4 - 6*a^3*b^2 - 20*a^2*b^3 - 20*a*b^4 - 6*b^5 + (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*
b^5)*d*x - 15*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh
(d*x + c)^2)*sinh(d*x + c)^4 + 16*(7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 - 5*(3*a^3*b^2
+ 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^3 - (6*a^3*b^2
+ 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*d*x)*cosh(d*
x + c))*sinh(d*x + c)^3 + 2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x - 4*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2
*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2 + 4*(14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a
^2*b^3)*d*x*cosh(d*x + c)^6 - 3*a^3*b^2 - 5*a^2*b^3 - 2*a*b^4 - 15*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 +
5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^4 + 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 +
2*a*b^4)*d*x - 6*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32
*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^8 + 8*(
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(d*x + c)^8
+ 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^6 + 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^
4 + 7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*
cosh(d*x + c)^3 + 3*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^4*b + 3*a
^3*b^2 + a^2*b^3 + 2*(9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(d*x + c)^4 + 2*(9*a^4*b + 33*
a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5 + 35*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^4 + 30*(3*a^4*b + 9
*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d
*x + c)^5 + 10*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^3 + (9*a^4*b + 33*a^3*b^2 + 51*a^2*b^
3 + 32*a*b^4 + 8*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x
+ c)^2 + 4*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^6 + 3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 15
*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^4 + 3*(9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4
+ 8*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^7 + 3*(3*a^4*b +
9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^5 + (9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cos
h(d*x + c)^3 + (3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c
)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*(a
^5*cosh(d*x + c)^8 + 8*a^5*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*sinh(d*x + c)^8 + 4*(a^5 + 2*a^4*b)*cosh(d*x +
c)^6 + 4*(7*a^5*cosh(d*x + c)^2 + a^5 + 2*a^4*b)*sinh(d*x + c)^6 + 8*(7*a^5*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b
)*cosh(d*x + c))*sinh(d*x + c)^5 + a^5 + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^4 + 2*(35*a^5*cosh(d*x
+ c)^4 + 3*a^5 + 8*a^4*b + 8*a^3*b^2 + 30*(a^5 + 2*a^4*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^5*cosh(d*x
+ c)^5 + 10*(a^5 + 2*a^4*b)*cosh(d*x + c)^3 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(a^5 + 2*a^4*b)*cosh(d*x + c)^2 + 4*(7*a^5*cosh(d*x + c)^6 + a^5 + 2*a^4*b + 15*(a^5 + 2*a^4*b)*cosh(d*x + c
)^4 + 3*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^5*cosh(d*x + c)^7 + 3*(a^5 + 2*a
^4*b)*cosh(d*x + c)^5 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^3 + (a^5 + 2*a^4*b)*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^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*
cosh(d*x + c)^7 - 3*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x
)*cosh(d*x + c)^5 - 2*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3
+ 32*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 - (3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 +
7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)
^8 + 8*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^
5*b^3)*d*sinh(d*x + c)^8 + 4*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^8
+ 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*s
inh(d*x + c)^6 + 2*(3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8
*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^3 + 3*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4
*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a
^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*
b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cos
h(d*x + c)^2 + 8*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 10*(a^8 + 5*a^7*b + 9*a^6*b^2 +
7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^
5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^8 +
5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^
3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*
x + c)^2 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d + 8*((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^
7 + 3*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 +
51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^3 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*
d*cosh(d*x + c))*sinh(d*x + c))

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giac [B]  time = 0.72, size = 475, normalized size = 3.65 \[ \frac {\frac {2 \, {\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}} + \frac {4 \, e^{\left (2 \, c\right )} \log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{3} e^{\left (2 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}} - \frac {4 \, d x}{a^{3}} - \frac {9 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 36 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 84 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 44 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 54 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 150 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 146 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 36 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 84 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(3*a^2*b + 3*a*b^2 + b^3)*log(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)/(a^6 +
3*a^5*b + 3*a^4*b^2 + a^3*b^3) + 4*e^(2*c)*log(abs(-e^(2*d*x + 2*c) + 1))/(a^3*e^(2*c) + 3*a^2*b*e^(2*c) + 3*
a*b^2*e^(2*c) + b^3*e^(2*c)) - 4*d*x/a^3 - (9*a^3*b*e^(8*d*x + 8*c) + 9*a^2*b^2*e^(8*d*x + 8*c) + 3*a*b^3*e^(8
*d*x + 8*c) + 36*a^3*b*e^(6*d*x + 6*c) + 84*a^2*b^2*e^(6*d*x + 6*c) + 44*a*b^3*e^(6*d*x + 6*c) + 8*b^4*e^(6*d*
x + 6*c) + 54*a^3*b*e^(4*d*x + 4*c) + 150*a^2*b^2*e^(4*d*x + 4*c) + 146*a*b^3*e^(4*d*x + 4*c) + 32*b^4*e^(4*d*
x + 4*c) + 36*a^3*b*e^(2*d*x + 2*c) + 84*a^2*b^2*e^(2*d*x + 2*c) + 44*a*b^3*e^(2*d*x + 2*c) + 8*b^4*e^(2*d*x +
2*c) + 9*a^3*b + 9*a^2*b^2 + 3*a*b^3)/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*
x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2))/d

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maple [B]  time = 0.47, size = 1046, normalized size = 8.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)-1/d/a^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+
b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^6-8/d*b
^3/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*
b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^6-2/d*b^4/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d
*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/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+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^4+
4/d*b^3/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*
c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^4+4/d*b^4/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(
1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a^2*tanh(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+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c
)^2-8/d*b^3/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+
1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^2-2/d*b^4/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*t
anh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a^2*tanh(1/2*d*x+1/2*c)^2+3/2/d*b/a/(a+b)^3*ln(tanh(1/
2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)+3/2/d*b^2/a^
2/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2
*b+a+b)+1/2/d*b^3/a^3/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*t
anh(1/2*d*x+1/2*c)^2*b+a+b)+1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c))

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maxima [B]  time = 0.72, size = 419, normalized size = 3.22 \[ \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} d} + \frac {2 \, {\left ({\left (3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 3 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{7} + 14 \, a^{6} b + 27 \, a^{5} b^{2} + 24 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {d x + c}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a^2*b + 3*a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^6 + 3*a^5*b + 3*a
^4*b^2 + a^3*b^3)*d) + 2*((3*a^2*b^2 + 2*a*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^2*b^2 + 7*a*b^3 + 3*b^4)*e^(-4*d*x -
4*c) + (3*a^2*b^2 + 2*a*b^3)*e^(-6*d*x - 6*c))/((a^7 + 2*a^6*b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a
^4*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*e^(-4*d*x - 4*c) + 4*(a^
7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-6*d*x - 6*c) + (a^7 + 2*a^6*b + a^5*b^2)*e^(-8*d*x - 8*c))*d) + log(e
^(-d*x - c) + 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + log(e^(-d*x - c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
d) + (d*x + c)/(a^3*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,\mathrm {coth}\left (c+d\,x\right )}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(c + d*x)^6*coth(c + d*x))/(b + a*cosh(c + d*x)^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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