3.84 \(\int \frac{1}{(a+b \coth (c+d x))^4} \, dx\)

Optimal. Leaf size=169 \[ \frac{b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \coth (c+d x))}+\frac{a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))^2}+\frac{b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}-\frac{4 a b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^4}+\frac{x \left (6 a^2 b^2+a^4+b^4\right )}{\left (a^2-b^2\right )^4} \]

[Out]

((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2)^4 + b/(3*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^3) + (a*b)/((a^2 - b^2)^2
*d*(a + b*Coth[c + d*x])^2) + (b*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(a + b*Coth[c + d*x])) - (4*a*b*(a^2 + b^2)*L
og[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^4*d)

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Rubi [A]  time = 0.264259, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ \frac{b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \coth (c+d x))}+\frac{a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))^2}+\frac{b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}-\frac{4 a b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^4}+\frac{x \left (6 a^2 b^2+a^4+b^4\right )}{\left (a^2-b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x])^(-4),x]

[Out]

((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2)^4 + b/(3*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^3) + (a*b)/((a^2 - b^2)^2
*d*(a + b*Coth[c + d*x])^2) + (b*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(a + b*Coth[c + d*x])) - (4*a*b*(a^2 + b^2)*L
og[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^4*d)

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+b \coth (c+d x))^4} \, dx &=\frac{b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac{\int \frac{a-b \coth (c+d x)}{(a+b \coth (c+d x))^3} \, dx}{a^2-b^2}\\ &=\frac{b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac{a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac{\int \frac{a^2+b^2-2 a b \coth (c+d x)}{(a+b \coth (c+d x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac{a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac{b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}+\frac{\int \frac{a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac{\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac{b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac{a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac{b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac{\left (4 i a b \left (a^2+b^2\right )\right ) \int \frac{-i b-i a \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^4}\\ &=\frac{\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac{b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac{a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac{b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac{4 a b \left (a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}

Mathematica [A]  time = 6.20822, size = 214, normalized size = 1.27 \[ -\frac{b^4}{3 a^3 d \left (a^2-b^2\right ) (a \tanh (c+d x)+b)^3}+\frac{b^3 \left (2 a^2-b^2\right )}{a^3 d \left (a^2-b^2\right )^2 (a \tanh (c+d x)+b)^2}-\frac{b^2 \left (-3 a^2 b^2+6 a^4+b^4\right )}{a^3 d \left (a^2-b^2\right )^3 (a \tanh (c+d x)+b)}-\frac{4 a b \left (a^2+b^2\right ) \log (a \tanh (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\log (1-\tanh (c+d x))}{2 d (a+b)^4}+\frac{\log (\tanh (c+d x)+1)}{2 d (a-b)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[c + d*x])^(-4),x]

[Out]

-Log[1 - Tanh[c + d*x]]/(2*(a + b)^4*d) + Log[1 + Tanh[c + d*x]]/(2*(a - b)^4*d) - (4*a*b*(a^2 + b^2)*Log[b +
a*Tanh[c + d*x]])/((a^2 - b^2)^4*d) - b^4/(3*a^3*(a^2 - b^2)*d*(b + a*Tanh[c + d*x])^3) + (b^3*(2*a^2 - b^2))/
(a^3*(a^2 - b^2)^2*d*(b + a*Tanh[c + d*x])^2) - (b^2*(6*a^4 - 3*a^2*b^2 + b^4))/(a^3*(a^2 - b^2)^3*d*(b + a*Ta
nh[c + d*x]))

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Maple [A]  time = 0.034, size = 230, normalized size = 1.4 \begin{align*}{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) }{2\,d \left ( a-b \right ) ^{4}}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) }{2\,d \left ( a+b \right ) ^{4}}}+{\frac{b}{3\,d \left ( a-b \right ) \left ( a+b \right ) \left ( a+b{\rm coth} \left (dx+c\right ) \right ) ^{3}}}+{\frac{ab}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( a+b{\rm coth} \left (dx+c\right ) \right ) ^{2}}}+3\,{\frac{{a}^{2}b}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }}+{\frac{{b}^{3}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }}-4\,{\frac{{a}^{3}b\ln \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-4\,{\frac{a{b}^{3}\ln \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*coth(d*x+c))^4,x)

[Out]

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

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Maxima [B]  time = 1.3454, size = 705, normalized size = 4.17 \begin{align*} -\frac{4 \,{\left (a^{3} b + a b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d} - \frac{4 \,{\left (9 \, a^{4} b^{2} + 18 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 4 \, a b^{5} + 2 \, b^{6} - 3 \,{\left (6 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 5 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \,{\left (3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{3 \,{\left (a^{10} + 2 \, a^{9} b - 3 \, a^{8} b^{2} - 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} + 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 2 \, a b^{9} + b^{10} - 3 \,{\left (a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \,{\left (a^{10} - 2 \, a^{9} b - 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} - 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} + 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} - 2 \, a b^{9} + b^{10}\right )} e^{\left (-4 \, d x - 4 \, c\right )} -{\left (a^{10} - 4 \, a^{9} b + 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} - 14 \, a^{6} b^{4} + 14 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 4 \, a b^{9} - b^{10}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac{d x + c}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="maxima")

[Out]

-4*(a^3*b + a*b^3)*log(-(a - b)*e^(-2*d*x - 2*c) + a + b)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d)
- 4/3*(9*a^4*b^2 + 18*a^3*b^3 + 11*a^2*b^4 + 4*a*b^5 + 2*b^6 - 3*(6*a^4*b^2 + 2*a^3*b^3 - 5*a^2*b^4 - 2*a*b^5
- b^6)*e^(-2*d*x - 2*c) + 3*(3*a^4*b^2 - 4*a^3*b^3 + b^6)*e^(-4*d*x - 4*c))/((a^10 + 2*a^9*b - 3*a^8*b^2 - 8*a
^7*b^3 + 2*a^6*b^4 + 12*a^5*b^5 + 2*a^4*b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 2*a*b^9 + b^10 - 3*(a^10 - 5*a^8*b^2 + 1
0*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*e^(-2*d*x - 2*c) + 3*(a^10 - 2*a^9*b - 3*a^8*b^2 + 8*a^7*b^3 + 2*a^
6*b^4 - 12*a^5*b^5 + 2*a^4*b^6 + 8*a^3*b^7 - 3*a^2*b^8 - 2*a*b^9 + b^10)*e^(-4*d*x - 4*c) - (a^10 - 4*a^9*b +
3*a^8*b^2 + 8*a^7*b^3 - 14*a^6*b^4 + 14*a^4*b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 4*a*b^9 - b^10)*e^(-6*d*x - 6*c))*d)
+ (d*x + c)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)

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Fricas [B]  time = 3.4678, size = 8128, normalized size = 48.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*(3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^6
+ 18*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)*si
nh(d*x + c)^5 + 3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*sinh
(d*x + c)^6 - 36*a^5*b^2 + 108*a^4*b^3 - 116*a^3*b^4 + 60*a^2*b^5 - 24*a*b^6 + 8*b^7 - 3*(12*a^5*b^2 + 4*a^4*b
^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6
- b^7)*d*x)*cosh(d*x + c)^4 - 3*(12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 - 15*(a^7 + 7*a^6*b +
21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^2 + 3*(a^7 + 5*a^6*b + 9*
a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*sinh(d*x + c)^4 + 12*(5*(a^7 + 7*a^6*b + 21*
a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^3 - (12*a^5*b^2 + 4*a^4*b^3
- 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 -
b^7)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*
b^6 - b^7)*d*x + 3*(24*a^5*b^2 - 32*a^4*b^3 - 12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 3*(a^7 + 3*a^6*b +
a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*d*x)*cosh(d*x + c)^2 + 3*(24*a^5*b^2 - 32*a^4*b^3 -
12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 15*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^
2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^4 + 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*
a*b^6 + b^7)*d*x - 6*(12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5
*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*(a^6*b - 3*a^5*b^
2 + 4*a^4*b^3 - 4*a^3*b^4 + 3*a^2*b^5 - a*b^6 - (a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6
)*cosh(d*x + c)^6 - 6*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)*sinh(d*x +
c)^5 - (a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*sinh(d*x + c)^6 + 3*(a^6*b + a^5*b^2 -
a^2*b^5 - a*b^6)*cosh(d*x + c)^4 + 3*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6 - 5*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 +
4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 4*(5*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*
b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^3 - 3*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)*cosh(d*x + c))*sinh(d*x + c)^
3 - 3*(a^6*b - a^5*b^2 - a^2*b^5 + a*b^6)*cosh(d*x + c)^2 - 3*(a^6*b - a^5*b^2 - a^2*b^5 + a*b^6 + 5*(a^6*b +
3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^4 - 6*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 6*((a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*
x + c)^5 - 2*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)*cosh(d*x + c)^3 + (a^6*b - a^5*b^2 - a^2*b^5 + a*b^6)*cosh(d*
x + c))*sinh(d*x + c))*log(2*(b*cosh(d*x + c) + a*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))) + 6*(3*(a^7
+ 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^5 - 2*(12*a^5
*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2
*b^5 - 5*a*b^6 - b^7)*d*x)*cosh(d*x + c)^3 + (24*a^5*b^2 - 32*a^4*b^3 - 12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4
*b^7 + 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*d*x)*cosh(d*x + c))*sinh(
d*x + c))/((a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*
b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^6 + 6*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 1
4*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)*sinh(d*x + c)^5 +
(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*
b^9 + 3*a*b^10 + b^11)*d*sinh(d*x + c)^6 - 3*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5
- 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c)^4 + 3*(5*(a^11 + 3*a^10*b -
a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^
11)*d*cosh(d*x + c)^2 - (a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 - 10*a^5*b^6 - 10*a^4
*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d)*sinh(d*x + c)^4 + 3*(a^11 - a^10*b - 5*a^9*b^2 + 5*a^8*b^3 +
10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*d*cosh(d*x + c)^2 +
4*(5*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 -
a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^3 - 3*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6
*b^5 - 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(
5*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2
*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^4 - 6*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5
- 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c)^2 + (a^11 - a^10*b - 5*a^9
*b^2 + 5*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*
d)*sinh(d*x + c)^2 - (a^11 - 3*a^10*b - a^9*b^2 + 11*a^8*b^3 - 6*a^7*b^4 - 14*a^6*b^5 + 14*a^5*b^6 + 6*a^4*b^7
- 11*a^3*b^8 + a^2*b^9 + 3*a*b^10 - b^11)*d + 6*((a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6
*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^5 - 2*(a^11 + a^10*b -
5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 - 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 -
b^11)*d*cosh(d*x + c)^3 + (a^11 - a^10*b - 5*a^9*b^2 + 5*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a
^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*d*cosh(d*x + c))*sinh(d*x + c))

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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(1/(a+b*coth(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.21574, size = 424, normalized size = 2.51 \begin{align*} -\frac{4 \,{\left (a^{3} b + a b^{3}\right )} \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{8} d - 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d - 4 \, a^{2} b^{6} d + b^{8} d} + \frac{d x + c}{a^{4} d - 4 \, a^{3} b d + 6 \, a^{2} b^{2} d - 4 \, a b^{3} d + b^{4} d} - \frac{4 \,{\left (3 \,{\left (3 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} + 2 \, a b^{5} - b^{6}\right )} e^{\left (4 \, d x + 4 \, c\right )} - 3 \,{\left (6 \, a^{4} b^{2} - 14 \, a^{3} b^{3} + 11 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac{9 \, a^{5} b^{2} - 27 \, a^{4} b^{3} + 29 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6} - 2 \, b^{7}}{a + b}\right )}}{3 \,{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )}^{3}{\left (a + b\right )}^{3}{\left (a - b\right )}^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="giac")

[Out]

-4*(a^3*b + a*b^3)*log(abs(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) - a + b))/(a^8*d - 4*a^6*b^2*d + 6*a^4*b^4*d
- 4*a^2*b^6*d + b^8*d) + (d*x + c)/(a^4*d - 4*a^3*b*d + 6*a^2*b^2*d - 4*a*b^3*d + b^4*d) - 4/3*(3*(3*a^4*b^2 -
2*a^3*b^3 - 2*a^2*b^4 + 2*a*b^5 - b^6)*e^(4*d*x + 4*c) - 3*(6*a^4*b^2 - 14*a^3*b^3 + 11*a^2*b^4 - 4*a*b^5 + b
^6)*e^(2*d*x + 2*c) + (9*a^5*b^2 - 27*a^4*b^3 + 29*a^3*b^4 - 15*a^2*b^5 + 6*a*b^6 - 2*b^7)/(a + b))/((a*e^(2*d
*x + 2*c) + b*e^(2*d*x + 2*c) - a + b)^3*(a + b)^3*(a - b)^4*d)