\(\int (a+b \coth (c+d x))^5 \, dx\) [77]

Optimal result

Integrand size = 12, antiderivative size = 142 \[ \int (a+b \coth (c+d x))^5 \, dx=a \left (a^4+10 a^2 b^2+5 b^4\right ) x-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\sinh (c+d x))}{d} \]

[Out]

a*(a^4+10*a^2*b^2+5*b^4)*x-4*a*b^2*(a^2+b^2)*coth(d*x+c)/d-1/2*b*(3*a^2+b^2)*(a+b*coth(d*x+c))^2/d-2/3*a*b*(a+
b*coth(d*x+c))^3/d-1/4*b*(a+b*coth(d*x+c))^4/d+b*(5*a^4+10*a^2*b^2+b^4)*ln(sinh(d*x+c))/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3563, 3609, 3606, 3556} \[ \int (a+b \coth (c+d x))^5 \, dx=-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\sinh (c+d x))}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d} \]

[In]

Int[(a + b*Coth[c + d*x])^5,x]

[Out]

a*(a^4 + 10*a^2*b^2 + 5*b^4)*x - (4*a*b^2*(a^2 + b^2)*Coth[c + d*x])/d - (b*(3*a^2 + b^2)*(a + b*Coth[c + d*x]
)^2)/(2*d) - (2*a*b*(a + b*Coth[c + d*x])^3)/(3*d) - (b*(a + b*Coth[c + d*x])^4)/(4*d) + (b*(5*a^4 + 10*a^2*b^
2 + b^4)*Log[Sinh[c + d*x]])/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3563

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

Rule 3606

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

Rule 3609

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

Rubi steps \begin{align*} \text {integral}& = -\frac {b (a+b \coth (c+d x))^4}{4 d}+\int (a+b \coth (c+d x))^3 \left (a^2+b^2+2 a b \coth (c+d x)\right ) \, dx \\ & = -\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\int (a+b \coth (c+d x))^2 \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \coth (c+d x)\right ) \, dx \\ & = -\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\int (a+b \coth (c+d x)) \left (a^4+6 a^2 b^2+b^4+4 a b \left (a^2+b^2\right ) \coth (c+d x)\right ) \, dx \\ & = a \left (a^4+10 a^2 b^2+5 b^4\right ) x-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\left (b \left (5 a^4+10 a^2 b^2+b^4\right )\right ) \int \coth (c+d x) \, dx \\ & = a \left (a^4+10 a^2 b^2+5 b^4\right ) x-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\sinh (c+d x))}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99 \[ \int (a+b \coth (c+d x))^5 \, dx=-\frac {60 a b^2 \left (2 a^2+b^2\right ) \coth (c+d x)+6 b^3 \left (10 a^2+b^2\right ) \coth ^2(c+d x)+20 a b^4 \coth ^3(c+d x)+3 b^5 \coth ^4(c+d x)+6 (a+b)^5 \log (1-\tanh (c+d x))-12 b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\tanh (c+d x))-6 (a-b)^5 \log (1+\tanh (c+d x))}{12 d} \]

[In]

Integrate[(a + b*Coth[c + d*x])^5,x]

[Out]

-1/12*(60*a*b^2*(2*a^2 + b^2)*Coth[c + d*x] + 6*b^3*(10*a^2 + b^2)*Coth[c + d*x]^2 + 20*a*b^4*Coth[c + d*x]^3
+ 3*b^5*Coth[c + d*x]^4 + 6*(a + b)^5*Log[1 - Tanh[c + d*x]] - 12*b*(5*a^4 + 10*a^2*b^2 + b^4)*Log[Tanh[c + d*
x]] - 6*(a - b)^5*Log[1 + Tanh[c + d*x]])/d

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(\frac {\left (-60 a^{4} b -120 a^{2} b^{3}-12 b^{5}\right ) \ln \left (1-\tanh \left (d x +c \right )\right )+\left (60 a^{4} b +120 a^{2} b^{3}+12 b^{5}\right ) \ln \left (\tanh \left (d x +c \right )\right )-3 b^{5} \coth \left (d x +c \right )^{4}-20 a \,b^{4} \coth \left (d x +c \right )^{3}+\left (-60 a^{2} b^{3}-6 b^{5}\right ) \coth \left (d x +c \right )^{2}+\left (-120 a^{3} b^{2}-60 a \,b^{4}\right ) \coth \left (d x +c \right )+12 d x \left (a -b \right )^{5}}{12 d}\)	\(150\)
derivativedivides	\(\frac {-10 a^{3} b^{2} \coth \left (d x +c \right )-5 a \,b^{4} \coth \left (d x +c \right )-\frac {5 a \,b^{4} \coth \left (d x +c \right )^{3}}{3}-5 a^{2} b^{3} \coth \left (d x +c \right )^{2}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \coth \left (d x +c \right )^{2}}{2}-\frac {b^{5} \coth \left (d x +c \right )^{4}}{4}+\frac {\left (a^{5}-5 a^{4} b +10 a^{3} b^{2}-10 a^{2} b^{3}+5 a \,b^{4}-b^{5}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}}{d}\)	\(182\)
default	\(\frac {-10 a^{3} b^{2} \coth \left (d x +c \right )-5 a \,b^{4} \coth \left (d x +c \right )-\frac {5 a \,b^{4} \coth \left (d x +c \right )^{3}}{3}-5 a^{2} b^{3} \coth \left (d x +c \right )^{2}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \coth \left (d x +c \right )^{2}}{2}-\frac {b^{5} \coth \left (d x +c \right )^{4}}{4}+\frac {\left (a^{5}-5 a^{4} b +10 a^{3} b^{2}-10 a^{2} b^{3}+5 a \,b^{4}-b^{5}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}}{d}\)	\(182\)
parts	\(a^{5} x +\frac {b^{5} \left (-\frac {\coth \left (d x +c \right )^{4}}{4}-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {5 a \,b^{4} \left (-\frac {\coth \left (d x +c \right )^{3}}{3}-\coth \left (d x +c \right )-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {10 a^{2} b^{3} \left (-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {10 a^{3} b^{2} \left (-\coth \left (d x +c \right )-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {5 a^{4} b \ln \left (\sinh \left (d x +c \right )\right )}{d}\)	\(209\)
risch	\(a^{5} x -5 b \,a^{4} x +10 a^{3} b^{2} x -10 b^{3} a^{2} x +5 a \,b^{4} x -b^{5} x -\frac {10 b \,a^{4} c}{d}-\frac {20 b^{3} a^{2} c}{d}-\frac {2 b^{5} c}{d}-\frac {4 b^{2} \left (15 a^{3} {\mathrm e}^{6 d x +6 c}+15 a^{2} b \,{\mathrm e}^{6 d x +6 c}+15 a \,b^{2} {\mathrm e}^{6 d x +6 c}+3 \,{\mathrm e}^{6 d x +6 c} b^{3}-45 a^{3} {\mathrm e}^{4 d x +4 c}-30 a^{2} b \,{\mathrm e}^{4 d x +4 c}-30 a \,b^{2} {\mathrm e}^{4 d x +4 c}-3 \,{\mathrm e}^{4 d x +4 c} b^{3}+45 a^{3} {\mathrm e}^{2 d x +2 c}+15 a^{2} b \,{\mathrm e}^{2 d x +2 c}+25 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+3 \,{\mathrm e}^{2 d x +2 c} b^{3}-15 a^{3}-10 a \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {5 b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{4}}{d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}\)	\(346\)

[In]

int((a+b*coth(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

1/12*((-60*a^4*b-120*a^2*b^3-12*b^5)*ln(1-tanh(d*x+c))+(60*a^4*b+120*a^2*b^3+12*b^5)*ln(tanh(d*x+c))-3*b^5*cot
h(d*x+c)^4-20*a*b^4*coth(d*x+c)^3+(-60*a^2*b^3-6*b^5)*coth(d*x+c)^2+(-120*a^3*b^2-60*a*b^4)*coth(d*x+c)+12*d*x
*(a-b)^5)/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2748 vs. \(2 (136) = 272\).

Time = 0.28 (sec) , antiderivative size = 2748, normalized size of antiderivative = 19.35 \[ \int (a+b \coth (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate((a+b*coth(d*x+c))^5,x, algorithm="fricas")

[Out]

1/3*(3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)^8 + 24*(a^5 - 5*a^4*b + 10*
a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a
^2*b^3 + 5*a*b^4 - b^5)*d*x*sinh(d*x + c)^8 - 12*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5 + (a^5 - 5*a^4*b + 10*
a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^6 - 12*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5 - 7*(a^
5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)^2 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*
a^2*b^3 + 5*a*b^4 - b^5)*d*x)*sinh(d*x + c)^6 + 24*(7*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5
)*d*x*cosh(d*x + c)^3 - 3*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 +
5*a*b^4 - b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 60*a^3*b^2 + 40*a*b^4 + 6*(30*a^3*b^2 + 20*a^2*b^3 + 20*a
*b^4 + 2*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^4 + 6*(35*(a^5 -
5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)^4 + 30*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 +
2*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x - 30*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4
+ b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 24*
(7*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)^5 - 10*(5*a^3*b^2 + 5*a^2*b^3 +
5*a*b^4 + b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^3 + (30*a^3*b^2
+ 20*a^2*b^3 + 20*a*b^4 + 2*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x +
c))*sinh(d*x + c)^3 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x - 4*(45*a^3*b^2 + 15*a^2
*b^3 + 25*a*b^4 + 3*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^2 + 4
*(21*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*x + c)^6 - 45*a^3*b^2 - 15*a^2*b^3 -
25*a*b^4 - 3*b^5 - 45*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a
*b^4 - b^5)*d*x)*cosh(d*x + c)^4 - 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x + 9*(30*a^3
*b^2 + 20*a^2*b^3 + 20*a*b^4 + 2*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d
*x + c)^2)*sinh(d*x + c)^2 + 3*((5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^8 + 8*(5*a^4*b + 10*a^2*b^3 + b^5)*
cosh(d*x + c)*sinh(d*x + c)^7 + (5*a^4*b + 10*a^2*b^3 + b^5)*sinh(d*x + c)^8 - 4*(5*a^4*b + 10*a^2*b^3 + b^5)*
cosh(d*x + c)^6 - 4*(5*a^4*b + 10*a^2*b^3 + b^5 - 7*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c
)^6 + 8*(7*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^3 - 3*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c))*sinh(d
*x + c)^5 + 5*a^4*b + 10*a^2*b^3 + b^5 + 6*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^4 + 2*(15*a^4*b + 30*a^2
*b^3 + 3*b^5 + 35*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^4 - 30*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)
^2)*sinh(d*x + c)^4 + 8*(7*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^5 - 10*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh
(d*x + c)^3 + 3*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(5*a^4*b + 10*a^2*b^3 + b^5)*c
osh(d*x + c)^2 + 4*(7*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^6 - 5*a^4*b - 10*a^2*b^3 - b^5 - 15*(5*a^4*b
+ 10*a^2*b^3 + b^5)*cosh(d*x + c)^4 + 9*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((5*
a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^7 - 3*(5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c)^5 + 3*(5*a^4*b + 10*a
^2*b^3 + b^5)*cosh(d*x + c)^3 - (5*a^4*b + 10*a^2*b^3 + b^5)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)
/(cosh(d*x + c) - sinh(d*x + c))) + 8*(3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d*x*cosh(d*
x + c)^7 - 9*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5
)*d*x)*cosh(d*x + c)^5 + 3*(30*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 2*b^5 + 3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^
2*b^3 + 5*a*b^4 - b^5)*d*x)*cosh(d*x + c)^3 - (45*a^3*b^2 + 15*a^2*b^3 + 25*a*b^4 + 3*b^5 + 3*(a^5 - 5*a^4*b +
10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (133) = 266\).

Time = 1.79 (sec) , antiderivative size = 588, normalized size of antiderivative = 4.14 \[ \int (a+b \coth (c+d x))^5 \, dx=\begin {cases} x \left (a + b \coth {\left (c \right )}\right )^{5} & \text {for}\: d = 0 \\- \frac {a^{5} \log {\left (- e^{- d x} \right )}}{d} - \frac {5 a^{4} b \log {\left (- e^{- d x} \right )} \coth {\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {10 a^{3} b^{2} \log {\left (- e^{- d x} \right )} \coth ^{2}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {10 a^{2} b^{3} \log {\left (- e^{- d x} \right )} \coth ^{3}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {5 a b^{4} \log {\left (- e^{- d x} \right )} \coth ^{4}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {b^{5} \log {\left (- e^{- d x} \right )} \coth ^{5}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} & \text {for}\: c = \log {\left (- e^{- d x} \right )} \\a^{5} x + 5 a^{4} b x \coth {\left (d x + \log {\left (e^{- d x} \right )} \right )} + 10 a^{3} b^{2} x \coth ^{2}{\left (d x + \log {\left (e^{- d x} \right )} \right )} + 10 a^{2} b^{3} x \coth ^{3}{\left (d x + \log {\left (e^{- d x} \right )} \right )} + 5 a b^{4} x \coth ^{4}{\left (d x + \log {\left (e^{- d x} \right )} \right )} + b^{5} x \coth ^{5}{\left (d x + \log {\left (e^{- d x} \right )} \right )} & \text {for}\: c = \log {\left (e^{- d x} \right )} \\a^{5} x + 5 a^{4} b x - \frac {5 a^{4} b \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} + \frac {5 a^{4} b \log {\left (\tanh {\left (c + d x \right )} \right )}}{d} + 10 a^{3} b^{2} x - \frac {10 a^{3} b^{2}}{d \tanh {\left (c + d x \right )}} + 10 a^{2} b^{3} x - \frac {10 a^{2} b^{3} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} + \frac {10 a^{2} b^{3} \log {\left (\tanh {\left (c + d x \right )} \right )}}{d} - \frac {5 a^{2} b^{3}}{d \tanh ^{2}{\left (c + d x \right )}} + 5 a b^{4} x - \frac {5 a b^{4}}{d \tanh {\left (c + d x \right )}} - \frac {5 a b^{4}}{3 d \tanh ^{3}{\left (c + d x \right )}} + b^{5} x - \frac {b^{5} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} + \frac {b^{5} \log {\left (\tanh {\left (c + d x \right )} \right )}}{d} - \frac {b^{5}}{2 d \tanh ^{2}{\left (c + d x \right )}} - \frac {b^{5}}{4 d \tanh ^{4}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*coth(d*x+c))**5,x)

[Out]

Piecewise((x*(a + b*coth(c))**5, Eq(d, 0)), (-a**5*log(-exp(-d*x))/d - 5*a**4*b*log(-exp(-d*x))*coth(d*x + log
(-exp(-d*x)))/d - 10*a**3*b**2*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**2/d - 10*a**2*b**3*log(-exp(-d*x))
*coth(d*x + log(-exp(-d*x)))**3/d - 5*a*b**4*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**4/d - b**5*log(-exp(
-d*x))*coth(d*x + log(-exp(-d*x)))**5/d, Eq(c, log(-exp(-d*x)))), (a**5*x + 5*a**4*b*x*coth(d*x + log(exp(-d*x
))) + 10*a**3*b**2*x*coth(d*x + log(exp(-d*x)))**2 + 10*a**2*b**3*x*coth(d*x + log(exp(-d*x)))**3 + 5*a*b**4*x
*coth(d*x + log(exp(-d*x)))**4 + b**5*x*coth(d*x + log(exp(-d*x)))**5, Eq(c, log(exp(-d*x)))), (a**5*x + 5*a**
4*b*x - 5*a**4*b*log(tanh(c + d*x) + 1)/d + 5*a**4*b*log(tanh(c + d*x))/d + 10*a**3*b**2*x - 10*a**3*b**2/(d*t
anh(c + d*x)) + 10*a**2*b**3*x - 10*a**2*b**3*log(tanh(c + d*x) + 1)/d + 10*a**2*b**3*log(tanh(c + d*x))/d - 5
*a**2*b**3/(d*tanh(c + d*x)**2) + 5*a*b**4*x - 5*a*b**4/(d*tanh(c + d*x)) - 5*a*b**4/(3*d*tanh(c + d*x)**3) +
b**5*x - b**5*log(tanh(c + d*x) + 1)/d + b**5*log(tanh(c + d*x))/d - b**5/(2*d*tanh(c + d*x)**2) - b**5/(4*d*t
anh(c + d*x)**4), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (136) = 272\).

Time = 0.20 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.45 \[ \int (a+b \coth (c+d x))^5 \, dx=\frac {5}{3} \, a b^{4} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + b^{5} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 10 \, a^{2} b^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + 10 \, a^{3} b^{2} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{5} x + \frac {5 \, a^{4} b \log \left (\sinh \left (d x + c\right )\right )}{d} \]

[In]

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

[Out]

5/3*a*b^4*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - 2)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x
- 4*c) + e^(-6*d*x - 6*c) - 1))) + b^5*(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))) + 10*a^2*b^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))) + 10*a^3*b^2*(x + c/d + 2/(d*(e^(-2*d*x - 2*
c) - 1))) + a^5*x + 5*a^4*b*log(sinh(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.59 \[ \int (a+b \coth (c+d x))^5 \, dx=\frac {3 \, {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} {\left (d x + c\right )} + 3 \, {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {4 \, {\left (15 \, a^{3} b^{2} + 10 \, a b^{4} - 3 \, {\left (5 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, {\left (15 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 10 \, a b^{4} + b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (45 \, a^{3} b^{2} + 15 \, a^{2} b^{3} + 25 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{3 \, d} \]

[In]

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

[Out]

1/3*(3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*(d*x + c) + 3*(5*a^4*b + 10*a^2*b^3 + b^5)*lo
g(abs(e^(2*d*x + 2*c) - 1)) + 4*(15*a^3*b^2 + 10*a*b^4 - 3*(5*a^3*b^2 + 5*a^2*b^3 + 5*a*b^4 + b^5)*e^(6*d*x +
6*c) + 3*(15*a^3*b^2 + 10*a^2*b^3 + 10*a*b^4 + b^5)*e^(4*d*x + 4*c) - (45*a^3*b^2 + 15*a^2*b^3 + 25*a*b^4 + 3*
b^5)*e^(2*d*x + 2*c))/(e^(2*d*x + 2*c) - 1)^4)/d

Mupad [B] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.72 \[ \int (a+b \coth (c+d x))^5 \, dx=x\,{\left (a-b\right )}^5-\frac {4\,\left (5\,a^3\,b^2+5\,a^2\,b^3+5\,a\,b^4+b^5\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (5\,a^4\,b+10\,a^2\,b^3+b^5\right )}{d}-\frac {4\,\left (5\,a^2\,b^3+5\,a\,b^4+2\,b^5\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,b^5}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,\left (3\,b^5+5\,a\,b^4\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \]

[In]

int((a + b*coth(c + d*x))^5,x)

[Out]

x*(a - b)^5 - (4*(5*a*b^4 + b^5 + 5*a^2*b^3 + 5*a^3*b^2))/(d*(exp(2*c + 2*d*x) - 1)) + (log(exp(2*c)*exp(2*d*x
) - 1)*(5*a^4*b + b^5 + 10*a^2*b^3))/d - (4*(5*a*b^4 + 2*b^5 + 5*a^2*b^3))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c +
2*d*x) + 1)) - (4*b^5)/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1
)) - (8*(5*a*b^4 + 3*b^5))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1))