Integrand size = 12, antiderivative size = 142 \[ \int (a+b \coth (c+d x))^5 \, dx=a \left (a^4+10 a^2 b^2+5 b^4\right ) x-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\sinh (c+d x))}{d} \]
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
Time = 0.85 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99 \[ \int (a+b \coth (c+d x))^5 \, dx=-\frac {60 a b^2 \left (2 a^2+b^2\right ) \coth (c+d x)+6 b^3 \left (10 a^2+b^2\right ) \coth ^2(c+d x)+20 a b^4 \coth ^3(c+d x)+3 b^5 \coth ^4(c+d x)+6 (a+b)^5 \log (1-\tanh (c+d x))-12 b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (\tanh (c+d x))-6 (a-b)^5 \log (1+\tanh (c+d x))}{12 d} \]
-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
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3963, 3042, 4011, 3042, 4011, 3042, 4008, 26, 3042, 26, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \coth (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^5dx\) |
| \(\Big \downarrow \) 3963 |
| \(\displaystyle \int (a+b \coth (c+d x))^3 \left (a^2+2 b \coth (c+d x) a+b^2\right )dx-\frac {b (a+b \coth (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b (a+b \coth (c+d x))^4}{4 d}+\int \left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^3 \left (a^2-2 i b \tan \left (i c+i d x+\frac {\pi }{2}\right ) a+b^2\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \coth (c+d x))^2 \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \coth (c+d x)\right )dx-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^2 \left (a \left (a^2+3 b^2\right )-i b \left (3 a^2+b^2\right ) \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )dx-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \coth (c+d x)) \left (a^4+6 b^2 a^2+4 b \left (a^2+b^2\right ) \coth (c+d x) a+b^4\right )dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right ) \left (a^4+6 b^2 a^2-4 i b \left (a^2+b^2\right ) \tan \left (i c+i d x+\frac {\pi }{2}\right ) a+b^4\right )dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle -i b \left (5 a^4+10 a^2 b^2+b^4\right ) \int i \coth (c+d x)dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle b \left (5 a^4+10 a^2 b^2+b^4\right ) \int \coth (c+d x)dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (5 a^4+10 a^2 b^2+b^4\right ) \int -i \tan \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i b \left (5 a^4+10 a^2 b^2+b^4\right ) \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx-\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {b \left (3 a^2+b^2\right ) (a+b \coth (c+d x))^2}{2 d}-\frac {4 a b^2 \left (a^2+b^2\right ) \coth (c+d x)}{d}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right ) \log (-i \sinh (c+d x))}{d}+a x \left (a^4+10 a^2 b^2+5 b^4\right )-\frac {b (a+b \coth (c+d x))^4}{4 d}-\frac {2 a b (a+b \coth (c+d x))^3}{3 d}\) |
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[(-I)*Sinh[c + d*x]])/d
3.1.77.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
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]
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 | \(a^{5} x -5 b \,a^{4} x +10 a^{3} b^{2} x -10 b^{3} a^{2} x +5 a \,b^{4} x -b^{5} x -\frac {10 b \,a^{4} c}{d}-\frac {20 b^{3} a^{2} c}{d}-\frac {2 b^{5} c}{d}-\frac {4 b^{2} \left (15 a^{3} {\mathrm e}^{6 d x +6 c}+15 a^{2} b \,{\mathrm e}^{6 d x +6 c}+15 a \,b^{2} {\mathrm e}^{6 d x +6 c}+3 \,{\mathrm e}^{6 d x +6 c} b^{3}-45 a^{3} {\mathrm e}^{4 d x +4 c}-30 a^{2} b \,{\mathrm e}^{4 d x +4 c}-30 a \,b^{2} {\mathrm e}^{4 d x +4 c}-3 \,{\mathrm e}^{4 d x +4 c} b^{3}+45 a^{3} {\mathrm e}^{2 d x +2 c}+15 a^{2} b \,{\mathrm e}^{2 d x +2 c}+25 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+3 \,{\mathrm e}^{2 d x +2 c} b^{3}-15 a^{3}-10 a \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {5 b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{4}}{d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}\) | \(346\) |
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*coth(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
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 \[ \int (a+b \coth (c+d x))^5 \, dx=\text {Too large to display} \]
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)*co sh(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 - ...
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} \]
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(-ex p(-d*x))*coth(d*x + log(-exp(-d*x)))**2/d - 10*a**2*b**3*log(-exp(-d*x))*c oth(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*co th(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*tanh(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*tanh(c + d* x)**4), True))
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 \[ \int (a+b \coth (c+d x))^5 \, dx=\frac {5}{3} \, a b^{4} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + b^{5} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 10 \, a^{2} b^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + 10 \, a^{3} b^{2} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{5} x + \frac {5 \, a^{4} b \log \left (\sinh \left (d x + c\right )\right )}{d} \]
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
Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.59 \[ \int (a+b \coth (c+d x))^5 \, dx=\frac {3 \, {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} {\left (d x + c\right )} + 3 \, {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {4 \, {\left (15 \, a^{3} b^{2} + 10 \, a b^{4} - 3 \, {\left (5 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, {\left (15 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 10 \, a b^{4} + b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (45 \, a^{3} b^{2} + 15 \, a^{2} b^{3} + 25 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{3 \, d} \]
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)*log(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
Time = 1.97 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.72 \[ \int (a+b \coth (c+d x))^5 \, dx=x\,{\left (a-b\right )}^5-\frac {4\,\left (5\,a^3\,b^2+5\,a^2\,b^3+5\,a\,b^4+b^5\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (5\,a^4\,b+10\,a^2\,b^3+b^5\right )}{d}-\frac {4\,\left (5\,a^2\,b^3+5\,a\,b^4+2\,b^5\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,b^5}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,\left (3\,b^5+5\,a\,b^4\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \]
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))