Integrand size = 23, antiderivative size = 232 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} d}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^ (1/2))/a^3/(a+b)^(9/2)/d-1/8*(8*a^3+32*a^2*b-15*a*b^2-4*b^3)*coth(d*x+c)/a ^2/(a+b)^4/d-1/24*(8*a^2-39*a*b-12*b^2)*coth(d*x+c)^3/a^2/(a+b)^3/d-1/4*b* coth(d*x+c)^3/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(11*a+4*b)*coth(d*x+ c)^3/a^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 8.50 (sec) , antiderivative size = 3334, normalized size of antiderivative = 14.37 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \]
((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x] ^6*(((I/64)*b^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b *Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4* c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]* Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b ^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4 *c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a ^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^4*(a + b*Sech [c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]^3 *Sech[2*c]*Sech[c + d*x]^6*(-36*a^6*d*x*Cosh[d*x] - 336*a^5*b*d*x*Cosh[d*x ] - 1560*a^4*b^2*d*x*Cosh[d*x] - 3600*a^3*b^3*d*x*Cosh[d*x] - 4260*a^2*b^4 *d*x*Cosh[d*x] - 2496*a*b^5*d*x*Cosh[d*x] - 576*b^6*d*x*Cosh[d*x] + 36*a^6 *d*x*Cosh[3*d*x] + 240*a^5*b*d*x*Cosh[3*d*x] + 408*a^4*b^2*d*x*Cosh[3*d*x] - 48*a^3*b^3*d*x*Cosh[3*d*x] - 732*a^2*b^4*d*x*Cosh[3*d*x] - 672*a*b^5*d* x*Cosh[3*d*x] - 192*b^6*d*x*Cosh[3*d*x] + 36*a^6*d*x*Cosh[2*c - d*x] + 336 *a^5*b*d*x*Cosh[2*c - d*x] + 1560*a^4*b^2*d*x*Cosh[2*c - d*x] + 3600*a^3*b ^3*d*x*Cosh[2*c - d*x] + 4260*a^2*b^4*d*x*Cosh[2*c - d*x] + 2496*a*b^5*d*x *Cosh[2*c - d*x] + 576*b^6*d*x*Cosh[2*c - d*x] + 36*a^6*d*x*Cosh[2*c + d*x ] + 336*a^5*b*d*x*Cosh[2*c + d*x] + 1560*a^4*b^2*d*x*Cosh[2*c + d*x] + ...
Time = 0.65 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 4629, 2075, 374, 25, 441, 25, 445, 27, 445, 25, 397, 219, 221}
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.
| \(\displaystyle \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i c+i d x)^4 \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {-\frac {\int -\frac {\coth ^4(c+d x) \left (7 b \tanh ^2(c+d x)+4 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\coth ^4(c+d x) \left (7 b \tanh ^2(c+d x)+4 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\coth ^4(c+d x) \left (8 a^2-39 b a-12 b^2+5 b (11 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\coth ^4(c+d x) \left (8 a^2-39 b a-12 b^2+5 b (11 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 \coth ^2(c+d x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3-b \left (8 a^2-39 b a-12 b^2\right ) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{3 (a+b)}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\coth ^2(c+d x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3-b \left (8 a^2-39 b a-12 b^2\right ) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int -\frac {8 a^4+40 b a^3+80 b^2 a^2+17 b^3 a+4 b^4-b \left (8 a^3+32 b a^2-15 b^2 a-4 b^3\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {8 a^4+40 b a^3+80 b^2 a^2+17 b^3 a+4 b^4-b \left (8 a^3+32 b a^2-15 b^2 a-4 b^3\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
(-1/4*(b*Coth[c + d*x]^3)/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + ((-1 /3*((8*a^2 - 39*a*b - 12*b^2)*Coth[c + d*x]^3)/(a + b) + (((8*(a + b)^4*Ar cTanh[Tanh[c + d*x]])/a - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt [b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b) - ((8*a^3 + 32*a ^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(a + b))/(a + b))/(2*a*(a + b)) - (b*(11*a + 4*b)*Coth[c + d*x]^3)/(2*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)) )/(4*a*(a + b)))/d
3.2.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f _.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/f Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 )), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(467\) vs. \(2(214)=428\).
Time = 268.04 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.02
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}+5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +17 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +17 b}{8 \left (a +b \right )^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{4} a^{3}}}{d}\) | \(468\) |
| default | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}+5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +17 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +17 b}{8 \left (a +b \right )^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{4} a^{3}}}{d}\) | \(468\) |
| risch | \(\frac {x}{a^{3}}-\frac {480 a^{5} b \,{\mathrm e}^{2 d x +2 c}+48 a^{6} {\mathrm e}^{12 d x +12 c}+144 a^{6} {\mathrm e}^{10 d x +10 c}+2073 a^{3} b^{3} {\mathrm e}^{8 d x +8 c}+642 a^{2} b^{4} {\mathrm e}^{8 d x +8 c}+1416 a \,b^{5} {\mathrm e}^{8 d x +8 c}-384 a^{5} b \,{\mathrm e}^{6 d x +6 c}-144 b^{6} {\mathrm e}^{10 d x +10 c}+128 a^{6} {\mathrm e}^{8 d x +8 c}+432 b^{6} {\mathrm e}^{8 d x +8 c}+440 a^{5} b \,{\mathrm e}^{8 d x +8 c}+1152 a^{4} b^{2} {\mathrm e}^{8 d x +8 c}+672 a^{5} b \,{\mathrm e}^{10 d x +10 c}+120 a^{5} b \,{\mathrm e}^{12 d x +12 c}-51 a^{3} b^{3} {\mathrm e}^{12 d x +12 c}-132 a^{2} b^{4} {\mathrm e}^{12 d x +12 c}-48 a \,b^{5} {\mathrm e}^{12 d x +12 c}+960 a^{4} b^{2} {\mathrm e}^{10 d x +10 c}-66 a^{2} b^{4} {\mathrm e}^{10 d x +10 c}-408 a \,b^{5} {\mathrm e}^{10 d x +10 c}+18 a^{2} b^{4}+104 a^{5} b +51 a^{3} b^{3}-2048 a^{4} b^{2} {\mathrm e}^{6 d x +6 c}-3072 a^{3} b^{3} {\mathrm e}^{6 d x +6 c}-228 a^{2} b^{4} {\mathrm e}^{6 d x +6 c}-1320 a \,b^{5} {\mathrm e}^{6 d x +6 c}+104 a^{5} b \,{\mathrm e}^{4 d x +4 c}+640 a^{4} b^{2} {\mathrm e}^{4 d x +4 c}+1511 a^{3} b^{3} {\mathrm e}^{4 d x +4 c}-528 a^{2} b^{4} {\mathrm e}^{4 d x +4 c}+264 a \,b^{5} {\mathrm e}^{4 d x +4 c}+832 a^{4} b^{2} {\mathrm e}^{2 d x +2 c}+294 a^{2} b^{4} {\mathrm e}^{2 d x +2 c}+96 a \,b^{5} {\mathrm e}^{2 d x +2 c}+32 a^{6}+80 a^{6} {\mathrm e}^{2 d x +2 c}+32 a^{6} {\mathrm e}^{6 d x +6 c}-432 b^{6} {\mathrm e}^{6 d x +6 c}+48 a^{6} {\mathrm e}^{4 d x +4 c}+144 b^{6} {\mathrm e}^{4 d x +4 c}}{12 a^{3} d \left (a +b \right )^{4} \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {63 \sqrt {\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{16 \left (a +b \right )^{5} d a}+\frac {9 \sqrt {\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{4 \left (a +b \right )^{5} d \,a^{2}}+\frac {\sqrt {\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{2 \left (a +b \right )^{5} d \,a^{3}}-\frac {63 \sqrt {\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{16 \left (a +b \right )^{5} d a}-\frac {9 \sqrt {\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{4 \left (a +b \right )^{5} d \,a^{2}}-\frac {\sqrt {\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{2 \left (a +b \right )^{5} d \,a^{3}}\) | \(996\) |
1/d*(1/a^3*ln(1+tanh(1/2*d*x+1/2*c))-1/8/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*( 1/3*tanh(1/2*d*x+1/2*c)^3*a+1/3*tanh(1/2*d*x+1/2*c)^3*b+5*tanh(1/2*d*x+1/2 *c)*a+17*b*tanh(1/2*d*x+1/2*c))-1/24/(a+b)^3/tanh(1/2*d*x+1/2*c)^3-1/8*(5* a+17*b)/(a+b)^4/tanh(1/2*d*x+1/2*c)-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+2*b^3/ (a+b)^4/a^3*(((-17/8*a^3-21/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*( 51*a^2+3*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(51*a^2+3*a*b-4*b^2)*a*tan h(1/2*d*x+1/2*c)^3+(-17/8*a^3-21/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c))/( tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+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+1/8*(63*a^2+36*a*b+8*b^2)*(-1/4/b^(1/2)/( a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1 /2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c )^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 11993 vs. \(2 (220) = 440\).
Time = 0.56 (sec) , antiderivative size = 24263, normalized size of antiderivative = 104.58 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 4920 vs. \(2 (220) = 440\).
Time = 0.85 (sec) , antiderivative size = 4920, normalized size of antiderivative = 21.21 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
1/8*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3 + 2*b^4)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3* b^4)*d) - 3/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/( (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/8*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3 + 2*b^4)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) + 3/4*b*log(2*( a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^4 + 4*a^3*b + 6*a^ 2*b^2 + 4*a*b^3 + b^4)*d) + 1/4*(2*a + 5*b)*log(e^(2*d*x + 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/2*b*log(e^(2*d*x + 2*c) - 1 )/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/4*(2*a + 5*b)*log(e^ (-2*d*x - 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 3/2* b*log(e^(-2*d*x - 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d ) - 1/256*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*log( (a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b ^4)*sqrt((a + b)*b)*d) + 5/64*(3*a*b + 10*b^2)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b )))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 1/25 6*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*log((a*e^(-2 *d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*...
\[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\coth \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]