Integrand size = 23, antiderivative size = 182 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{7/2} d}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 (a+b)^3 d}-\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
x/a^3-1/8*b^(3/2)*(35*a^2+28*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^ (1/2))/a^3/(a+b)^(7/2)/d-1/8*(8*a^2-11*a*b-4*b^2)*coth(d*x+c)/a^2/(a+b)^3/ d-1/4*b*coth(d*x+c)/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(9*a+4*b)*coth (d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 8.75 (sec) , antiderivative size = 2083, normalized size of antiderivative = 11.45 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \]
((35*a^2 + 28*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x] ^6*(((I/64)*b^2*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 ^2*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)^3*(a + b*Sech [c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*S ech[2*c]*Sech[c + d*x]^6*(8*a^5*d*x*Cosh[d*x] + 56*a^4*b*d*x*Cosh[d*x] + 1 84*a^3*b^2*d*x*Cosh[d*x] + 296*a^2*b^3*d*x*Cosh[d*x] + 224*a*b^4*d*x*Cosh[ d*x] + 64*b^5*d*x*Cosh[d*x] - 12*a^5*d*x*Cosh[3*d*x] - 68*a^4*b*d*x*Cosh[3 *d*x] - 132*a^3*b^2*d*x*Cosh[3*d*x] - 108*a^2*b^3*d*x*Cosh[3*d*x] - 32*a*b ^4*d*x*Cosh[3*d*x] - 8*a^5*d*x*Cosh[2*c - d*x] - 56*a^4*b*d*x*Cosh[2*c - d *x] - 184*a^3*b^2*d*x*Cosh[2*c - d*x] - 296*a^2*b^3*d*x*Cosh[2*c - d*x] - 224*a*b^4*d*x*Cosh[2*c - d*x] - 64*b^5*d*x*Cosh[2*c - d*x] - 8*a^5*d*x*Cos h[2*c + d*x] - 56*a^4*b*d*x*Cosh[2*c + d*x] - 184*a^3*b^2*d*x*Cosh[2*c + d *x] - 296*a^2*b^3*d*x*Cosh[2*c + d*x] - 224*a*b^4*d*x*Cosh[2*c + d*x] - 64 *b^5*d*x*Cosh[2*c + d*x] + 8*a^5*d*x*Cosh[4*c + d*x] + 56*a^4*b*d*x*Cos...
Time = 0.57 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 25, 4629, 25, 2075, 374, 25, 441, 25, 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.
| \(\displaystyle \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^3 \tan (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle -\frac {\int -\frac {\coth ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle -\frac {\frac {\int -\frac {\coth ^2(c+d x) \left (5 b \tanh ^2(c+d x)+4 a-b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}+\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {\coth ^2(c+d x) \left (5 b \tanh ^2(c+d x)+4 a-b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {\coth ^2(c+d x) \left (8 a^2-11 b a-4 b^2+3 b (9 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {\coth ^2(c+d x) \left (8 a^2-11 b a-4 b^2+3 b (9 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {-\frac {\int -\frac {8 a^3+32 b a^2+13 b^2 a+4 b^3-b \left (8 a^2-11 b a-4 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {8 a^3+32 b a^2+13 b^2 a+4 b^3-b \left (8 a^2-11 b a-4 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^2 \left (35 a^2+28 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^2 \left (35 a^2+28 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
-(((b*Coth[c + d*x])/(4*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) - ((((8*( a + b)^3*ArcTanh[Tanh[c + d*x]])/a - (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*Ar cTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b) - ((8 *a^2 - 11*a*b - 4*b^2)*Coth[c + d*x])/(a + b))/(2*a*(a + b)) - (b*(9*a + 4 *b)*Coth[c + d*x])/(2*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)))/(4*a*(a + b) ))/d)
3.2.66.3.1 Defintions of rubi rules used
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. \(393\) vs. \(2(166)=332\).
Time = 136.06 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.16
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{2} \left (\frac {\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (35 a^{2}+28 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{3} a^{3}}}{d}\) | \(394\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{2} \left (\frac {\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (35 a^{2}+28 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{3} a^{3}}}{d}\) | \(394\) |
| risch | \(\frac {x}{a^{3}}-\frac {8 a^{5} {\mathrm e}^{8 d x +8 c}-13 a^{3} b^{2} {\mathrm e}^{8 d x +8 c}-36 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}-16 a \,b^{4} {\mathrm e}^{8 d x +8 c}+32 a^{5} {\mathrm e}^{6 d x +6 c}+64 a^{4} b \,{\mathrm e}^{6 d x +6 c}-26 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}-86 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}-136 a \,b^{4} {\mathrm e}^{6 d x +6 c}-48 b^{5} {\mathrm e}^{6 d x +6 c}+48 a^{5} {\mathrm e}^{4 d x +4 c}+128 a^{4} b \,{\mathrm e}^{4 d x +4 c}+128 a^{3} b^{2} {\mathrm e}^{4 d x +4 c}+30 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+120 a \,b^{4} {\mathrm e}^{4 d x +4 c}+48 b^{5} {\mathrm e}^{4 d x +4 c}+32 a^{5} {\mathrm e}^{2 d x +2 c}+64 a^{4} b \,{\mathrm e}^{2 d x +2 c}+26 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}+86 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}+32 a \,b^{4} {\mathrm e}^{2 d x +2 c}+8 a^{5}+13 a^{3} b^{2}+6 a^{2} b^{3}}{4 a^{3} d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right ) \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}}+\frac {35 \sqrt {\left (a +b \right ) b}\, b \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 )^{4} d a}+\frac {7 \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 )}{4 \left (a +b \right )^{4} d \,a^{2}}+\frac {\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 )}{2 \left (a +b \right )^{4} d \,a^{3}}-\frac {35 \sqrt {\left (a +b \right ) b}\, b \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 )^{4} d a}-\frac {7 \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 )}{4 \left (a +b \right )^{4} d \,a^{2}}-\frac {\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 )}{2 \left (a +b \right )^{4} d \,a^{3}}\) | \(736\) |
1/d*(-1/2/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)+1/a^3*ln(1+tanh(1/ 2*d*x+1/2*c))-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)-1/2/(a+b)^3/tanh(1/2*d*x+1/2 *c)+2*b^2/(a+b)^3/a^3*(((-13/8*a^3-17/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2* c)^7-1/8*(39*a^2+7*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(39*a^2+7*a*b-4* b^2)*a*tanh(1/2*d*x+1/2*c)^3+(-13/8*a^3-17/8*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*(35*a^2+28*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. 5665 vs. \(2 (172) = 344\).
Time = 0.39 (sec) , antiderivative size = 11606, normalized size of antiderivative = 63.77 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1971 vs. \(2 (172) = 344\).
Time = 0.46 (sec) , antiderivative size = 1971, normalized size of antiderivative = 10.83 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
1/4*(3*a^2*b + 3*a*b^2 + b^3)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d) - 1/4*(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) - 1/64*(15*a^3*b + 70*a^2*b^2 + 56 *a*b^3 + 16*b^4)*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^6 + 3*a^5*b + 3*a^4*b^ 2 + a^3*b^3)*sqrt((a + b)*b)*d) + 1/64*(15*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*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^6 + 3*a^5*b + 3*a^4*b^2 + a^ 3*b^3)*sqrt((a + b)*b)*d) - 15/32*b*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^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/16*(8*a^5 + 9*a^4*b + 2 8*a^3*b^2 + 12*a^2*b^3 + (8*a^5 - 9*a^4*b - 98*a^3*b^2 - 160*a^2*b^3 - 64* a*b^4)*e^(8*d*x + 8*c) + 2*(16*a^5 + 23*a^4*b - 77*a^3*b^2 - 246*a^2*b^3 - 288*a*b^4 - 96*b^5)*e^(6*d*x + 6*c) + 2*(24*a^5 + 64*a^4*b + 99*a^3*b^2 + 190*a^2*b^3 + 272*a*b^4 + 96*b^5)*e^(4*d*x + 4*c) + 2*(16*a^5 + 41*a^4*b + 77*a^3*b^2 + 130*a^2*b^3 + 48*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 - (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^(10*d*x + 10 *c) - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(8*d*x + 8*c) - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*...
\[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]