Integrand size = 23, antiderivative size = 139 \[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 \sqrt {b} (a+b)^{3/2} d}-\frac {\tanh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(3 a+4 b) \tanh (c+d x)}{8 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]
x/a^3-1/8*(3*a^2+12*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^ 3/(a+b)^(3/2)/d/b^(1/2)-1/4*tanh(d*x+c)/a/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*(3 *a+4*b)*tanh(d*x+c)/a^2/(a+b)/d/(a+b-b*tanh(d*x+c)^2)
Leaf count is larger than twice the leaf count of optimal. \(1317\) vs. \(2(139)=278\).
Time = 11.78 (sec) , antiderivative size = 1317, normalized size of antiderivative = 9.47 \[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (\frac {6 a (a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {4 \left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {4 a \sqrt {b} \left (3 a^2+16 a b+16 b^2+3 a (a+2 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a+b)^2 (a+2 b+a \cosh (2 (c+d x)))^2}-\frac {2 \sqrt {b} \left (3 a^3+14 a^2 b+24 a b^2+16 b^3+a \left (3 a^2+4 a b+4 b^2\right ) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a+b)^2 (a+2 b+a \cosh (2 (c+d x)))^2}+\frac {\sqrt {b} \left (-\frac {2 \left (3 a^5-10 a^4 b+80 a^3 b^2+480 a^2 b^3+640 a b^4+256 b^5\right ) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\text {sech}(2 c) \left (256 b^2 (a+b)^2 \left (3 a^2+8 a b+8 b^2\right ) d x \cosh (2 c)+512 a b^2 (a+b)^2 (a+2 b) d x \cosh (2 d x)+128 a^4 b^2 d x \cosh (2 (c+2 d x))+256 a^3 b^3 d x \cosh (2 (c+2 d x))+128 a^2 b^4 d x \cosh (2 (c+2 d x))+512 a^4 b^2 d x \cosh (4 c+2 d x)+2048 a^3 b^3 d x \cosh (4 c+2 d x)+2560 a^2 b^4 d x \cosh (4 c+2 d x)+1024 a b^5 d x \cosh (4 c+2 d x)+128 a^4 b^2 d x \cosh (6 c+4 d x)+256 a^3 b^3 d x \cosh (6 c+4 d x)+128 a^2 b^4 d x \cosh (6 c+4 d x)-9 a^6 \sinh (2 c)+12 a^5 b \sinh (2 c)+684 a^4 b^2 \sinh (2 c)+2880 a^3 b^3 \sinh (2 c)+5280 a^2 b^4 \sinh (2 c)+4608 a b^5 \sinh (2 c)+1536 b^6 \sinh (2 c)+9 a^6 \sinh (2 d x)-14 a^5 b \sinh (2 d x)-608 a^4 b^2 \sinh (2 d x)-2112 a^3 b^3 \sinh (2 d x)-2560 a^2 b^4 \sinh (2 d x)-1024 a b^5 \sinh (2 d x)+3 a^6 \sinh (2 (c+2 d x))-12 a^5 b \sinh (2 (c+2 d x))-204 a^4 b^2 \sinh (2 (c+2 d x))-384 a^3 b^3 \sinh (2 (c+2 d x))-192 a^2 b^4 \sinh (2 (c+2 d x))-3 a^6 \sinh (4 c+2 d x)+10 a^5 b \sinh (4 c+2 d x)+304 a^4 b^2 \sinh (4 c+2 d x)+1056 a^3 b^3 \sinh (4 c+2 d x)+1280 a^2 b^4 \sinh (4 c+2 d x)+512 a b^5 \sinh (4 c+2 d x)\right )}{(a+2 b+a \cosh (2 (c+d x)))^2}\right )}{a^3 (a+b)^2}+\frac {2 \sqrt {b} \left (\frac {6 a^2 \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {a \text {sech}(2 c) \left (\left (-9 a^4-16 a^3 b+48 a^2 b^2+128 a b^3+64 b^4\right ) \sinh (2 d x)+a \left (-3 a^3+2 a^2 b+24 a b^2+16 b^3\right ) \sinh (2 (c+2 d x))+\left (3 a^4-64 a^2 b^2-128 a b^3-64 b^4\right ) \sinh (4 c+2 d x)\right )+\left (9 a^5+18 a^4 b-64 a^3 b^2-256 a^2 b^3-320 a b^4-128 b^5\right ) \tanh (2 c)}{a^2 (a+2 b+a \cosh (2 (c+d x)))^2}\right )}{(a+b)^2}\right )}{4096 b^{5/2} d \left (a+b \text {sech}^2(c+d x)\right )^3} \]
((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((6*a*(a + 2*b)*ArcTanh [(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (4*(3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (4*a *Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[ 2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[b]*( 3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (Sqrt[b]*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cos h[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2 *c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*( a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 12 8*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 6 84*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4 608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b...
Time = 0.42 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 25, 4629, 25, 2075, 373, 402, 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.
| \(\displaystyle \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan (i c+i d x)^2}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan (i c+i d x)^2}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle -\frac {\int -\frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {3 \tanh ^2(c+d x)+1}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {(3 a+4 b) \tanh ^2(c+d x)+5 a+4 b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {(3 a+4 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {(3 a+4 b) \tanh ^2(c+d x)+5 a+4 b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {(3 a+4 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {\left (3 a^2+12 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {(3 a+4 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b) \text {arctanh}(\tanh (c+d x))}{a}-\frac {\left (3 a^2+12 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {(3 a+4 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\tanh (c+d x)}{4 a \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b) \text {arctanh}(\tanh (c+d x))}{a}-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b} \sqrt {a+b}}}{2 a (a+b)}-\frac {(3 a+4 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
-((Tanh[c + d*x]/(4*a*(a + b - b*Tanh[c + d*x]^2)^2) - (((8*(a + b)*ArcTan h[Tanh[c + d*x]])/a - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]*Sqrt[a + b]))/(2*a*(a + b)) - ((3*a + 4*b)* Tanh[c + d*x])/(2*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)))/(4*a))/d)
3.2.62.3.1 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[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -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. \(334\) vs. \(2(125)=250\).
Time = 41.62 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.41
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (-\frac {5}{8} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {a \left (15 a^{2}+15 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )}-\frac {a \left (15 a^{2}+15 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )}+\left (-\frac {5}{8} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {2 \left (3 a^{2}+12 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8 a +8 b}}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(335\) |
| default | \(\frac {\frac {\frac {2 \left (\left (-\frac {5}{8} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {a \left (15 a^{2}+15 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )}-\frac {a \left (15 a^{2}+15 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )}+\left (-\frac {5}{8} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {2 \left (3 a^{2}+12 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8 a +8 b}}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(335\) |
| risch | \(\frac {x}{a^{3}}+\frac {5 a^{3} {\mathrm e}^{6 d x +6 c}+20 a^{2} b \,{\mathrm e}^{6 d x +6 c}+16 a \,b^{2} {\mathrm e}^{6 d x +6 c}+15 a^{3} {\mathrm e}^{4 d x +4 c}+58 a^{2} b \,{\mathrm e}^{4 d x +4 c}+88 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 \,{\mathrm e}^{4 d x +4 c} b^{3}+15 a^{3} {\mathrm e}^{2 d x +2 c}+44 a^{2} b \,{\mathrm e}^{2 d x +2 c}+32 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+5 a^{3}+6 a^{2} b}{4 a^{3} d \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, \left (a +b \right ) d a}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) b}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) b^{2}}{2 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, \left (a +b \right ) d a}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) b}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) b^{2}}{2 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,a^{3}}\) | \(699\) |
1/d*(2/a^3*(((-5/8*a^2-1/2*a*b)*tanh(1/2*d*x+1/2*c)^7-1/8*a*(15*a^2+15*a*b -4*b^2)/(a+b)*tanh(1/2*d*x+1/2*c)^5-1/8*a*(15*a^2+15*a*b-4*b^2)/(a+b)*tanh (1/2*d*x+1/2*c)^3+(-5/8*a^2-1/2*a*b)*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*(3*a^2+12*a*b+8*b^2)/(a+b)*(-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))))-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/a ^3*ln(1+tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 3459 vs. \(2 (131) = 262\).
Time = 0.36 (sec) , antiderivative size = 7158, normalized size of antiderivative = 51.50 \[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\tanh ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1255 vs. \(2 (131) = 262\).
Time = 0.43 (sec) , antiderivative size = 1255, normalized size of antiderivative = 9.03 \[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-1/64*(3*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*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^5 + 2*a^4*b + a^3*b^2)*sqrt((a + b)*b)*d) + 1/64*(3*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*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^5 + 2*a^4*b + a^3 *b^2)*sqrt((a + b)*b)*d) + 1/16*(5*a^4 + 20*a^3*b + 12*a^2*b^2 + (5*a^4 + 66*a^3*b + 128*a^2*b^2 + 64*a*b^3)*e^(6*d*x + 6*c) + (15*a^4 + 164*a^3*b + 460*a^2*b^2 + 512*a*b^3 + 192*b^4)*e^(4*d*x + 4*c) + (15*a^4 + 118*a^3*b + 208*a^2*b^2 + 96*a*b^3)*e^(2*d*x + 2*c))/((a^7 + 2*a^6*b + a^5*b^2 + (a^ 7 + 2*a^6*b + a^5*b^2)*e^(8*d*x + 8*c) + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2* a^4*b^3)*e^(6*d*x + 6*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*e^(4*d*x + 4*c) + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^ (2*d*x + 2*c))*d) - 1/16*(5*a^4 + 20*a^3*b + 12*a^2*b^2 + (15*a^4 + 118*a^ 3*b + 208*a^2*b^2 + 96*a*b^3)*e^(-2*d*x - 2*c) + (15*a^4 + 164*a^3*b + 460 *a^2*b^2 + 512*a*b^3 + 192*b^4)*e^(-4*d*x - 4*c) + (5*a^4 + 66*a^3*b + 128 *a^2*b^2 + 64*a*b^3)*e^(-6*d*x - 6*c))/((a^7 + 2*a^6*b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*e^(-4*d*x - 4*c) + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-6*d*x - 6*c) + (a^7 + 2*a^6*b + a^5*b^2)*e^( -8*d*x - 8*c))*d) - 1/8*(5*a^3 + 2*a^2*b + (15*a^3 + 32*a^2*b + 8*a*b^2...
\[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\tanh \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\tanh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,\left ({\mathrm {cosh}\left (c+d\,x\right )}^2-1\right )}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]