Integrand size = 23, antiderivative size = 132 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {(a-3 b) x}{2 (a+b)^3}-\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} (a+b)^3 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac {b \tanh (c+d x)}{(a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]
-1/2*(a-3*b)*x/(a+b)^3-1/2*(3*a-b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))*b^( 1/2)/(a+b)^3/d/a^(1/2)+1/2*cosh(d*x+c)*sinh(d*x+c)/(a+b)/d/(a+b*tanh(d*x+c )^2)-b*tanh(d*x+c)/(a+b)^2/d/(a+b*tanh(d*x+c)^2)
Time = 1.50 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {-2 (a-3 b) (c+d x)+\frac {2 \sqrt {b} (-3 a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}+(a+b) \sinh (2 (c+d x))-\frac {2 b (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}}{4 (a+b)^3 d} \]
(-2*(a - 3*b)*(c + d*x) + (2*Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d *x])/Sqrt[a]])/Sqrt[a] + (a + b)*Sinh[2*(c + d*x)] - (2*b*(a + b)*Sinh[2*( c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(4*(a + b)^3*d)
Time = 0.35 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 25, 4146, 373, 402, 27, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\int \frac {a-3 b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {2 b \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\int -\frac {2 a \left (-2 b \tanh ^2(c+d x)+a-b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {\int \frac {-2 b \tanh ^2(c+d x)+a-b}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{a+b}+\frac {2 b \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {(a-3 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {b (3 a-b) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}}{a+b}+\frac {2 b \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {(a-3 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{a+b}+\frac {2 b \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {(a-3 b) \text {arctanh}(\tanh (c+d x))}{a+b}}{a+b}+\frac {2 b \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}}{d}\) |
(Tanh[c + d*x]/(2*(a + b)*(1 - Tanh[c + d*x]^2)*(a + b*Tanh[c + d*x]^2)) - ((((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ((a - 3*b)*ArcTanh[Tanh[c + d*x]])/(a + b))/(a + b) + (2*b*Tanh[c + d*x])/((a + b)*(a + b*Tanh[c + d*x]^2)))/(2*(a + b)))/d
3.1.35.3.1 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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((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[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(380\) vs. \(2(118)=236\).
Time = 4.50 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.89
| method | result | size |
| risch | \(-\frac {x a}{2 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {3 x b}{2 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left ({\mathrm e}^{2 d x +2 c} a -b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}{d \left (a +b \right )^{3} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{4 \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{4 a \left (a +b \right )^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{4 \left (a +b \right )^{3} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{4 a \left (a +b \right )^{3} d}\) | \(381\) |
| derivativedivides | \(\frac {-\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (-a +3 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{3}}+\frac {4 b \left (\frac {\left (-\frac {a}{4}-\frac {b}{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{4}-\frac {b}{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}+\frac {\left (3 a -b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{4}\right )}{\left (a +b \right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(387\) |
| default | \(\frac {-\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (-a +3 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{3}}+\frac {4 b \left (\frac {\left (-\frac {a}{4}-\frac {b}{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{4}-\frac {b}{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}+\frac {\left (3 a -b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{4}\right )}{\left (a +b \right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(387\) |
-1/2*x/(a+b)/(a^2+2*a*b+b^2)*a+3/2*x/(a+b)/(a^2+2*a*b+b^2)*b+1/8/d/(a^2+2* a*b+b^2)*exp(2*d*x+2*c)-1/8/d/(a^2+2*a*b+b^2)*exp(-2*d*x-2*c)+b*(exp(2*d*x +2*c)*a-b*exp(2*d*x+2*c)+a+b)/d/(a+b)^3/(a*exp(4*d*x+4*c)+b*exp(4*d*x+4*c) +2*exp(2*d*x+2*c)*a-2*b*exp(2*d*x+2*c)+a+b)+3/4*(-a*b)^(1/2)/(a+b)^3/d*ln( exp(2*d*x+2*c)-(2*(-a*b)^(1/2)-a+b)/(a+b))-1/4/a*(-a*b)^(1/2)/(a+b)^3/d*ln (exp(2*d*x+2*c)-(2*(-a*b)^(1/2)-a+b)/(a+b))*b-3/4*(-a*b)^(1/2)/(a+b)^3/d*l n(exp(2*d*x+2*c)+(2*(-a*b)^(1/2)+a-b)/(a+b))+1/4/a*(-a*b)^(1/2)/(a+b)^3/d* ln(exp(2*d*x+2*c)+(2*(-a*b)^(1/2)+a-b)/(a+b))*b
Leaf count of result is larger than twice the leaf count of optimal. 1798 vs. \(2 (118) = 236\).
Time = 0.34 (sec) , antiderivative size = 3918, normalized size of antiderivative = 29.68 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^6 - 2*(2*(a^2 - 2*a*b - 3*b^ 2)*d*x - 14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - a^2 + b^2)*sinh(d*x + c) ^6 + 4*(14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(2*(a^2 - 2*a*b - 3*b^2 )*d*x - a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*((a^2 - 4*a*b + 3*b^ 2)*d*x - a*b + b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 - 4*(a^2 - 4*a*b + 3*b^2)*d*x - 15*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^ 2 + b^2)*cosh(d*x + c)^2 + 4*a*b - 4*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 2* a*b + b^2)*cosh(d*x + c)^5 - 5*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*c osh(d*x + c)^3 - 4*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c))* sinh(d*x + c)^3 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x + a^2 - 4*a*b - 5*b^2)*co sh(d*x + c)^2 + 2*(14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 15*(2*(a^2 - 2 *a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*d *x - 24*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^2 - a^2 + 4* a*b + 5*b^2)*sinh(d*x + c)^2 - 2*((3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^6 + 6*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^2 + 2*a*b - b ^2)*sinh(d*x + c)^6 + 2*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^4 + (15*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^2 + 6*a^2 - 8*a*b + 2*b^2)*sinh(d*x + c)^4 + 4*(5*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^3 + 2*(3*a^2 - 4*a*b + b^2)*c...
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (118) = 236\).
Time = 0.40 (sec) , antiderivative size = 840, normalized size of antiderivative = 6.36 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {b \log \left ({\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a - b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} - \frac {b \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} - \frac {{\left (3 \, a^{2} b - 6 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b} d} + \frac {{\left (3 \, a^{2} b - 6 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b} d} + \frac {{\left (3 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{4 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b} d} + \frac {a^{2} b - b^{3} + {\left (a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{4 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{2} b^{3} - a b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} - \frac {a^{2} b - b^{3} + {\left (a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{2} b^{3} - a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {a b + b^{2} + {\left (a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + 2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {d x + c}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} \]
1/2*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a ^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/2*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/ 8*(3*a^2*b - 6*a*b^2 - b^3)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/s qrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)*d) + 1/8*(3*a^2*b - 6*a*b^2 - b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b)) /((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)*d) + 1/4*(3*a*b + b^2)*arc tan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^3 + 2*a^2*b + a* b^2)*sqrt(a*b)*d) + 1/4*(a^2*b - b^3 + (a^2*b - 6*a*b^2 + b^3)*e^(2*d*x + 2*c))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + (a^5 + 4*a^4*b + 6 *a^3*b^2 + 4*a^2*b^3 + a*b^4)*e^(4*d*x + 4*c) + 2*(a^5 + 2*a^4*b - 2*a^2*b ^3 - a*b^4)*e^(2*d*x + 2*c))*d) - 1/4*(a^2*b - b^3 + (a^2*b - 6*a*b^2 + b^ 3)*e^(-2*d*x - 2*c))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + 2*( a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4)*e^(-2*d*x - 2*c) + (a^5 + 4*a^4*b + 6*a ^3*b^2 + 4*a^2*b^3 + a*b^4)*e^(-4*d*x - 4*c))*d) - 1/2*(a*b + b^2 + (a*b - b^2)*e^(-2*d*x - 2*c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + 2*(a^4 + a^3 *b - a^2*b^2 - a*b^3)*e^(-2*d*x - 2*c) + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^ 3)*e^(-4*d*x - 4*c))*d) - 1/2*(d*x + c)/((a^2 + 2*a*b + b^2)*d) + 1/8*e^(2 *d*x + 2*c)/((a^2 + 2*a*b + b^2)*d) - 1/8*e^(-2*d*x - 2*c)/((a^2 + 2*a*b + b^2)*d)
\[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]