Integrand size = 23, antiderivative size = 140 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {(a+5 b) x}{2 (a+b)^3}+\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^3 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-b) b \tanh (c+d x)}{2 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
1/2*(a+5*b)*x/(a+b)^3+1/2*b^(3/2)*(5*a+b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/ 2))/a^(3/2)/(a+b)^3/d+1/2*cosh(d*x+c)*sinh(d*x+c)/(a+b)/d/(a+b*tanh(d*x+c) ^2)-1/2*(a-b)*b*tanh(d*x+c)/a/(a+b)^2/d/(a+b*tanh(d*x+c)^2)
Time = 0.90 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {2 (a+5 b) (c+d x)+\frac {2 b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+(a+b) \sinh (2 (c+d x))+\frac {2 b^2 (a+b) \sinh (2 (c+d x))}{a (a-b+(a+b) \cosh (2 (c+d x)))}}{4 (a+b)^3 d} \] Input:
Integrate[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(2*(a + 5*b)*(c + d*x) + (2*b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x ])/Sqrt[a]])/a^(3/2) + (a + b)*Sinh[2*(c + d*x)] + (2*b^2*(a + b)*Sinh[2*( c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(4*(a + b)^3*d)
Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4158, 316, 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.
| \(\displaystyle \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4158 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \tanh ^2(c+d x)+a+2 b}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (a^2+4 b a+b^2+(a-b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (a-b) \tanh (c+d x)}{a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^2+4 b a+b^2+(a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{a (a+b)}-\frac {b (a-b) \tanh (c+d x)}{a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {\frac {b^2 (5 a+b) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}+\frac {a (a+5 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}}{a (a+b)}-\frac {b (a-b) \tanh (c+d x)}{a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (a+5 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{a (a+b)}-\frac {b (a-b) \tanh (c+d x)}{a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {b^{3/2} (5 a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {a (a+5 b) \text {arctanh}(\tanh (c+d x))}{a+b}}{a (a+b)}-\frac {b (a-b) \tanh (c+d x)}{a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
Input:
Int[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(Tanh[c + d*x]/(2*(a + b)*(1 - Tanh[c + d*x]^2)*(a + b*Tanh[c + d*x]^2)) + (((b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + (a*(a + 5*b)*ArcTanh[Tanh[c + d*x]])/(a + b))/(a*(a + b)) - ((a - b)*b*Tanh[c + d*x])/(a*(a + b)*(a + b*Tanh[c + d*x]^2)))/(2*(a + b)))/d
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[sec[(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[ff/(c^(m - 1)*f) Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^ p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && I ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(124)=248\).
Time = 9.18 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.76
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {2 b^{2} \left (\frac {-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (5 a +b \right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\) | \(386\) |
| default | \(\frac {\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {2 b^{2} \left (\frac {-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (5 a +b \right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\) | \(386\) |
| risch | \(\frac {x a}{2 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {5 x b}{2 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {b^{2} \left ({\mathrm e}^{2 d x +2 c} a -{\mathrm e}^{2 d x +2 c} b +a +b \right )}{d \left (a +b \right )^{3} a \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}+\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{4 a \left (a +b \right )^{3} d}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{4 a^{2} \left (a +b \right )^{3} d}-\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{4 a \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{4 a^{2} \left (a +b \right )^{3} d}\) | \(399\) |
Input:
int(cosh(d*x+c)^2/(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c )-1)+1/2/(a+b)^3*(-a-5*b)*ln(tanh(1/2*d*x+1/2*c)-1)-1/2/(a+b)^2/(tanh(1/2* d*x+1/2*c)+1)^2+1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)+1/2*(a+5*b)/(a+b)^3*ln (tanh(1/2*d*x+1/2*c)+1)-2/(a+b)^3*b^2*((-1/2*(a+b)/a*tanh(1/2*d*x+1/2*c)^3 -1/2*(a+b)/a*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+ 1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/2*(5*a+b)*(1/2*(a+((a+b)*b)^(1/2 )+b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1 /2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))-1/2*(-a+((a+b)*b)^(1/2) -b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1 /2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 2001 vs. \(2 (124) = 248\).
Time = 0.18 (sec) , antiderivative size = 4324, normalized size of antiderivative = 30.89 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**2/(a+b*tanh(d*x+c)**2)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (124) = 248\).
Time = 0.28 (sec) , antiderivative size = 840, normalized size of antiderivative = 6.00 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (124) = 248\).
Time = 0.74 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.97 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {12 \, {\left (d x + c\right )} {\left (a + 5 \, b\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {12 \, {\left (5 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} + \frac {3 \, e^{\left (2 \, d x + 2 \, c\right )}}{a^{2} + 2 \, a b + b^{2}} - \frac {2 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 12 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 10 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 7 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 22 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 7 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 24 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 28 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 6 \, a^{2} b + 3 \, a b^{2}}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 2 \, a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, b e^{\left (4 \, d x + 4 \, c\right )} + a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )}\right )}}}{24 \, d} \] Input:
integrate(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/24*(12*(d*x + c)*(a + 5*b)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 12*(5*a*b^2 + b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + 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)) + 3*e^(2*d*x + 2*c)/(a ^2 + 2*a*b + b^2) - (2*a^3*e^(6*d*x + 6*c) + 12*a^2*b*e^(6*d*x + 6*c) + 10 *a*b^2*e^(6*d*x + 6*c) + 7*a^3*e^(4*d*x + 4*c) + 22*a^2*b*e^(4*d*x + 4*c) + 7*a*b^2*e^(4*d*x + 4*c) - 24*b^3*e^(4*d*x + 4*c) + 8*a^3*e^(2*d*x + 2*c) + 12*a^2*b*e^(2*d*x + 2*c) + 28*a*b^2*e^(2*d*x + 2*c) + 24*b^3*e^(2*d*x + 2*c) + 3*a^3 + 6*a^2*b + 3*a*b^2)/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*(a *e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c) + 2*a*e^(4*d*x + 4*c) - 2*b*e^(4*d*x + 4*c) + a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c))))/d
Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:
int(cosh(c + d*x)^2/(a + b*tanh(c + d*x)^2)^2,x)
Output:
int(cosh(c + d*x)^2/(a + b*tanh(c + d*x)^2)^2, x)
Time = 0.29 (sec) , antiderivative size = 1269, normalized size of antiderivative = 9.06 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x)
Output:
(20*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt (b))/sqrt(a))*a**2*b + 24*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d *x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**2 + 4*e**(6*c + 6*d*x)*sqrt(b)*sq rt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**3 + 40*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a ))*a**2*b - 32*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**2 - 8*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan( (e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**3 + 20*e**(2*c + 2*d*x)*s qrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b + 24*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt (b))/sqrt(a))*a*b**2 + 4*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d* x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**3 - 20*e**(6*c + 6*d*x)*sqrt(b)*sqrt (a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a**2*b - 24*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a ))*a*b**2 - 4*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b**3 - 40*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e **(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a**2*b + 32*e**(4*c + 4*d*x)*s qrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a*b**2 + 8*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt( b))/sqrt(a))*b**3 - 20*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d...