Integrand size = 23, antiderivative size = 198 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {(a+7 b) x}{2 (a+b)^4}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \]
1/2*(a+7*b)*x/(a+b)^4+1/8*b^(3/2)*(35*a^2+14*a*b+3*b^2)*arctan(b^(1/2)*tan h(d*x+c)/a^(1/2))/a^(5/2)/(a+b)^4/d+1/2*cosh(d*x+c)*sinh(d*x+c)/(a+b)/d/(a +b*tanh(d*x+c)^2)^2-1/4*(2*a-b)*b*tanh(d*x+c)/a/(a+b)^2/d/(a+b*tanh(d*x+c) ^2)^2-1/8*(a-3*b)*b*(4*a+b)*tanh(d*x+c)/a^2/(a+b)^3/d/(a+b*tanh(d*x+c)^2)
Time = 2.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {4 (a+7 b) (c+d x)+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2}}+2 (a+b) \sinh (2 (c+d x))+\frac {4 b^3 (a+b) \sinh (2 (c+d x))}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {b^2 (a+b) (13 a+3 b) \sinh (2 (c+d x))}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{8 (a+b)^4 d} \]
(4*(a + 7*b)*(c + d*x) + (b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b ]*Tanh[c + d*x])/Sqrt[a]])/a^(5/2) + 2*(a + b)*Sinh[2*(c + d*x)] + (4*b^3* (a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (b^ 2*(a + b)*(13*a + 3*b)*Sinh[2*(c + d*x)])/(a^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^4*d)
Time = 0.47 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4158, 316, 402, 27, 402, 25, 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.
| \(\displaystyle \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4158 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {5 b \tanh ^2(c+d x)+a+2 b}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (2 a^2+8 b a+3 b^2+3 (2 a-b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 a^2+8 b a+3 b^2+3 (2 a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {4 a^3+24 b a^2+11 b^2 a+3 b^3+(a-3 b) b (4 a+b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {4 a^3+24 b a^2+11 b^2 a+3 b^3+(a-3 b) b (4 a+b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {b^2 \left (35 a^2+14 a b+3 b^2\right ) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}+\frac {4 a^2 (a+7 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {4 a^2 (a+7 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{2 a (a+b)}-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {4 a^2 (a+7 b) \text {arctanh}(\tanh (c+d x))}{a+b}}{2 a (a+b)}-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )}}{2 a (a+b)}-\frac {b (2 a-b) \tanh (c+d x)}{2 a (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}+\frac {\tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
(Tanh[c + d*x]/(2*(a + b)*(1 - Tanh[c + d*x]^2)*(a + b*Tanh[c + d*x]^2)^2) + (-1/2*((2*a - b)*b*Tanh[c + d*x])/(a*(a + b)*(a + b*Tanh[c + d*x]^2)^2) + (((b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqr t[a]])/(Sqrt[a]*(a + b)) + (4*a^2*(a + 7*b)*ArcTanh[Tanh[c + d*x]])/(a + b ))/(2*a*(a + b)) - ((a - 3*b)*b*(4*a + b)*Tanh[c + d*x])/(2*a*(a + b)*(a + b*Tanh[c + d*x]^2)))/(2*a*(a + b)))/(2*(a + b)))/d
3.2.25.3.1 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[((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. \(500\) vs. \(2(180)=360\).
Time = 27.89 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.53
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\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 \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\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 )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\) | \(501\) |
| default | \(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\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 \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\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 )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\) | \(501\) |
| risch | \(\frac {x a}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {7 x b}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {b^{2} \left (13 a^{3} {\mathrm e}^{6 d x +6 c}-a^{2} b \,{\mathrm e}^{6 d x +6 c}-17 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 \,{\mathrm e}^{6 d x +6 c} b^{3}+39 a^{3} {\mathrm e}^{4 d x +4 c}-17 a^{2} b \,{\mathrm e}^{4 d x +4 c}+33 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 \,{\mathrm e}^{4 d x +4 c} b^{3}+39 a^{3} {\mathrm e}^{2 d x +2 c}+13 a^{2} b \,{\mathrm e}^{2 d x +2 c}-35 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-9 \,{\mathrm e}^{2 d x +2 c} b^{3}+13 a^{3}+29 a^{2} b +19 a \,b^{2}+3 b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d \,a^{2}}+\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}+\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}+\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}-\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}-\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}-\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}\) | \(728\) |
1/d*(-1/2/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))^2+1/2/(a+b)^3/(1+tanh(1/2*d*x+1/ 2*c))+1/2*(a+7*b)/(a+b)^4*ln(1+tanh(1/2*d*x+1/2*c))+1/2/(a+b)^3/(tanh(1/2* d*x+1/2*c)-1)^2+1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+1/2/(a+b)^4*(-a-7*b)*l n(tanh(1/2*d*x+1/2*c)-1)-2*b^2/(a+b)^4*((-1/8*(13*a^2+18*a*b+5*b^2)/a*tanh (1/2*d*x+1/2*c)^7-1/8*(39*a^3+98*a^2*b+71*a*b^2+12*b^3)/a^2*tanh(1/2*d*x+1 /2*c)^5-1/8*(39*a^3+98*a^2*b+71*a*b^2+12*b^3)/a^2*tanh(1/2*d*x+1/2*c)^3-1/ 8*(13*a^2+18*a*b+5*b^2)/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*tanh(1/2*d*x+1/2*c)^2*b+a)^2+1/8/a*(35*a^2+14*a* b+3*b^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)*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. 6780 vs. \(2 (180) = 360\).
Time = 0.46 (sec) , antiderivative size = 13887, normalized size of antiderivative = 70.14 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (180) = 360\).
Time = 0.57 (sec) , antiderivative size = 1806, normalized size of antiderivative = 9.12 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
3/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a ^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 3/4*b*log(2*(a - b)*e^(-2*d *x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 3/32*(5*a^3*b - 15*a^2*b^2 - 5*a*b^3 - b^4)*arctan(1 /2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^ 2 + 4*a^3*b^3 + a^2*b^4)*sqrt(a*b)*d) + 3/32*(5*a^3*b - 15*a^2*b^2 - 5*a*b ^3 - b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*sqrt(a*b)*d) - 1/16*(15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b ))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) + 1/16*(9*a^4*b + 4 *a^3*b^2 - 22*a^2*b^3 - 20*a*b^4 - 3*b^5 + 3*(3*a^4*b - 22*a^3*b^2 - 20*a^ 2*b^3 + 6*a*b^4 + b^5)*e^(6*d*x + 6*c) + (27*a^4*b - 156*a^3*b^2 + 110*a^2 *b^3 - 36*a*b^4 - 9*b^5)*e^(4*d*x + 4*c) + (27*a^4*b - 86*a^3*b^2 - 84*a^2 *b^3 + 38*a*b^4 + 9*b^5)*e^(2*d*x + 2*c))/((a^8 + 6*a^7*b + 15*a^6*b^2 + 2 0*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6 + (a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(6*d*x + 6*c) + 2*(3*a^8 + 10*a^7*b + 13*a^6*b^2 + 12*a^5*b^3 + 13*a^4*b^4 + 10*a^3*b^5 + 3*a^2*b^6)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(2*d*x + 2*c))*d) - 1/16*(9*a^4*b + 4*a^3*b^2 - ...
\[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )^{2}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]