Integrand size = 23, antiderivative size = 120 \[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\left (3 a^2+10 a b+15 b^2\right ) x}{8 (a+b)^3}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3 d}+\frac {(3 a+7 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d} \]
1/8*(3*a^2+10*a*b+15*b^2)*x/(a+b)^3+1/8*(3*a+7*b)*cosh(d*x+c)*sinh(d*x+c)/ (a+b)^2/d+1/4*cosh(d*x+c)^3*sinh(d*x+c)/(a+b)/d+b^(5/2)*arctan(b^(1/2)*tan h(d*x+c)/a^(1/2))/(a+b)^3/d/a^(1/2)
Time = 0.95 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\left (3 a^2+10 a b+15 b^2\right ) (c+d x)}{8 (a+b)^3 d}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3 d}+\frac {(a+2 b) \sinh (2 (c+d x))}{4 (a+b)^2 d}+\frac {\sinh (4 (c+d x))}{32 (a+b) d} \]
((3*a^2 + 10*a*b + 15*b^2)*(c + d*x))/(8*(a + b)^3*d) + (b^(5/2)*ArcTan[(S qrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3*d) + ((a + 2*b)*Sinh[2* (c + d*x)])/(4*(a + b)^2*d) + Sinh[4*(c + d*x)]/(32*(a + b)*d)
Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4158, 316, 402, 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.
| \(\displaystyle \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4158 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^3 \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \tanh ^2(c+d x)+3 a+4 b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{4 (a+b)}+\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 a^2+7 b a+8 b^2+b (3 a+7 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 (a+b)}+\frac {(3 a+7 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right )}}{4 (a+b)}+\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2+10 a b+15 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {8 b^3 \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}}{2 (a+b)}+\frac {(3 a+7 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right )}}{4 (a+b)}+\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2+10 a b+15 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{2 (a+b)}+\frac {(3 a+7 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right )}}{4 (a+b)}+\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a+b}+\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{2 (a+b)}+\frac {(3 a+7 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right )}}{4 (a+b)}+\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
(Tanh[c + d*x]/(4*(a + b)*(1 - Tanh[c + d*x]^2)^2) + (((8*b^(5/2)*ArcTan[( Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ((3*a^2 + 10*a*b + 15 *b^2)*ArcTanh[Tanh[c + d*x]])/(a + b))/(2*(a + b)) + ((3*a + 7*b)*Tanh[c + d*x])/(2*(a + b)*(1 - Tanh[c + d*x]^2)))/(4*(a + b)))/d
3.2.5.3.1 Defintions of rubi rules used
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. \(272\) vs. \(2(106)=212\).
Time = 11.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.28
| method | result | size |
| risch | \(\frac {3 a^{2} x}{8 \left (a +b \right )^{3}}+\frac {5 a x b}{4 \left (a +b \right )^{3}}+\frac {15 x \,b^{2}}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 d \left (a +b \right )}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{2 d x +2 c} b}{4 \left (a +b \right )^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{4 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 d \left (a +b \right )}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{2 a \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 a \left (a +b \right )^{3} d}\) | \(273\) |
| derivativedivides | \(\frac {\frac {1}{2 \left (2 a +2 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a -11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {1}{2 \left (2 a +2 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {7 a +11 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{3}}-\frac {2 b^{3} a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(438\) |
| default | \(\frac {\frac {1}{2 \left (2 a +2 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a -11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {1}{2 \left (2 a +2 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {7 a +11 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{3}}-\frac {2 b^{3} a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(438\) |
3/8*a^2*x/(a+b)^3+5/4*a*x/(a+b)^3*b+15/8*x/(a+b)^3*b^2+1/64/d/(a+b)*exp(4* d*x+4*c)+1/8/(a+b)^2/d*exp(2*d*x+2*c)*a+1/4/(a+b)^2/d*exp(2*d*x+2*c)*b-1/8 /(a^2+2*a*b+b^2)/d*exp(-2*d*x-2*c)*a-1/4/(a^2+2*a*b+b^2)/d*exp(-2*d*x-2*c) *b-1/64/d/(a+b)*exp(-4*d*x-4*c)+1/2/a*(-a*b)^(1/2)*b^2/(a+b)^3/d*ln(exp(2* d*x+2*c)+(2*(-a*b)^(1/2)+a-b)/(a+b))-1/2/a*(-a*b)^(1/2)*b^2/(a+b)^3/d*ln(e xp(2*d*x+2*c)-(2*(-a*b)^(1/2)-a+b)/(a+b))
Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (106) = 212\).
Time = 0.30 (sec) , antiderivative size = 2180, normalized size of antiderivative = 18.17 \[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
[1/64*((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 + 8*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^4 + 8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 + 6*a*b + 4*b^2)*s inh(d*x + c)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 6*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2 + 2*a*b + b^2)*cosh( d*x + c)^4 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x + 60*(a^2 + 3*a*b + 2*b^2)*co sh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c) + 20*(a^2 + 3*a*b + 2*b^2) *cosh(d*x + c)^3)*sinh(d*x + c)^3 - 8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^ 2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 12*(3*a^2 + 10*a*b + 15*b^2 )*d*x*cosh(d*x + c)^2 + 30*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^4 - 2*a^2 - 6*a*b - 4*b^2)*sinh(d*x + c)^2 + 32*(b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^2*cosh (d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4)*sqrt(-b/a)*log(((a^2 + 2* a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c) ^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2 )*cosh(d*x + c))*sinh(d*x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^...
\[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\cosh ^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (106) = 212\).
Time = 0.33 (sec) , antiderivative size = 514, normalized size of antiderivative = 4.28 \[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {{\left (a b - b^{2}\right )} {\left (d x + c\right )}}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \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 )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\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 )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {{\left (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^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {{\left (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^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b} d} + \frac {{\left (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^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {3 \, b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} {\left (a + b\right )} d} - \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {3 \, {\left (d x + c\right )}}{8 \, {\left (a + b\right )} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, {\left (a + b\right )} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, {\left (a + b\right )} d} \]
-1/2*(a*b - b^2)*(d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 1/64*(8*b *e^(-2*d*x - 2*c) + a + b)*e^(4*d*x + 4*c)/((a^2 + 2*a*b + b^2)*d) + 1/4*b *log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^2 + 2*a*b + b^2)*d) - 1/4*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^2 + 2*a*b + b^2)*d) - 1/4*(a*b - b^2)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b)*d) - 1/8*(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^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a*b)*d) + 1/4*(a*b - b ^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b)*d) - 3/8*b*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b) /sqrt(a*b))/(sqrt(a*b)*(a + b)*d) - 1/64*(8*b*e^(-2*d*x - 2*c) + (a + b)*e ^(-4*d*x - 4*c))/((a^2 + 2*a*b + b^2)*d) + 3/8*(d*x + c)/((a + b)*d) + 1/8 *e^(2*d*x + 2*c)/((a + b)*d) - 1/8*e^(-2*d*x - 2*c)/((a + b)*d)
\[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{4}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.50 (sec) , antiderivative size = 967, normalized size of antiderivative = 8.06 \[ \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {x\,\left (3\,a^2+10\,a\,b+15\,b^2\right )}{8\,{\left (a+b\right )}^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d\,\left (a+b\right )}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d\,\left (a+b\right )}+\frac {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {4\,b^3}{d\,{\left (a+b\right )}^5\,\sqrt {b^5}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {\left (a-b\right )\,\left (a^4\,d\,\sqrt {b^5}-b^4\,d\,\sqrt {b^5}-2\,a\,b^3\,d\,\sqrt {b^5}+2\,a^3\,b\,d\,\sqrt {b^5}\right )}{b^3\,{\left (a+b\right )}^2\,\sqrt {a\,d^2\,{\left (a+b\right )}^6}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}\right )+\frac {\left (a-b\right )\,\left (a^4\,d\,\sqrt {b^5}+b^4\,d\,\sqrt {b^5}+4\,a\,b^3\,d\,\sqrt {b^5}+4\,a^3\,b\,d\,\sqrt {b^5}+6\,a^2\,b^2\,d\,\sqrt {b^5}\right )}{b^3\,{\left (a+b\right )}^2\,\sqrt {a\,d^2\,{\left (a+b\right )}^6}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}\right )\,\left (\frac {a^4\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}{2}+\frac {b^4\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}{2}+2\,a\,b^3\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}+2\,a^3\,b\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}+3\,a^2\,b^2\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}\right )\right )\,\sqrt {b^5}}{\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a+2\,b\right )}{8\,d\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{8\,d\,{\left (a+b\right )}^2} \]
(x*(10*a*b + 3*a^2 + 15*b^2))/(8*(a + b)^3) - exp(- 4*c - 4*d*x)/(64*d*(a + b)) + exp(4*c + 4*d*x)/(64*d*(a + b)) + (atan((exp(2*c)*exp(2*d*x)*((4*b ^3)/(d*(a + b)^5*(b^5)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + ((a - b)*( a^4*d*(b^5)^(1/2) - b^4*d*(b^5)^(1/2) - 2*a*b^3*d*(b^5)^(1/2) + 2*a^3*b*d* (b^5)^(1/2)))/(b^3*(a + b)^2*(a*d^2*(a + b)^6)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4 *d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2))) + ((a - b)*(a^4*d*(b^5)^(1 /2) + b^4*d*(b^5)^(1/2) + 4*a*b^3*d*(b^5)^(1/2) + 4*a^3*b*d*(b^5)^(1/2) + 6*a^2*b^2*d*(b^5)^(1/2)))/(b^3*(a + b)^2*(a*d^2*(a + b)^6)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2)))*((a^4*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d ^2 + 15*a^5*b^2*d^2)^(1/2))/2 + (b^4*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2))/2 + 2*a*b^3*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4 *d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) + 2*a^3*b*(a^7*d^2 + a*b^6*d ^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^ 5*b^2*d^2)^(1/2) + 3*a^2*b^2*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^ 5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2)))*(b^5)^(1 /2))/(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^...