Integrand size = 23, antiderivative size = 117 \[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\left (3 a^2-4 a b+8 b^2\right ) x}{8 a^3}-\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b} d}+\frac {(3 a-4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d} \]
1/8*(3*a^2-4*a*b+8*b^2)*x/a^3+1/8*(3*a-4*b)*cosh(d*x+c)*sinh(d*x+c)/a^2/d+ 1/4*cosh(d*x+c)^3*sinh(d*x+c)/a/d-b^(5/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b )^(1/2))/a^3/d/(a+b)^(1/2)
Time = 0.84 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {4 \left (3 a^2-4 a b+8 b^2\right ) (c+d x)-\frac {32 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+8 a (a-b) \sinh (2 (c+d x))+a^2 \sinh (4 (c+d x))}{32 a^3 d} \]
(4*(3*a^2 - 4*a*b + 8*b^2)*(c + d*x) - (32*b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/Sqrt[a + b] + 8*a*(a - b)*Sinh[2*(c + d*x)] + a^2*S inh[4*(c + d*x)])/(32*a^3*d)
Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4634, 316, 402, 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.
| \(\displaystyle \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^4 \left (a+b \sec (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4634 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^3 \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b \tanh ^2(c+d x)+3 a-b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{4 a}+\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 a^2-b a+4 b^2-(3 a-4 b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a}+\frac {(3 a-4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}}{4 a}+\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2-4 a b+8 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {8 b^3 \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a}+\frac {(3 a-4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}}{4 a}+\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {8 b^3 \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a}+\frac {(3 a-4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}}{4 a}+\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {8 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a}+\frac {(3 a-4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}}{4 a}+\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
(Tanh[c + d*x]/(4*a*(1 - Tanh[c + d*x]^2)^2) + ((((3*a^2 - 4*a*b + 8*b^2)* ArcTanh[Tanh[c + d*x]])/a - (8*b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqr t[a + b]])/(a*Sqrt[a + b]))/(2*a) + ((3*a - 4*b)*Tanh[c + d*x])/(2*a*(1 - Tanh[c + d*x]^2)))/(4*a))/d
3.1.73.3.1 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[((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_.)*sec[(e_.) + (f_.)*(x_)]^(n_) )^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ [m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(103)=206\).
Time = 1.17 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\frac {3 x}{8 a}-\frac {b x}{2 a^{2}}+\frac {x \,b^{2}}{a^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 d a}+\frac {{\mathrm e}^{2 d x +2 c}}{8 d a}-\frac {{\mathrm e}^{2 d x +2 c} b}{8 d \,a^{2}}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 d a}+\frac {{\mathrm e}^{-2 d x -2 c} b}{8 d \,a^{2}}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 d a}+\frac {\sqrt {\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{2 \left (a +b \right ) d \,a^{3}}-\frac {\sqrt {\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{2 \left (a +b \right ) d \,a^{3}}\) | \(234\) |
| derivativedivides | \(\frac {\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+4 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}+\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}-\frac {1}{4 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-5 a +4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {7 a -4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-4 a b +8 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3}}}{d}\) | \(347\) |
| default | \(\frac {\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+4 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}+\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}-\frac {1}{4 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-5 a +4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {7 a -4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-4 a b +8 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3}}}{d}\) | \(347\) |
3/8*x/a-1/2*b*x/a^2+x/a^3*b^2+1/64/d/a*exp(4*d*x+4*c)+1/8/d/a*exp(2*d*x+2* c)-1/8/d/a^2*exp(2*d*x+2*c)*b-1/8/d/a*exp(-2*d*x-2*c)+1/8/d/a^2*exp(-2*d*x -2*c)*b-1/64/d/a*exp(-4*d*x-4*c)+1/2*((a+b)*b)^(1/2)/(a+b)*b^2/d/a^3*ln(ex p(2*d*x+2*c)+(2*((a+b)*b)^(1/2)+a+2*b)/a)-1/2*((a+b)*b)^(1/2)/(a+b)*b^2/d/ a^3*ln(exp(2*d*x+2*c)-(2*((a+b)*b)^(1/2)-a-2*b)/a)
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (103) = 206\).
Time = 0.29 (sec) , antiderivative size = 1713, normalized size of antiderivative = 14.64 \[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \]
[1/64*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sin h(d*x + c)^8 + 8*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^4 + 8*(a^2 - a* b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + 2*a^2 - 2*a*b)*sinh(d*x + c)^6 + 8*(7*a^2*cosh(d*x + c)^3 + 6*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 4*(3*a^2 - 4*a*b + 8*b^2)*d*x + 60*(a^2 - a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 4*(3 *a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c) + 20*(a^2 - a*b)*cosh(d*x + c)^3)* sinh(d*x + c)^3 - 8*(a^2 - a*b)*cosh(d*x + c)^2 + 4*(7*a^2*cosh(d*x + c)^6 + 12*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^2 + 30*(a^2 - a*b)*cosh(d* x + c)^4 - 2*a^2 + 2*a*b)*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 + b)) *log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh (d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c) ^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh(d*x + c)^4 + 4*a*co sh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c )^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x...
\[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\cosh ^{4}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (103) = 206\).
Time = 0.31 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.50 \[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {3 \, {\left (d x + c\right )}}{8 \, a d} - \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, a^{2} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} - \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} {\left (d x + c\right )}}{2 \, a^{3} d} + \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{2} d} + \frac {{\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a^{3} d} \]
3/16*b*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)))/(sqrt((a + b)*b)*a*d) + 3/8*(d*x + c)/(a*d) - 1/64*(8*b*e^(-2*d*x - 2*c) - a)*e^(4*d*x + 4*c)/(a^2*d) + 1/8* e^(2*d*x + 2*c)/(a*d) - 1/8*e^(-2*d*x - 2*c)/(a*d) - 1/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^2*d) + 1/4*b*log(2*(a + 2*b)* e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^2*d) + 1/8*(a*b + 2*b^2)*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)))/(sqrt((a + b)*b)*a^2*d) - 1/8*(a*b + 2*b^2)*lo g((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)))/(sqrt((a + b)*b)*a^2*d) + 1/2*(a*b + 2*b^2) *(d*x + c)/(a^3*d) + 1/64*(8*b*e^(-2*d*x - 2*c) - a*e^(-4*d*x - 4*c))/(a^2 *d) + 1/16*(a^2*b + 8*a*b^2 + 8*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)))/(sqr t((a + b)*b)*a^3*d)
\[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{4}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.61 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.22 \[ \int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {x\,\left (3\,a^2-4\,a\,b+8\,b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,a\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a-b\right )}{8\,a^2\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-b\right )}{8\,a^2\,d}+\frac {b^{5/2}\,\ln \left (\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^4}-\frac {2\,b^{5/2}\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,d\,\sqrt {a+b}}\right )}{2\,a^3\,d\,\sqrt {a+b}}-\frac {b^{5/2}\,\ln \left (\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^4}+\frac {2\,b^{5/2}\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,d\,\sqrt {a+b}}\right )}{2\,a^3\,d\,\sqrt {a+b}} \]
(x*(3*a^2 - 4*a*b + 8*b^2))/(8*a^3) - exp(- 4*c - 4*d*x)/(64*a*d) + exp(4* c + 4*d*x)/(64*a*d) - (exp(- 2*c - 2*d*x)*(a - b))/(8*a^2*d) + (exp(2*c + 2*d*x)*(a - b))/(8*a^2*d) + (b^(5/2)*log((4*b^3*exp(2*c + 2*d*x))/a^4 - (2 *b^(5/2)*(a*d + a*d*exp(2*c + 2*d*x) + 2*b*d*exp(2*c + 2*d*x)))/(a^4*d*(a + b)^(1/2))))/(2*a^3*d*(a + b)^(1/2)) - (b^(5/2)*log((4*b^3*exp(2*c + 2*d* x))/a^4 + (2*b^(5/2)*(a*d + a*d*exp(2*c + 2*d*x) + 2*b*d*exp(2*c + 2*d*x)) )/(a^4*d*(a + b)^(1/2))))/(2*a^3*d*(a + b)^(1/2))