Integrand size = 14, antiderivative size = 207 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{16 a^4 (a+b)^{7/2} d}-\frac {b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac {b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:
x/a^4-1/16*b^(1/2)*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctanh(b^(1/2)*tanh( d*x+c)/(a+b)^(1/2))/a^4/(a+b)^(7/2)/d-1/6*b*tanh(d*x+c)/a/(a+b)/d/(a+b-b*t anh(d*x+c)^2)^3-1/24*b*(11*a+6*b)*tanh(d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(d* x+c)^2)^2-1/16*b*(19*a^2+22*a*b+8*b^2)*tanh(d*x+c)/a^3/(a+b)^3/d/(a+b-b*ta nh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 8.57 (sec) , antiderivative size = 1405, normalized size of antiderivative = 6.79 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Sech[c + d*x]^2)^(-4),x]
Output:
((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cosh[2*c + 2*d*x])^4 *Sech[c + d*x]^8*(((I/256)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[ a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*S qrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[ 2*c + d*x])]*Cosh[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]] ) - ((I/256)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b* Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c ] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*S inh[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3 *(a + b*Sech[c + d*x]^2)^4) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*S ech[c + d*x]^8*(480*a^6*d*x*Cosh[2*c] + 3168*a^5*b*d*x*Cosh[2*c] + 8928*a^ 4*b^2*d*x*Cosh[2*c] + 14112*a^3*b^3*d*x*Cosh[2*c] + 13248*a^2*b^4*d*x*Cosh [2*c] + 6912*a*b^5*d*x*Cosh[2*c] + 1536*b^6*d*x*Cosh[2*c] + 360*a^6*d*x*Co sh[2*d*x] + 2232*a^5*b*d*x*Cosh[2*d*x] + 5688*a^4*b^2*d*x*Cosh[2*d*x] + 72 72*a^3*b^3*d*x*Cosh[2*d*x] + 4608*a^2*b^4*d*x*Cosh[2*d*x] + 1152*a*b^5*d*x *Cosh[2*d*x] + 360*a^6*d*x*Cosh[4*c + 2*d*x] + 2232*a^5*b*d*x*Cosh[4*c + 2 *d*x] + 5688*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cosh[4*c + 2 *d*x] + 4608*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1152*a*b^5*d*x*Cosh[4*c + 2*d *x] + 144*a^6*d*x*Cosh[2*c + 4*d*x] + 720*a^5*b*d*x*Cosh[2*c + 4*d*x] + 12 96*a^4*b^2*d*x*Cosh[2*c + 4*d*x] + 1008*a^3*b^3*d*x*Cosh[2*c + 4*d*x] +...
Time = 0.46 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 4616, 316, 25, 402, 27, 402, 25, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sec (i c+i d x)^2\right )^4}dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^4}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {-\frac {\int -\frac {5 b \tanh ^2(c+d x)+6 a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 b \tanh ^2(c+d x)+6 a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (8 a^2+5 b a+2 b^2+b (11 a+6 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {8 a^2+5 b a+2 b^2+b (11 a+6 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int -\frac {16 a^3+29 b a^2+26 b^2 a+8 b^3+b \left (19 a^2+22 b a+8 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {16 a^3+29 b a^2+26 b^2 a+8 b^3+b \left (19 a^2+22 b a+8 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
Input:
Int[(a + b*Sech[c + d*x]^2)^(-4),x]
Output:
(-1/6*(b*Tanh[c + d*x])/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^3) + (-1/4* (b*(11*a + 6*b)*Tanh[c + d*x])/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + (3*(((16*(a + b)^3*ArcTanh[Tanh[c + d*x]])/a - (Sqrt[b]*(35*a^3 + 70*a^2* b + 56*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sq rt[a + b]))/(2*a*(a + b)) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tanh[c + d*x])/(2 *a*(a + b)*(a + b - b*Tanh[c + d*x]^2))))/(4*a*(a + b)))/(6*a*(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[(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[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(191)=382\).
Time = 4.16 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16 \left (a +b \right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{16 a^{3}+48 a^{2} b +48 a \,b^{2}+16 b^{3}}\right )}{a^{4}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}}{d}\) | \(545\) |
| default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16 \left (a +b \right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{16 a^{3}+48 a^{2} b +48 a \,b^{2}+16 b^{3}}\right )}{a^{4}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}}{d}\) | \(545\) |
| risch | \(\frac {x}{a^{4}}+\frac {b \left (116 a^{4} b +44 a^{3} b^{2}+435 a^{5} {\mathrm e}^{8 d x +8 c}+87 a^{5}+87 a^{5} {\mathrm e}^{10 d x +10 c}+1408 b^{5} {\mathrm e}^{6 d x +6 c}+870 a^{5} {\mathrm e}^{4 d x +4 c}+435 a^{5} {\mathrm e}^{2 d x +2 c}+870 a^{5} {\mathrm e}^{6 d x +6 c}+366 a^{4} b \,{\mathrm e}^{10 d x +10 c}+408 a^{3} b^{2} {\mathrm e}^{10 d x +10 c}+144 a^{2} b^{3} {\mathrm e}^{10 d x +10 c}+2124 a^{4} b \,{\mathrm e}^{8 d x +8 c}+384 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}+3792 a^{4} b \,{\mathrm e}^{4 d x +4 c}+6432 a^{3} b^{2} {\mathrm e}^{4 d x +4 c}+4608 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+1248 a \,b^{4} {\mathrm e}^{4 d x +4 c}+1374 a^{4} b \,{\mathrm e}^{2 d x +2 c}+1248 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}+3972 a^{3} b^{2} {\mathrm e}^{8 d x +8 c}+3072 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}+864 a \,b^{4} {\mathrm e}^{8 d x +8 c}+4292 a^{4} b \,{\mathrm e}^{6 d x +6 c}+8792 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}+9936 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}+5824 a \,b^{4} {\mathrm e}^{6 d x +6 c}\right )}{24 a^{4} \left (a +b \right )^{3} d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{3}}+\frac {35 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{32 \left (a +b \right )^{4} d a}+\frac {35 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right ) b}{16 \left (a +b \right )^{4} d \,a^{2}}+\frac {7 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right ) b^{2}}{4 \left (a +b \right )^{4} d \,a^{3}}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right ) b^{3}}{2 \left (a +b \right )^{4} d \,a^{4}}-\frac {35 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{32 \left (a +b \right )^{4} d a}-\frac {35 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right ) b}{16 \left (a +b \right )^{4} d \,a^{2}}-\frac {7 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right ) b^{2}}{4 \left (a +b \right )^{4} d \,a^{3}}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right ) b^{3}}{2 \left (a +b \right )^{4} d \,a^{4}}\) | \(872\) |
Input:
int(1/(a+b*sech(d*x+c)^2)^4,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a^4*ln(tanh(1/2*d*x+1/2*c)-1)+2*b/a^4*((-1/16*a*(29*a^2+26*a*b+8*b ^2)/(a+b)*tanh(1/2*d*x+1/2*c)^11-1/48*(435*a^3+281*a^2*b-66*a*b^2-72*b^3)* a/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-1/8*a*(145*a^4+148*a^3*b+37*a^2*b^ 2+2*a*b^3+8*b^4)/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*a*(145*a^ 4+148*a^3*b+37*a^2*b^2+2*a*b^3+8*b^4)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d *x+1/2*c)^5-1/48*(435*a^3+281*a^2*b-66*a*b^2-72*b^3)*a/(a^2+2*a*b+b^2)*tan h(1/2*d*x+1/2*c)^3-1/16*a*(29*a^2+26*a*b+8*b^2)/(a+b)*tanh(1/2*d*x+1/2*c)) /(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2* a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^3+1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)/ (a^3+3*a^2*b+3*a*b^2+b^3)*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/ 2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b )^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2) +(a+b)^(1/2))))+1/a^4*ln(tanh(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 8503 vs. \(2 (200) = 400\).
Time = 0.30 (sec) , antiderivative size = 17283, normalized size of antiderivative = 83.49 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*sech(d*x+c)^2)^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sech(d*x+c)**2)**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (200) = 400\).
Time = 0.20 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.47 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*sech(d*x+c)^2)^4,x, algorithm="maxima")
Output:
1/32*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*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)))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt((a + b)*b)*d) - 1/24*( 87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3 + 3*(145*a^5*b + 458*a^4*b^2 + 416*a^3 *b^3 + 128*a^2*b^4)*e^(-2*d*x - 2*c) + 6*(145*a^5*b + 632*a^4*b^2 + 1072*a ^3*b^3 + 768*a^2*b^4 + 208*a*b^5)*e^(-4*d*x - 4*c) + 2*(435*a^5*b + 2146*a ^4*b^2 + 4396*a^3*b^3 + 4968*a^2*b^4 + 2912*a*b^5 + 704*b^6)*e^(-6*d*x - 6 *c) + 3*(145*a^5*b + 708*a^4*b^2 + 1324*a^3*b^3 + 1024*a^2*b^4 + 288*a*b^5 )*e^(-8*d*x - 8*c) + 3*(29*a^5*b + 122*a^4*b^2 + 136*a^3*b^3 + 48*a^2*b^4) *e^(-10*d*x - 10*c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3 + 6*(a^10 + 5* a^9*b + 9*a^8*b^2 + 7*a^7*b^3 + 2*a^6*b^4)*e^(-2*d*x - 2*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(-4*d*x - 4*c) + 4*(5*a^10 + 33*a^9*b + 93*a^8*b^2 + 147*a^7*b^3 + 138*a^6*b^4 + 72 *a^5*b^5 + 16*a^4*b^6)*e^(-6*d*x - 6*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^ 2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(-8*d*x - 8*c) + 6*(a^10 + 5* a^9*b + 9*a^8*b^2 + 7*a^7*b^3 + 2*a^6*b^4)*e^(-10*d*x - 10*c) + (a^10 + 3* a^9*b + 3*a^8*b^2 + a^7*b^3)*e^(-12*d*x - 12*c))*d) + (d*x + c)/(a^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (200) = 400\).
Time = 0.26 (sec) , antiderivative size = 594, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=-\frac {\frac {3 \, {\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (87 \, a^{5} b e^{\left (10 \, d x + 10 \, c\right )} + 366 \, a^{4} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 408 \, a^{3} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 144 \, a^{2} b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 435 \, a^{5} b e^{\left (8 \, d x + 8 \, c\right )} + 2124 \, a^{4} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3972 \, a^{3} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3072 \, a^{2} b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 864 \, a b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 870 \, a^{5} b e^{\left (6 \, d x + 6 \, c\right )} + 4292 \, a^{4} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8792 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9936 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 5824 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 1408 \, b^{6} e^{\left (6 \, d x + 6 \, c\right )} + 870 \, a^{5} b e^{\left (4 \, d x + 4 \, c\right )} + 3792 \, a^{4} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6432 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4608 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 1248 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 435 \, a^{5} b e^{\left (2 \, d x + 2 \, c\right )} + 1374 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 1248 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 384 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{3}} - \frac {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \] Input:
integrate(1/(a+b*sech(d*x+c)^2)^4,x, algorithm="giac")
Output:
-1/48*(3*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*arctan(1/2*(a*e^(2*d* x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^ 3)*sqrt(-a*b - b^2)) - 2*(87*a^5*b*e^(10*d*x + 10*c) + 366*a^4*b^2*e^(10*d *x + 10*c) + 408*a^3*b^3*e^(10*d*x + 10*c) + 144*a^2*b^4*e^(10*d*x + 10*c) + 435*a^5*b*e^(8*d*x + 8*c) + 2124*a^4*b^2*e^(8*d*x + 8*c) + 3972*a^3*b^3 *e^(8*d*x + 8*c) + 3072*a^2*b^4*e^(8*d*x + 8*c) + 864*a*b^5*e^(8*d*x + 8*c ) + 870*a^5*b*e^(6*d*x + 6*c) + 4292*a^4*b^2*e^(6*d*x + 6*c) + 8792*a^3*b^ 3*e^(6*d*x + 6*c) + 9936*a^2*b^4*e^(6*d*x + 6*c) + 5824*a*b^5*e^(6*d*x + 6 *c) + 1408*b^6*e^(6*d*x + 6*c) + 870*a^5*b*e^(4*d*x + 4*c) + 3792*a^4*b^2* e^(4*d*x + 4*c) + 6432*a^3*b^3*e^(4*d*x + 4*c) + 4608*a^2*b^4*e^(4*d*x + 4 *c) + 1248*a*b^5*e^(4*d*x + 4*c) + 435*a^5*b*e^(2*d*x + 2*c) + 1374*a^4*b^ 2*e^(2*d*x + 2*c) + 1248*a^3*b^3*e^(2*d*x + 2*c) + 384*a^2*b^4*e^(2*d*x + 2*c) + 87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^3) - 48*(d*x + c)/a^4)/d
Timed out. \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^4} \,d x \] Input:
int(1/(a + b/cosh(c + d*x)^2)^4,x)
Output:
int(1/(a + b/cosh(c + d*x)^2)^4, x)
Time = 0.59 (sec) , antiderivative size = 8934, normalized size of antiderivative = 43.16 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx =\text {Too large to display} \] Input:
int(1/(a+b*sech(d*x+c)^2)^4,x)
Output:
( - 105*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt( a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**7 - 420*e**(12*c + 12*d*x)*sq rt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d *x)*sqrt(a))*a**6*b - 588*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sq rt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5*b**2 - 38 4*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4*b**3 - 96*e**(12*c + 12*d*x)*sqrt (b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x )*sqrt(a))*a**3*b**4 - 105*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt (2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**7 - 420*e**(1 2*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b ) + e**(c + d*x)*sqrt(a))*a**6*b - 588*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 *b**2 - 384*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt (a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4*b**3 - 96*e**(12*c + 12*d* x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**4 + 105*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( 2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**7 + 420*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x) *a + a + 2*b)*a**6*b + 588*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(2...