Integrand size = 10, antiderivative size = 197 \[ \int x^4 \sec ^{-1}(a+b x) \, dx=\frac {a \left (20+53 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5} \] Output:
1/30*a*(53*a^2+20)*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^5+11/60*a*x^2*(b*x+a)*( 1-1/(b*x+a)^2)^(1/2)/b^3-1/20*x^3*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^2-1/120* (58*a^2+9)*(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)/b^5+1/5*a^5*arcsec(b*x+a)/b^5+1 /5*x^5*arcsec(b*x+a)-1/40*(40*a^4+40*a^2+3)*arctanh((1-1/(b*x+a)^2)^(1/2)) /b^5
Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.88 \[ \int x^4 \sec ^{-1}(a+b x) \, dx=\frac {\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (a^2 \left (71+154 a^2\right )+2 a \left (31+48 a^2\right ) b x-9 \left (1+4 a^2\right ) b^2 x^2+16 a b^3 x^3-6 b^4 x^4\right )+24 b^5 x^5 \sec ^{-1}(a+b x)-24 a^5 \arcsin \left (\frac {1}{a+b x}\right )-3 \left (3+40 a^2+40 a^4\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{120 b^5} \] Input:
Integrate[x^4*ArcSec[a + b*x],x]
Output:
(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^2*(71 + 154*a^2) + 2* a*(31 + 48*a^2)*b*x - 9*(1 + 4*a^2)*b^2*x^2 + 16*a*b^3*x^3 - 6*b^4*x^4) + 24*b^5*x^5*ArcSec[a + b*x] - 24*a^5*ArcSin[(a + b*x)^(-1)] - 3*(3 + 40*a^2 + 40*a^4)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x )^2])])/(120*b^5)
Time = 0.72 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5781, 4926, 3042, 4269, 3042, 4544, 3042, 4536, 2009}
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.
| \(\displaystyle \int x^4 \sec ^{-1}(a+b x) \, dx\) |
| \(\Big \downarrow \) 5781 |
| \(\displaystyle \frac {\int b^4 x^4 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)d\sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 4926 |
| \(\displaystyle \frac {\frac {1}{5} \int -b^5 x^5d\sec ^{-1}(a+b x)+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \left (a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^5d\sec ^{-1}(a+b x)+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 4269 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int b^2 x^2 \left (4 a^3+11 (a+b x)^2 a-3 \left (4 a^2+1\right ) (a+b x)\right )d\sec ^{-1}(a+b x)-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int \left (a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^2 \left (4 a^3+11 \csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )^2 a-3 \left (4 a^2+1\right ) \csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right )d\sec ^{-1}(a+b x)-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 4544 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int -b x \left (12 a^4-\left (48 a^2+31\right ) (a+b x) a+\left (58 a^2+9\right ) (a+b x)^2\right )d\sec ^{-1}(a+b x)+\frac {11}{3} a b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \left (a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right ) \left (12 a^4-\left (48 a^2+31\right ) \csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right ) a+\left (58 a^2+9\right ) \csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )^2\right )d\sec ^{-1}(a+b x)+\frac {11}{3} a b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (24 a^5+4 \left (53 a^2+20\right ) (a+b x)^2 a-3 \left (40 a^4+40 a^2+3\right ) (a+b x)\right )d\sec ^{-1}(a+b x)-\frac {1}{2} \left (58 a^2+9\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {11}{3} a b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (24 a^5 \sec ^{-1}(a+b x)+4 \left (53 a^2+20\right ) a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}-3 \left (40 a^4+40 a^2+3\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )\right )-\frac {1}{2} \left (58 a^2+9\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {11}{3} a b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^3 x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{5} b^5 x^5 \sec ^{-1}(a+b x)}{b^5}\) |
Input:
Int[x^4*ArcSec[a + b*x],x]
Output:
((b^5*x^5*ArcSec[a + b*x])/5 + (-1/4*(b^3*x^3*(a + b*x)*Sqrt[1 - (a + b*x) ^(-2)]) + ((11*a*b^2*x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/3 + (-1/2*((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)]) + (4*a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)] + 24*a^5*ArcSec[a + b*x] - 3*(3 + 40*a^2 + 40*a^4)*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/2)/3)/4)/5)/b^5
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[1/(n - 1) Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3* a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2 Int[Simp[2*A*a + (2*B*a + b*(2* A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, e, f, A, B, C}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot [e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[( a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m )*Csc[e + f*x] + (b*B*(m + 1) + a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*Sec[(c _.) + (d_.)*(x_)])^(n_.)*Tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e + f* x)^m*((a + b*Sec[c + d*x])^(n + 1)/(b*d*(n + 1))), x] - Simp[f*(m/(b*d*(n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Sec[c + d*x])^(n + 1), x], x] /; FreeQ [{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcSec[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d^(m + 1) Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d *e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]], x] /; FreeQ[{a, b, c, d, e , f}, x] && IGtQ[p, 0] && IntegerQ[m]
Time = 0.24 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.67
| method | result | size |
| derivativedivides | \(\frac {-\frac {\operatorname {arcsec}\left (b x +a \right ) a^{5}}{5}+\operatorname {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\operatorname {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\operatorname {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\operatorname {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\operatorname {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\) | \(329\) |
| default | \(\frac {-\frac {\operatorname {arcsec}\left (b x +a \right ) a^{5}}{5}+\operatorname {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\operatorname {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\operatorname {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\operatorname {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\operatorname {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\) | \(329\) |
| parts | \(\frac {x^{5} \operatorname {arcsec}\left (b x +a \right )}{5}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (-6 x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b^{3} \sqrt {b^{2}}+22 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\, a \,b^{2} x^{2}-24 a^{5} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {b^{2}}-58 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\, a^{2} b x -120 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{4} b +154 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\, a^{3}-9 x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b \sqrt {b^{2}}-120 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{2} b +71 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\, a -9 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b \right )}{120 b^{5} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(425\) |
Input:
int(x^4*arcsec(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b^5*(-1/5*arcsec(b*x+a)*a^5+arcsec(b*x+a)*a^4*(b*x+a)-2*arcsec(b*x+a)*a^ 3*(b*x+a)^2+2*arcsec(b*x+a)*a^2*(b*x+a)^3-arcsec(b*x+a)*a*(b*x+a)^4+1/5*ar csec(b*x+a)*(b*x+a)^5-1/120*((b*x+a)^2-1)^(1/2)*(24*a^5*arctan(1/((b*x+a)^ 2-1)^(1/2))+120*a^4*ln(b*x+a+((b*x+a)^2-1)^(1/2))-240*a^3*((b*x+a)^2-1)^(1 /2)+120*a^2*(b*x+a)*((b*x+a)^2-1)^(1/2)-40*a*(b*x+a)^2*((b*x+a)^2-1)^(1/2) +6*(b*x+a)^3*((b*x+a)^2-1)^(1/2)+120*a^2*ln(b*x+a+((b*x+a)^2-1)^(1/2))-80* a*((b*x+a)^2-1)^(1/2)+9*(b*x+a)*((b*x+a)^2-1)^(1/2)+9*ln(b*x+a+((b*x+a)^2- 1)^(1/2)))/(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)/(b*x+a))
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.77 \[ \int x^4 \sec ^{-1}(a+b x) \, dx=\frac {24 \, b^{5} x^{5} \operatorname {arcsec}\left (b x + a\right ) + 48 \, a^{5} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 3 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, b^{3} x^{3} - 22 \, a b^{2} x^{2} - 154 \, a^{3} + {\left (58 \, a^{2} + 9\right )} b x - 71 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{120 \, b^{5}} \] Input:
integrate(x^4*arcsec(b*x+a),x, algorithm="fricas")
Output:
1/120*(24*b^5*x^5*arcsec(b*x + a) + 48*a^5*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 3*(40*a^4 + 40*a^2 + 3)*log(-b*x - a + sqrt(b^2*x^ 2 + 2*a*b*x + a^2 - 1)) - (6*b^3*x^3 - 22*a*b^2*x^2 - 154*a^3 + (58*a^2 + 9)*b*x - 71*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^5
\[ \int x^4 \sec ^{-1}(a+b x) \, dx=\int x^{4} \operatorname {asec}{\left (a + b x \right )}\, dx \] Input:
integrate(x**4*asec(b*x+a),x)
Output:
Integral(x**4*asec(a + b*x), x)
\[ \int x^4 \sec ^{-1}(a+b x) \, dx=\int { x^{4} \operatorname {arcsec}\left (b x + a\right ) \,d x } \] Input:
integrate(x^4*arcsec(b*x+a),x, algorithm="maxima")
Output:
1/5*x^5*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - integrate(1/5*(b^2*x ^6 + a*b*x^5)*e^(1/2*log(b*x + a + 1) + 1/2*log(b*x + a - 1))/(b^2*x^2 + 2 *a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 - 1)*e^(log(b*x + a + 1) + log(b*x + a - 1)) - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (173) = 346\).
Time = 0.16 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.09 \[ \int x^4 \sec ^{-1}(a+b x) \, dx=-\frac {1}{960} \, b {\left (\frac {192 \, {\left (b x + a\right )}^{5} {\left (\frac {5 \, a}{b x + a} - \frac {10 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {10 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {5 \, a^{4}}{{\left (b x + a\right )}^{4}} - 1\right )} \arccos \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{6}} - \frac {3 \, {\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 40 \, {\left (b x + a\right )}^{3} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 240 \, {\left (b x + a\right )}^{2} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 960 \, {\left (b x + a\right )} a^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 360 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (\frac {1}{2} \, {\left | b x + a \right |} {\left | -2 \, \sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 2 \right |}\right ) - \frac {120 \, {\left (8 \, a^{3} + 3 \, a\right )} {\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 24 \, {\left (10 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 40 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 3}{{\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4}}}{b^{6}}\right )} \] Input:
integrate(x^4*arcsec(b*x+a),x, algorithm="giac")
Output:
-1/960*b*(192*(b*x + a)^5*(5*a/(b*x + a) - 10*a^2/(b*x + a)^2 + 10*a^3/(b* x + a)^3 - 5*a^4/(b*x + a)^4 - 1)*arccos(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/b^6 - (3*(b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4 + 40*(b*x + a)^ 3*a*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 240*(b*x + a)^2*a^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 960*(b*x + a)*a^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 24* (b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 360*(b*x + a)*a*(sqrt(-1/(b *x + a)^2 + 1) - 1) + 24*(40*a^4 + 40*a^2 + 3)*log(1/2*abs(b*x + a)*abs(-2 *sqrt(-1/(b*x + a)^2 + 1) + 2)) - (120*(8*a^3 + 3*a)*(b*x + a)^3*(sqrt(-1/ (b*x + a)^2 + 1) - 1)^3 + 24*(10*a^2 + 1)*(b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 40*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 3)/((b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4))/b^6)
Timed out. \[ \int x^4 \sec ^{-1}(a+b x) \, dx=\int x^4\,\mathrm {acos}\left (\frac {1}{a+b\,x}\right ) \,d x \] Input:
int(x^4*acos(1/(a + b*x)),x)
Output:
int(x^4*acos(1/(a + b*x)), x)
\[ \int x^4 \sec ^{-1}(a+b x) \, dx=\int \mathit {asec} \left (b x +a \right ) x^{4}d x \] Input:
int(x^4*asec(b*x+a),x)
Output:
int(asec(a + b*x)*x**4,x)