\(\int x^2 \sec ^{-1}(a+b x)^3 \, dx\) [33]

Optimal result

Integrand size = 12, antiderivative size = 494 \[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\frac {(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac {3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac {i \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac {i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3} \] Output:

(b*x+a)*arcsec(b*x+a)/b^3-3*I*a*polylog(2,-(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1 
/2))^2)/b^3+3*a*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)*arcsec(b*x+a)^2/b^3-1/2*(b*x 
+a)^2*(1-1/(b*x+a)^2)^(1/2)*arcsec(b*x+a)^2/b^3+1/3*a^3*arcsec(b*x+a)^3/b^ 
3+1/3*x^3*arcsec(b*x+a)^3+I*arcsec(b*x+a)^2*arctan(1/(b*x+a)+I*(1-1/(b*x+a 
)^2)^(1/2))/b^3-I*arcsec(b*x+a)*polylog(2,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^ 
(1/2)))/b^3-arctanh((1-1/(b*x+a)^2)^(1/2))/b^3+6*a*arcsec(b*x+a)*ln(1+(1/( 
b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)/b^3+I*arcsec(b*x+a)*polylog(2,I*(1/(b*x 
+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^3-6*I*a^2*arcsec(b*x+a)*polylog(2,-I*(1/(b 
*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^3+6*I*a^2*arcsec(b*x+a)^2*arctan(1/(b*x+ 
a)+I*(1-1/(b*x+a)^2)^(1/2))/b^3-3*I*a*arcsec(b*x+a)^2/b^3+6*I*a^2*arcsec(b 
*x+a)*polylog(2,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^3+polylog(3,-I*(1 
/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^3+6*a^2*polylog(3,-I*(1/(b*x+a)+I*(1- 
1/(b*x+a)^2)^(1/2)))/b^3-polylog(3,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/ 
b^3-6*a^2*polylog(3,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))/b^3
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.90 \[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\frac {-\coth ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )+(a+b x) \sec ^{-1}(a+b x)-3 i a \sec ^{-1}(a+b x)^2+3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2-\frac {1}{2} (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+\frac {1}{3} a^3 \sec ^{-1}(a+b x)^3+\frac {1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3+i \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )+6 i a^2 \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )+6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )-i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )-6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )+i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )+6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )-3 i a \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )+6 a^2 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )-6 a^2 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3} \] Input:

Integrate[x^2*ArcSec[a + b*x]^3,x]
 

Output:

(-ArcCoth[Sqrt[1 - (a + b*x)^(-2)]] + (a + b*x)*ArcSec[a + b*x] - (3*I)*a* 
ArcSec[a + b*x]^2 + 3*a*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x] 
^2 - ((a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]^2)/2 + (a^3*Arc 
Sec[a + b*x]^3)/3 + (b^3*x^3*ArcSec[a + b*x]^3)/3 + I*ArcSec[a + b*x]^2*Ar 
cTan[E^(I*ArcSec[a + b*x])] + (6*I)*a^2*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcS 
ec[a + b*x])] + 6*a*ArcSec[a + b*x]*Log[1 + E^((2*I)*ArcSec[a + b*x])] - I 
*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] - (6*I)*a^2*ArcSec 
[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + I*ArcSec[a + b*x]*PolyL 
og[2, I*E^(I*ArcSec[a + b*x])] + (6*I)*a^2*ArcSec[a + b*x]*PolyLog[2, I*E^ 
(I*ArcSec[a + b*x])] - (3*I)*a*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])] + Po 
lyLog[3, (-I)*E^(I*ArcSec[a + b*x])] + 6*a^2*PolyLog[3, (-I)*E^(I*ArcSec[a 
 + b*x])] - PolyLog[3, I*E^(I*ArcSec[a + b*x])] - 6*a^2*PolyLog[3, I*E^(I* 
ArcSec[a + b*x])])/b^3
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5781, 4926, 3042, 4678, 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.

\(\displaystyle \int x^2 \sec ^{-1}(a+b x)^3 \, dx\)

\(\Big \downarrow \) 5781

\(\displaystyle \frac {\int b^2 x^2 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^3d\sec ^{-1}(a+b x)}{b^3}\)

\(\Big \downarrow \) 4926

\(\displaystyle \frac {\int -b^3 x^3 \sec ^{-1}(a+b x)^2d\sec ^{-1}(a+b x)+\frac {1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3}{b^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sec ^{-1}(a+b x)^2 \left (a-\csc \left (\sec ^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^3d\sec ^{-1}(a+b x)+\frac {1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3}{b^3}\)

\(\Big \downarrow \) 4678

\(\displaystyle \frac {\int \left (\sec ^{-1}(a+b x)^2 a^3-3 (a+b x) \sec ^{-1}(a+b x)^2 a^2+3 (a+b x)^2 \sec ^{-1}(a+b x)^2 a-(a+b x)^3 \sec ^{-1}(a+b x)^2\right )d\sec ^{-1}(a+b x)+\frac {1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3}{b^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{3} a^3 \sec ^{-1}(a+b x)^3+6 i a^2 \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )-6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )+6 i a^2 \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )+6 a^2 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )-6 a^2 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )+i \sec ^{-1}(a+b x)^2 \arctan \left (e^{i \sec ^{-1}(a+b x)}\right )-\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )+\frac {1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3-i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )+i \sec ^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )-3 i a \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )-3 i a \sec ^{-1}(a+b x)^2-\frac {1}{2} (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+(a+b x) \sec ^{-1}(a+b x)+6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}\)

Input:

Int[x^2*ArcSec[a + b*x]^3,x]
 

Output:

((a + b*x)*ArcSec[a + b*x] - (3*I)*a*ArcSec[a + b*x]^2 + 3*a*(a + b*x)*Sqr 
t[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]^2 - ((a + b*x)^2*Sqrt[1 - (a + b*x)^ 
(-2)]*ArcSec[a + b*x]^2)/2 + (a^3*ArcSec[a + b*x]^3)/3 + (b^3*x^3*ArcSec[a 
 + b*x]^3)/3 + I*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])] + (6*I)*a 
^2*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])] - ArcTanh[Sqrt[1 - (a + 
 b*x)^(-2)]] + 6*a*ArcSec[a + b*x]*Log[1 + E^((2*I)*ArcSec[a + b*x])] - I* 
ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] - (6*I)*a^2*ArcSec[ 
a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + I*ArcSec[a + b*x]*PolyLo 
g[2, I*E^(I*ArcSec[a + b*x])] + (6*I)*a^2*ArcSec[a + b*x]*PolyLog[2, I*E^( 
I*ArcSec[a + b*x])] - (3*I)*a*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])] + Pol 
yLog[3, (-I)*E^(I*ArcSec[a + b*x])] + 6*a^2*PolyLog[3, (-I)*E^(I*ArcSec[a 
+ b*x])] - PolyLog[3, I*E^(I*ArcSec[a + b*x])] - 6*a^2*PolyLog[3, I*E^(I*A 
rcSec[a + b*x])])/b^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4678

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

rule 4926

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]
 

rule 5781

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]
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arcsec}\left (b x +a \right ) \left (6 \operatorname {arcsec}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arcsec}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arcsec}\left (b x +a \right )^{2} \left (b x +a \right )^{3}+18 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )-3 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i \operatorname {arcsec}\left (b x +a \right ) a +6 b x +6 a \right )}{6}-i \operatorname {arcsec}\left (b x +a \right ) \operatorname {polylog}\left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+6 i \operatorname {polylog}\left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )+3 \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )^{2}+6 \operatorname {polylog}\left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2}-6 i \operatorname {arcsec}\left (b x +a \right )^{2} a +i \operatorname {arcsec}\left (b x +a \right ) \operatorname {polylog}\left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-3 \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )^{2}-6 \operatorname {polylog}\left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2}+6 \ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a \,\operatorname {arcsec}\left (b x +a \right )+2 i \arctan \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 i \operatorname {polylog}\left (2, -{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a +\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{2}-6 i \operatorname {polylog}\left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )+\operatorname {polylog}\left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{2}-\operatorname {polylog}\left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{b^{3}}\)	\(716\)
default	\(\frac {\frac {\operatorname {arcsec}\left (b x +a \right ) \left (6 \operatorname {arcsec}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arcsec}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arcsec}\left (b x +a \right )^{2} \left (b x +a \right )^{3}+18 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )-3 \,\operatorname {arcsec}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i \operatorname {arcsec}\left (b x +a \right ) a +6 b x +6 a \right )}{6}-i \operatorname {arcsec}\left (b x +a \right ) \operatorname {polylog}\left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+6 i \operatorname {polylog}\left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )+3 \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )^{2}+6 \operatorname {polylog}\left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2}-6 i \operatorname {arcsec}\left (b x +a \right )^{2} a +i \operatorname {arcsec}\left (b x +a \right ) \operatorname {polylog}\left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-3 \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )^{2}-6 \operatorname {polylog}\left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2}+6 \ln \left (1+{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a \,\operatorname {arcsec}\left (b x +a \right )+2 i \arctan \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 i \operatorname {polylog}\left (2, -{\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}^{2}\right ) a +\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{2}-6 i \operatorname {polylog}\left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right ) a^{2} \operatorname {arcsec}\left (b x +a \right )+\operatorname {polylog}\left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-\frac {\operatorname {arcsec}\left (b x +a \right )^{2} \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{2}-\operatorname {polylog}\left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )}{b^{3}}\)	\(716\)

Input:

int(x^2*arcsec(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(1/6*arcsec(b*x+a)*(6*arcsec(b*x+a)^2*a^2*(b*x+a)-6*arcsec(b*x+a)^2* 
a*(b*x+a)^2+2*arcsec(b*x+a)^2*(b*x+a)^3+18*arcsec(b*x+a)*(((b*x+a)^2-1)/(b 
*x+a)^2)^(1/2)*a*(b*x+a)-3*arcsec(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*( 
b*x+a)^2+18*I*a*arcsec(b*x+a)+6*b*x+6*a)-I*arcsec(b*x+a)*polylog(2,-I*(1/( 
b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))+6*I*polylog(2,I*(1/(b*x+a)+I*(1-1/(b*x+a) 
^2)^(1/2)))*a^2*arcsec(b*x+a)+3*ln(1+I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)) 
)*a^2*arcsec(b*x+a)^2+6*polylog(3,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))* 
a^2-6*I*arcsec(b*x+a)^2*a+I*arcsec(b*x+a)*polylog(2,I*(1/(b*x+a)+I*(1-1/(b 
*x+a)^2)^(1/2)))-3*ln(1-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a^2*arcsec( 
b*x+a)^2-6*polylog(3,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a^2+6*ln(1+(1/ 
(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2))^2)*a*arcsec(b*x+a)+2*I*arctan(1/(b*x+a)+I 
*(1-1/(b*x+a)^2)^(1/2))-3*I*polylog(2,-(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)) 
^2)*a+1/2*arcsec(b*x+a)^2*ln(1+I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))-6*I* 
polylog(2,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))*a^2*arcsec(b*x+a)+polylo 
g(3,-I*(1/(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))-1/2*arcsec(b*x+a)^2*ln(1-I*(1/ 
(b*x+a)+I*(1-1/(b*x+a)^2)^(1/2)))-polylog(3,I*(1/(b*x+a)+I*(1-1/(b*x+a)^2) 
^(1/2))))
 

Fricas [F]

\[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arcsec}\left (b x + a\right )^{3} \,d x } \] Input:

integrate(x^2*arcsec(b*x+a)^3,x, algorithm="fricas")
 

Output:

integral(x^2*arcsec(b*x + a)^3, x)
 

Sympy [F]

\[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int x^{2} \operatorname {asec}^{3}{\left (a + b x \right )}\, dx \] Input:

integrate(x**2*asec(b*x+a)**3,x)
 

Output:

Integral(x**2*asec(a + b*x)**3, x)
 

Maxima [F]

\[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arcsec}\left (b x + a\right )^{3} \,d x } \] Input:

integrate(x^2*arcsec(b*x+a)^3,x, algorithm="maxima")
 

Output:

1/3*x^3*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 1/4*x^3*arctan(sqr 
t(b*x + a + 1)*sqrt(b*x + a - 1))*log(b^2*x^2 + 2*a*b*x + a^2)^2 - integra 
te(1/4*((4*b*x^3*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b*x^3*log 
(b^2*x^2 + 2*a*b*x + a^2)^2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 4*(3*(b 
^3*x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*x^2)*log(b*x + a)^2 - 
 (b^3*x^5 + 2*a*b^2*x^4 + (a^2 - 1)*b*x^3 + 3*(b^3*x^5 + 3*a*b^2*x^4 + (3* 
a^2 - 1)*b*x^3 + (a^3 - a)*x^2)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2) 
)*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))/(b^3*x^3 + 3*a*b^2*x^2 + a^ 
3 + (3*a^2 - 1)*b*x - a), x)
 

Giac [F]

\[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arcsec}\left (b x + a\right )^{3} \,d x } \] Input:

integrate(x^2*arcsec(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate(x^2*arcsec(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int x^2\,{\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \] Input:

int(x^2*acos(1/(a + b*x))^3,x)
 

Output:

int(x^2*acos(1/(a + b*x))^3, x)
 

Reduce [F]

\[ \int x^2 \sec ^{-1}(a+b x)^3 \, dx=\int \mathit {asec} \left (b x +a \right )^{3} x^{2}d x \] Input:

int(x^2*asec(b*x+a)^3,x)
 

Output:

int(asec(a + b*x)**3*x**2,x)