\(\int \frac {(d+e x^2)^2 (a+b \sec ^{-1}(c x))}{x^2} \, dx\) [83]

Optimal result

Integrand size = 21, antiderivative size = 162 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e \left (12 c^2 d+e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \]

[Out]

-d^2*(a+b*arcsec(c*x))/x+2*d*e*x*(a+b*arcsec(c*x))+1/3*e^2*x^3*(a+b*arcsec(c*x))-1/6*b*e*(12*c^2*d+e)*x*arctan
h(c*x/(c^2*x^2-1)^(1/2))/c^2/(c^2*x^2)^(1/2)+b*c*d^2*(c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)-1/6*b*e^2*x^2*(c^2*x^2-
1)^(1/2)/c/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5346, 12, 1279, 396, 223, 212} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (12 c^2 d+e\right )}{6 c^2 \sqrt {c^2 x^2}}+\frac {b c d^2 \sqrt {c^2 x^2-1}}{\sqrt {c^2 x^2}}-\frac {b e^2 x^2 \sqrt {c^2 x^2-1}}{6 c \sqrt {c^2 x^2}} \]

[In]

Int[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^2,x]

[Out]

(b*c*d^2*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2] - (b*e^2*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) - (d^2*(a + b*
ArcSec[c*x]))/x + 2*d*e*x*(a + b*ArcSec[c*x]) + (e^2*x^3*(a + b*ArcSec[c*x]))/3 - (b*e*(12*c^2*d + e)*x*ArcTan
h[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^2*Sqrt[c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 5346

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}} \\ & = \frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {6 d e+e^2 x^2}{\sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}} \\ & = \frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{6 c \sqrt {c^2 x^2}} \\ & = \frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{6 c \sqrt {c^2 x^2}} \\ & = \frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e \left (12 c^2 d+e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=\frac {c^2 \left (b \sqrt {1-\frac {1}{c^2 x^2}} x \left (6 c^2 d^2-e^2 x^2\right )+2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )\right )+2 b c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \sec ^{-1}(c x)-b e \left (12 c^2 d+e\right ) x \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3 x} \]

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^2,x]

[Out]

(c^2*(b*Sqrt[1 - 1/(c^2*x^2)]*x*(6*c^2*d^2 - e^2*x^2) + 2*a*c*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d^
2 + 6*d*e*x^2 + e^2*x^4)*ArcSec[c*x] - b*e*(12*c^2*d + e)*x*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3*x)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54

method	result	size
parts	\(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+\frac {b \,e^{2} \operatorname {arcsec}\left (c x \right ) x^{3}}{3}+2 b e \,\operatorname {arcsec}\left (c x \right ) x d -\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2}}{x}-\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c \,x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e^{2}}{6 c^{4} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}\)	\(249\)
derivativedivides	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\operatorname {arcsec}\left (c x \right ) d e x}{c}+\frac {b \,\operatorname {arcsec}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2}}{c x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\)	\(272\)
default	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\operatorname {arcsec}\left (c x \right ) d e x}{c}+\frac {b \,\operatorname {arcsec}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2}}{c x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\)	\(272\)

[In]

int((e*x^2+d)^2*(a+b*arcsec(c*x))/x^2,x,method=_RETURNVERBOSE)

[Out]

a*(1/3*e^2*x^3+2*d*e*x-d^2/x)+1/3*b*e^2*arcsec(c*x)*x^3+2*b*e*arcsec(c*x)*x*d-b*arcsec(c*x)*d^2/x-1/6*b/c^3*(c
^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+b/c*(c^2*x^2-1)/x^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2-2*b/c^2*(c^2*x^2
-1)^(1/2)/x/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e*ln(c*x+(c^2*x^2-1)^(1/2))-1/6*b/c^4*(c^2*x^2-1)^(1/2)/x/((c^2*x^2-
1)/c^2/x^2)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))*e^2

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=\frac {2 \, a c^{3} e^{2} x^{4} + 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 4 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (12 \, b c^{2} d e + b e^{2}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \operatorname {arcsec}\left (c x\right ) + {\left (6 \, b c^{3} d^{2} - b c e^{2} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c^{3} x} \]

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x))/x^2,x, algorithm="fricas")

[Out]

1/6*(2*a*c^3*e^2*x^4 + 6*b*c^4*d^2*x + 12*a*c^3*d*e*x^2 - 6*a*c^3*d^2 - 4*(3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e
^2)*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (12*b*c^2*d*e + b*e^2)*x*log(-c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c^3*e^2
*x^4 + 6*b*c^3*d*e*x^2 - 3*b*c^3*d^2 + (3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x)*arcsec(c*x) + (6*b*c^3*d^2 -
b*c*e^2*x^2)*sqrt(c^2*x^2 - 1))/(c^3*x)

Sympy [A] (verification not implemented)

Time = 4.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=- \frac {a d^{2}}{x} + 2 a d e x + \frac {a e^{2} x^{3}}{3} + b c d^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {asec}{\left (c x \right )}}{x} + 2 b d e x \operatorname {asec}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asec}{\left (c x \right )}}{3} - \frac {2 b d e \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} - \frac {b e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \]

[In]

integrate((e*x**2+d)**2*(a+b*asec(c*x))/x**2,x)

[Out]

-a*d**2/x + 2*a*d*e*x + a*e**2*x**3/3 + b*c*d**2*sqrt(1 - 1/(c**2*x**2)) - b*d**2*asec(c*x)/x + 2*b*d*e*x*asec
(c*x) + b*e**2*x**3*asec(c*x)/3 - 2*b*d*e*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c
- b*e**2*Piecewise((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I*c*x**3/(2*sqrt
(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/(2*c**2), True))/(3*c)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} + {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {{\left (2 \, c x \operatorname {arcsec}\left (c x\right ) - \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x))/x^2,x, algorithm="maxima")

[Out]

1/3*a*e^2*x^3 + (c*sqrt(-1/(c^2*x^2) + 1) - arcsec(c*x)/x)*b*d^2 + 1/12*(4*x^3*arcsec(c*x) - (2*sqrt(-1/(c^2*x
^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)
/c^2)/c)*b*e^2 + 2*a*d*e*x + (2*c*x*arcsec(c*x) - log(sqrt(-1/(c^2*x^2) + 1) + 1) + log(-sqrt(-1/(c^2*x^2) + 1
) + 1))*b*d*e/c - a*d^2/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6018 vs. \(2 (144) = 288\).

Time = 2.47 (sec) , antiderivative size = 6018, normalized size of antiderivative = 37.15 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x))/x^2,x, algorithm="giac")

[Out]

-1/6*(6*b*c^4*d^2*arccos(1/(c*x))/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(
1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 6*a*c^4*d^2/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 24*b*c^4*d^2
*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3
/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 12*b*c^4*d^2*sqrt(-1/(c^2*x^2)
+ 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2
*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) + 24*a*c^4*d^2*(1/(c^2*x^2) - 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/
(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x
) + 1)^2) - 12*b*c^2*d*e*arccos(1/(c*x))/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 36*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*arccos(1/(
c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^
2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 12*b*c^2*d*e*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/
(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8) - 12*b*c^2*d*e*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^4 + 2*c^4*(1/(c^2*x^2
) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)
+ 36*b*c^4*d^2*(-1/(c^2*x^2) + 1)^(3/2)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) - 12*a*c^2*d*e/(c^4 + 2*c^4*
(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*
x) + 1)^8) + 36*a*c^4*d^2*(1/(c^2*x^2) - 1)^2/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*
x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 24*b*c^4*d^2*(1/(c^2
*x^2) - 1)^3*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c
*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 24*b*c^2*d*e*(1/(c^2*x^2) - 1)*log(ab
s(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2)
- 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 24*b*c^2*d*e*(1/(c^2*x^2)
- 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(
1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 36*b*c^4*d^2*
(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2
) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) + 24*a*c^4*d^2*(1/(c^2*x^
2) - 1)^3/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1
/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 2*b*e^2*arccos(1/(c*x))/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(
1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 24*b*c
^2*d*e*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2
) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 6*b*c^4*d^2*(1/(c^2*x^2
) - 1)^4*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x)
+ 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + b*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/
(c*x) + 1))/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(
1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - b*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^4 + 2*c^4*(1/(c^
2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1
)^8) - 12*b*c^4*d^2*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
- 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) - 2*a
*e^2/(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*
x^2) - 1)^4/(1/(c*x) + 1)^8) + 24*a*c^2*d*e*(1/(c^2*x^2) - 1)^2/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^
2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 6*
a*c^4*d^2*(1/(c^2*x^2) - 1)^4/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(
c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 8*b*e^2*(1/(c^2*x^2) - 1)*arccos(1/(
c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^
2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 2*b*e^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1
/(c*x) + 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 24*b*c^2*d*e*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2
*x^2) + 1) + 1/(c*x) + 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x
) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 2*b*e^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(
-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/
(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 24*b*c^2*d*e*(1/(c^2*x^2) - 1)^3
*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^
2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 2*b*e^2*sqrt(-1/(c
^2*x^2) + 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) + 8*a*e^2*(1/(c^2*x^2) - 1)/((c^4 + 2*c^4*(1/(c^2*x^2) -
1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(
c*x) + 1)^2) - 12*b*e^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 -
2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 12*b*c
^2*d*e*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2
) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 12*b*c^2*d*e*(1/(c^2*x^
2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c
^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 12*b*c^2*
d*e*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^
8) + 2*b*e^2*(-1/(c^2*x^2) + 1)^(3/2)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1
)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) - 12*a*e^2*(1/(c^2*x^2) - 1)^2
/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2
) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 12*a*c^2*d*e*(1/(c^2*x^2) - 1)^4/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/
(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x
) + 1)^8) + 8*b*e^2*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^
4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 2*b*e^2*(1
/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)
^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 2
*b*e^2*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(
c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) +
1)^6) - 2*b*e^2*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2
*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) + 8*a*e^2
*(1/(c^2*x^2) - 1)^3/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)
^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 2*b*e^2*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/(
(c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - b*e^2*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x)
+ 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^
2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + b*e^2*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1
/(c*x) - 1))/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 2*b*e^2*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)/((
c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) - 2*a*e^2*(1/(c^2*x^2) - 1)^4/((c^4 + 2*c^4*(1/(c^2*x^2) - 1)/(1/(c*x)
+ 1)^2 - 2*c^4*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^4*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8
))*c

Mupad [F(-1)]

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

[In]

int(((d + e*x^2)^2*(a + b*acos(1/(c*x))))/x^2,x)

[Out]

int(((d + e*x^2)^2*(a + b*acos(1/(c*x))))/x^2, x)