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

Optimal. Leaf size=186 \[ -\frac{1}{2} i b d^2 \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \sec ^{-1}(c x)\right )+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (6 c^2 d+e\right )}{6 c^3}-\frac{b e^2 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]

[Out]

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

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Rubi [A]  time = 0.411894, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {5240, 266, 43, 4732, 6742, 453, 264, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{1}{2} i b d^2 \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \sec ^{-1}(c x)\right )+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (6 c^2 d+e\right )}{6 c^3}-\frac{b e^2 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5240

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[
((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n
, 0] && IntegerQ[m] && IntegerQ[p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4732

Int[((a_.) + ArcCos[(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*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 264

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

Rule 2326

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSin[(Rt[-e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[-e, 2], x] - Dist[(b*n)/Rt[-e, 2], Int[ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]/x, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right )^2 \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{b \operatorname{Subst}\left (\int \left (-\frac{e \left (e+4 d x^2\right )}{4 x^4 \sqrt{1-\frac{x^2}{c^2}}}+\frac{d^2 \log (x)}{\sqrt{1-\frac{x^2}{c^2}}}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}+\frac{(b e) \operatorname{Subst}\left (\int \frac{e+4 d x^2}{x^4 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )+\frac{\left (b e \left (6 c^2 d+e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c^3}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\left (b d^2\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\left (2 i b d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\left (b d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{1}{2} \left (i b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}{6 c^3}-\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x^3}{12 c}-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{1}{2} i b d^2 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.333324, size = 160, normalized size = 0.86 \[ \frac{1}{2} i b d^2 \left (\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )+\sec ^{-1}(c x) \left (\sec ^{-1}(c x)+2 i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )\right )+a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e x \left (c x \sec ^{-1}(c x)-\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c}-\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 x^2+2\right )}{12 c^3}+\frac{1}{4} b e^2 x^4 \sec ^{-1}(c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.714, size = 242, normalized size = 1.3 \begin{align*}{\frac{a{x}^{4}{e}^{2}}{4}}+a{x}^{2}de+a{d}^{2}\ln \left ( cx \right ) +{\frac{i}{2}}b{d}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}+{\frac{b{\rm arcsec} \left (cx\right ){x}^{4}{e}^{2}}{4}}+b{\rm arcsec} \left (cx\right ){x}^{2}de-{\frac{b{x}^{3}{e}^{2}}{12\,c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{bxde}{c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{ibde}{{c}^{2}}}-{\frac{bx{e}^{2}}{6\,{c}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{{\frac{i}{6}}b{e}^{2}}{{c}^{4}}}-b{d}^{2}{\rm arcsec} \left (cx\right )\ln \left ( 1+ \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) ^{2} \right ) +{\frac{i}{2}}b{d}^{2}{\it polylog} \left ( 2,- \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) ^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4*e^2+a*x^2*d*e+a*d^2*ln(c*x)+1/2*I*b*d^2*arcsec(c*x)^2+1/4*b*arcsec(c*x)*x^4*e^2+b*arcsec(c*x)*x^2*d*
e-1/12*b/c*((c^2*x^2-1)/c^2/x^2)^(1/2)*x^3*e^2-b/c*((c^2*x^2-1)/c^2/x^2)^(1/2)*x*d*e-I*b/c^2*d*e-1/6*b/c^3*((c
^2*x^2-1)/c^2/x^2)^(1/2)*x*e^2-1/6*I*b/c^4*e^2-b*d^2*arcsec(c*x)*ln(1+(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)+1/2*I*b
*d^2*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \left (x\right ) - \frac{-2 i \, b c^{4} e^{2} x^{4} \log \left (c\right ) - 4 i \, b c^{4} d^{2} \log \left (-c x + 1\right ) \log \left (x\right ) - 4 i \, b c^{4} d^{2} \log \left (x\right )^{2} - 4 i \, b c^{4} d^{2}{\rm Li}_2\left (c x\right ) - 4 i \, b c^{4} d^{2}{\rm Li}_2\left (-c x\right ) + i \,{\left (4 \,{\left ({\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \left (x\right )\right )} \log \left (x\right ) - \log \left (c x - 1\right ) \log \left (x\right ) + \log \left (-c x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} +{\rm Li}_2\left (c x\right ) +{\rm Li}_2\left (-c x\right )\right )} b d^{2} + b e^{2}{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c x + 1\right )}{c^{4}} + \frac{\log \left (c x - 1\right )}{c^{4}}\right )} + 4 \, b d e{\left (\frac{\log \left (c x + 1\right )}{c^{2}} + \frac{\log \left (c x - 1\right )}{c^{2}}\right )}\right )} c^{4} + \frac{2}{3} \,{\left (12 \, b d^{2} \int \frac{\sqrt{c x + 1} \sqrt{c x - 1} \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} + \frac{12 \, \sqrt{c x + 1} \sqrt{c x - 1} b d e}{c^{2}} + \frac{{\left (c^{2} x^{2} + 2\right )} \sqrt{c x + 1} \sqrt{c x - 1} b e^{2}}{c^{4}}\right )} c^{4} +{\left (-8 i \, b c^{4} d e \log \left (c\right ) - i \, b c^{2} e^{2}\right )} x^{2} - 2 \,{\left (b c^{4} e^{2} x^{4} + 4 \, b c^{4} d e x^{2} + 4 \, b c^{4} d^{2} \log \left (x\right )\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left (i \, b c^{4} e^{2} x^{4} + 4 i \, b c^{4} d e x^{2} + 4 i \, b c^{4} d^{2} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right ) +{\left (-4 i \, b c^{4} d^{2} \log \left (x\right ) - 4 i \, b c^{2} d e - i \, b e^{2}\right )} \log \left (c x + 1\right ) +{\left (-4 i \, b c^{2} d e - i \, b e^{2}\right )} \log \left (c x - 1\right ) +{\left (-2 i \, b c^{4} e^{2} x^{4} - 8 i \, b c^{4} d e x^{2} - 8 i \, b c^{4} d^{2} \log \left (c\right )\right )} \log \left (x\right )}{8 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e^2*x^4 + a*d*e*x^2 + a*d^2*log(x) - 1/8*(-2*I*b*c^4*e^2*x^4*log(c) - 4*I*b*c^4*d^2*log(-c*x + 1)*log(x)
- 4*I*b*c^4*d^2*log(x)^2 - 4*I*b*c^4*d^2*dilog(c*x) - 4*I*b*c^4*d^2*dilog(-c*x) + I*(b*e^2*(x^2/c^2 + log(c*x
+ 1)/c^4 + log(c*x - 1)/c^4) + 4*b*d*e*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2) + 32*b*d^2*integrate(1/4*log(x)/
(c^2*x^3 - x), x))*c^4 + 8*c^4*integrate(1/4*(b*e^2*x^4 + 4*b*d*e*x^2 + 4*b*d^2*log(x))*sqrt(c*x + 1)*sqrt(c*x
- 1)/(c^2*x^3 - x), x) + (-8*I*b*c^4*d*e*log(c) - I*b*c^2*e^2)*x^2 - 2*(b*c^4*e^2*x^4 + 4*b*c^4*d*e*x^2 + 4*b
*c^4*d^2*log(x))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + (I*b*c^4*e^2*x^4 + 4*I*b*c^4*d*e*x^2 + 4*I*b*c^4*d^2*lo
g(x))*log(c^2*x^2) + (-4*I*b*c^4*d^2*log(x) - 4*I*b*c^2*d*e - I*b*e^2)*log(c*x + 1) + (-4*I*b*c^2*d*e - I*b*e^
2)*log(c*x - 1) + (-2*I*b*c^4*e^2*x^4 - 8*I*b*c^4*d*e*x^2 - 8*I*b*c^4*d^2*log(c))*log(x))/c^4

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname{arcsec}\left (c x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arcsec(c*x))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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