3.2.51 \(\int \frac {x^3 (a+b \text {sech}^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [151]

Optimal. Leaf size=251 \[ -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {b \left (3 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}}+\frac {2 b d^{3/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2} \]

[Out]

1/3*(e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/e^2+1/6*b*(3*c^2*d-e)*arctan(e^(1/2)*(-c^2*x^2+1)^(1/2)/c/(e*x^2+d)^(1/
2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^3/e^(3/2)+2/3*b*d^(3/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/
2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/e^2-d*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2)/e^2-1/6*b*(1/(c*x+1))^(1/2)*(c*x+
1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/c^2/e

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Rubi [A]
time = 0.22, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 6436, 12, 587, 159, 163, 65, 223, 209, 95, 213} \begin {gather*} -\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (3 c^2 d-e\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}}+\frac {2 b d^{3/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]

[Out]

-1/6*(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(c^2*e) - (d*Sqrt[d + e*x^2]*(a
+ b*ArcSech[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^2) + (b*(3*c^2*d - e)*Sqrt[(1 + c*x)^(-
1)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^3*e^(3/2)) + (2*b*d^(3/2)*Sqrt
[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*e^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 45

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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 272

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 587

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n],
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 6436

Int[((a_.) + ArcSech[(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*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), 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])) || (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{3 e^2 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{x \sqrt {1-c^2 x^2}} \, dx}{3 e^2}\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(-2 d+e x) \sqrt {d+e x}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {2 c^2 d^2+\frac {1}{2} \left (3 c^2 d-e\right ) e x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c^2 e^2}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2}-\frac {\left (b \left (3 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c^2 e}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (2 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{3 e^2}+\frac {\left (b \left (3 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{6 c^4 e}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {2 b d^{3/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2}+\frac {\left (b \left (3 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{6 c^4 e}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {b \left (3 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}}+\frac {2 b d^{3/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2}\\ \end {align*}

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Mathematica [A]
time = 20.86, size = 406, normalized size = 1.62 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (b e \sqrt {\frac {1-c x}{1+c x}} (1+c x)+2 a c^2 \left (2 d-e x^2\right )+2 b c^2 \left (2 d-e x^2\right ) \text {sech}^{-1}(c x)\right )}{6 c^2 e^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (-3 \left (-c^2\right )^{3/2} d \sqrt {-c^2 d-e} \sqrt {e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \text {ArcSin}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+\sqrt {-c^2} \sqrt {-c^2 d-e} e^{3/2} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \text {ArcSin}\left (\frac {\sqrt {-c^2} \sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {-c^2 d-e}}\right )+4 c^5 d^{3/2} \sqrt {-d-e x^2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{6 c^5 e^2 (-1+c x) \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]

[Out]

-1/6*(Sqrt[d + e*x^2]*(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 2*a*c^2*(2*d - e*x^2) + 2*b*c^2*(2*d - e*x^2)
*ArcSech[c*x]))/(c^2*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(-3*(-c^2)^(3/2)*d*Sqrt[-(c^2*d) -
e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d)
- e])] + Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*e^(3/2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(Sqrt[-c^2]*Sqrt[e]*
Sqrt[1 - c^2*x^2])/(c*Sqrt[-(c^2*d) - e])] + 4*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])
/Sqrt[-d - e*x^2]]))/(6*c^5*e^2*(-1 + c*x)*Sqrt[d + e*x^2])

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Maple [F]
time = 1.30, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x)

[Out]

int(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/3*(sqrt(x^2*e + d)*x^2*e^(-1) - 2*sqrt(x^2*e + d)*d*e^(-2))*a + 1/3*((x^4*e^2 - d*x^2*e - 2*d^2)*e^(-2)*log(
sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/sqrt(x^2*e + d) - 3*integrate(1/3*(6*(c^2*x^4*e^2 - x^2*e^2)*x^3*log(sqrt(x)
) + 3*(c^2*x^4*e^2*log(c) - x^2*e^2*log(c))*x^3 + (6*(c^2*x^4*e^2 - x^2*e^2)*x^3*log(sqrt(x)) + (c^2*x^4*(3*lo
g(c) + 1)*e^2 - 2*c^2*d^2 - (c^2*d*e + 3*e^2*log(c))*x^2)*x^3)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))/((c^2
*x^4*e^2 - x^2*e^2 + (c^2*x^4*e^2 - x^2*e^2)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))*sqrt(x^2*e + d)), x))*b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (159) = 318\).
time = 0.62, size = 1092, normalized size = 4.35 \begin {gather*} \left [\frac {2 \, b c^{3} d^{\frac {3}{2}} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} \cosh \left (1\right )^{2} + x^{4} \sinh \left (1\right )^{2} - 4 \, {\left (c^{3} d x^{3} - c x^{3} \cosh \left (1\right ) - c x^{3} \sinh \left (1\right ) - 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} \cosh \left (1\right ) - 2 \, {\left (3 \, c^{2} d x^{4} - x^{4} \cosh \left (1\right ) - 4 \, d x^{2}\right )} \sinh \left (1\right )}{x^{4}}\right ) + {\left (3 \, b c^{2} d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \arctan \left (\frac {{\left (c^{2} d x + {\left (2 \, c^{2} x^{3} - x\right )} \cosh \left (1\right ) + {\left (2 \, c^{2} x^{3} - x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )}}{2 \, {\left ({\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} x^{4} - x^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d x^{2} - d\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{2} + 2 \, {\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right ) - d\right )} \sinh \left (1\right )\right )}}\right ) + 4 \, {\left (b c^{3} x^{2} \cosh \left (1\right ) + b c^{3} x^{2} \sinh \left (1\right ) - 2 \, b c^{3} d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, {\left (2 \, a c^{3} x^{2} \cosh \left (1\right ) + 2 \, a c^{3} x^{2} \sinh \left (1\right ) - 4 \, a c^{3} d - {\left (b c^{2} x \cosh \left (1\right ) + b c^{2} x \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{12 \, {\left (c^{3} \cosh \left (1\right )^{2} + 2 \, c^{3} \cosh \left (1\right ) \sinh \left (1\right ) + c^{3} \sinh \left (1\right )^{2}\right )}}, \frac {4 \, b c^{3} \sqrt {-d} d \arctan \left (-\frac {{\left (c^{3} d x^{3} - c x^{3} \cosh \left (1\right ) - c x^{3} \sinh \left (1\right ) - 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{4} - d x^{2}\right )} \sinh \left (1\right )\right )}}\right ) + {\left (3 \, b c^{2} d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \arctan \left (\frac {{\left (c^{2} d x + {\left (2 \, c^{2} x^{3} - x\right )} \cosh \left (1\right ) + {\left (2 \, c^{2} x^{3} - x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )}}{2 \, {\left ({\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} x^{4} - x^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d x^{2} - d\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{2} + 2 \, {\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right ) - d\right )} \sinh \left (1\right )\right )}}\right ) + 4 \, {\left (b c^{3} x^{2} \cosh \left (1\right ) + b c^{3} x^{2} \sinh \left (1\right ) - 2 \, b c^{3} d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, {\left (2 \, a c^{3} x^{2} \cosh \left (1\right ) + 2 \, a c^{3} x^{2} \sinh \left (1\right ) - 4 \, a c^{3} d - {\left (b c^{2} x \cosh \left (1\right ) + b c^{2} x \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{12 \, {\left (c^{3} \cosh \left (1\right )^{2} + 2 \, c^{3} \cosh \left (1\right ) \sinh \left (1\right ) + c^{3} \sinh \left (1\right )^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(2*b*c^3*d^(3/2)*log((c^4*d^2*x^4 - 8*c^2*d^2*x^2 + x^4*cosh(1)^2 + x^4*sinh(1)^2 - 4*(c^3*d*x^3 - c*x^3
*cosh(1) - c*x^3*sinh(1) - 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
+ 8*d^2 - 2*(3*c^2*d*x^4 - 4*d*x^2)*cosh(1) - 2*(3*c^2*d*x^4 - x^4*cosh(1) - 4*d*x^2)*sinh(1))/x^4) + (3*b*c^
2*d - b*cosh(1) - b*sinh(1))*sqrt(cosh(1) + sinh(1))*arctan(1/2*(c^2*d*x + (2*c^2*x^3 - x)*cosh(1) + (2*c^2*x^
3 - x)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))*sqrt(cosh(1) + sinh(1))/((c
^2*x^4 - x^2)*cosh(1)^2 + (c^2*x^4 - x^2)*sinh(1)^2 + (c^2*d*x^2 - d)*cosh(1) + (c^2*d*x^2 + 2*(c^2*x^4 - x^2)
*cosh(1) - d)*sinh(1))) + 4*(b*c^3*x^2*cosh(1) + b*c^3*x^2*sinh(1) - 2*b*c^3*d)*sqrt(x^2*cosh(1) + x^2*sinh(1)
+ d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(2*a*c^3*x^2*cosh(1) + 2*a*c^3*x^2*sinh(1) - 4*a
*c^3*d - (b*c^2*x*cosh(1) + b*c^2*x*sinh(1))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sinh(1) +
d))/(c^3*cosh(1)^2 + 2*c^3*cosh(1)*sinh(1) + c^3*sinh(1)^2), 1/12*(4*b*c^3*sqrt(-d)*d*arctan(-1/2*(c^3*d*x^3 -
c*x^3*cosh(1) - c*x^3*sinh(1) - 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(-d)*sqrt(-(c^2*x^2 - 1)/(c^
2*x^2))/(c^2*d^2*x^2 - d^2 + (c^2*d*x^4 - d*x^2)*cosh(1) + (c^2*d*x^4 - d*x^2)*sinh(1))) + (3*b*c^2*d - b*cosh
(1) - b*sinh(1))*sqrt(cosh(1) + sinh(1))*arctan(1/2*(c^2*d*x + (2*c^2*x^3 - x)*cosh(1) + (2*c^2*x^3 - x)*sinh(
1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))*sqrt(cosh(1) + sinh(1))/((c^2*x^4 - x^2
)*cosh(1)^2 + (c^2*x^4 - x^2)*sinh(1)^2 + (c^2*d*x^2 - d)*cosh(1) + (c^2*d*x^2 + 2*(c^2*x^4 - x^2)*cosh(1) - d
)*sinh(1))) + 4*(b*c^3*x^2*cosh(1) + b*c^3*x^2*sinh(1) - 2*b*c^3*d)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c
*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(2*a*c^3*x^2*cosh(1) + 2*a*c^3*x^2*sinh(1) - 4*a*c^3*d - (b*
c^2*x*cosh(1) + b*c^2*x*sinh(1))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d))/(c^3*cos
h(1)^2 + 2*c^3*cosh(1)*sinh(1) + c^3*sinh(1)^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asech(c*x))/(e*x**2+d)**(1/2),x)

[Out]

Integral(x**3*(a + b*asech(c*x))/sqrt(d + e*x**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^3/sqrt(e*x^2 + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*acosh(1/(c*x))))/(d + e*x^2)^(1/2),x)

[Out]

int((x^3*(a + b*acosh(1/(c*x))))/(d + e*x^2)^(1/2), x)

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