3.3.65 \(\int \frac {\sqrt {d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2}{x^4} \, dx\) [265]

Optimal. Leaf size=294 \[ -\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {2 b c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \]

[Out]

-1/3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/d/x^3-1/3*b^2*c^2*(c^2*d*x^2+d)^(1/2)/x+1/3*b^2*c^3*arcsinh(c*x)
*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/3*c^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+2/3*
b*c^3*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/3*b^2*c^3*p
olylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/3*b*c*(a+b*arcsinh(c*x))*(c^2*x^
2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/x^2

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Rubi [A]
time = 0.22, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {5800, 5802, 283, 221, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {b c \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {2 b c^3 \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 \sqrt {c^2 d x^2+d}}{3 x}-\frac {b^2 c^3 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {b^2 c^3 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

-1/3*(b^2*c^2*Sqrt[d + c^2*d*x^2])/x + (b^2*c^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(3*Sqrt[1 + c^2*x^2]) - (b*c
*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (c^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh
[c*x])^2)/(3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*d*x^3) + (2*b*c^3*Sqrt[d +
c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (b^2*c^3*Sqrt[d + c^2*d
*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])

Rule 221

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

Rule 283

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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]

Rule 3797

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

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rule 5802

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*(d + e*x^2)^p*((a + b*ArcSinh[c*x])/(f*(m + 1))), x] + (-Dist[b*c*(d^p/(f*(m + 1))), Int[(f*x)^(m + 1
)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*A
rcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {\left (2 b c^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-\frac {\left (4 b c^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {2 b c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {2 b c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {2 b c^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 240, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {d+c^2 d x^2} \left (a b c x+a^2 \sqrt {1+c^2 x^2}+a^2 c^2 x^2 \sqrt {1+c^2 x^2}+b^2 c^2 x^2 \sqrt {1+c^2 x^2}+b^2 \left (-c^3 x^3+\sqrt {1+c^2 x^2}+c^2 x^2 \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)^2-b \sinh ^{-1}(c x) \left (-b c x-2 a \left (1+c^2 x^2\right )^{3/2}+2 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-2 a b c^3 x^3 \log (c x)+b^2 c^3 x^3 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )}{3 x^3 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

-1/3*(Sqrt[d + c^2*d*x^2]*(a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + a^2*c^2*x^2*Sqrt[1 + c^2*x^2] + b^2*c^2*x^2*Sqrt[
1 + c^2*x^2] + b^2*(-(c^3*x^3) + Sqrt[1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x
]*(-(b*c*x) - 2*a*(1 + c^2*x^2)^(3/2) + 2*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])]) - 2*a*b*c^3*x^3*Log[c*x] + b
^2*c^3*x^3*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(x^3*Sqrt[1 + c^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2556\) vs. \(2(274)=548\).
time = 4.07, size = 2557, normalized size = 8.70

method	result	size
default	\(\text {Expression too large to display}\)	\(2557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^2+1)*c^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*
x^4+3*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)^2+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*l
n(1-c*x-(c^2*x^2+1)^(1/2))*c^3+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5-b^2
*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3+1/3*b^2*(d*(c^2*x^2+1))^(1/2
)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+
1)*x^4/(c^2*x^2+1)^(1/2)*c^7+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*arcsinh(c*x)*c^6-2*a*b*
(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-10/3*a*b*(d*(c^2*x^2+1))^(1/2)/
(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)*c^2-20/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*
x/(c^2*x^2+1)*arcsinh(c*x)*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*arc
sinh(c*x)*c+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^7+1/3*b^2
*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+
3*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-a*b
*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*
x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c
^2*x^2+1)*c^4-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-2/3*a*b*(d*(c^2*x^
2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^
2*x^2+1)*x*arcsinh(c*x)*c^4+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^
(1/2))*c^3-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-5/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*
x^2+1)*c^4-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-10/3*b^2*(d*
(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)^2*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c
^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-5/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^
2*x^2+1)*arcsinh(c*x)^2*c^2-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)^2*c
^8-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-3*b^2*(d*(c^2*x^2+1)
)^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)^2*c^6+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*
x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^5-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2
+1)^(1/2)*arcsinh(c*x)*c^5-1/3*a^2/d/x^3*(c^2*d*x^2+d)^(3/2)+2*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+
1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^7-6*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*
arcsinh(c*x)*c^6+2*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5-1/
3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ar
csinh(c*x)^2*c^3+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c^3+2/3*b^2*
(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c^3-4/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c
^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3+1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6+1/3*a*b*(d*(c^2
*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*c^4-a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/
2)*c^3+2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*c^3

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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((a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-1/3*((-1)^(2*c^2*d*x^2 + 2*d)*c^2*d^(3/2)*log(2*c^2*d + 2*d/x^2) - c^2*d^(3/2)*log(x^2 + 1/c^2) + sqrt(c^4*d*
x^4 + 2*c^2*d*x^2 + d)*d/x^2)*a*b*c/d - 1/3*b^2*((c^2*sqrt(d)*x^2 + sqrt(d))*sqrt(c^2*x^2 + 1)*log(c*x + sqrt(
c^2*x^2 + 1))^2/x^3 - 3*integrate(2/3*((c^2*x^2 + 1)*c^2*sqrt(d)*x + (c^3*sqrt(d)*x^2 + c*sqrt(d))*sqrt(c^2*x^
2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c*x^4 + sqrt(c^2*x^2 + 1)*x^3), x)) - 2/3*(c^2*d*x^2 + d)^(3/2)*a*b*arcs
inh(c*x)/(d*x^3) - 1/3*(c^2*d*x^2 + d)^(3/2)*a^2/(d*x^3)

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Fricas [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((a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2*(c**2*d*x**2+d)**(1/2)/x**4,x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2/x**4, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2))/x^4,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2))/x^4, x)

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