3.293 \(\int \frac {x^2 (a+b \sinh ^{-1}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\)

Optimal. Leaf size=204 \[ \frac {x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt {c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.28, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5758, 5677, 5675, 5661, 321, 215} \[ -\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {c^2 d x^2+d}}+\frac {x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt {c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 215

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 321

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5675

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

Rule 5677

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e,
c^2*d] &&  !GtQ[d, 0]

Rule 5758

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

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d+c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.89, size = 198, normalized size = 0.97 \[ \frac {12 a^2 c x \left (c^2 d x^2+d\right )-12 a^2 \sqrt {d} \sqrt {c^2 d x^2+d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )-6 a b d \sqrt {c^2 x^2+1} \left (2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-\sinh \left (2 \sinh ^{-1}(c x)\right )\right )+\cosh \left (2 \sinh ^{-1}(c x)\right )\right )-b^2 d \sqrt {c^2 x^2+1} \left (4 \sinh ^{-1}(c x)^3-3 \left (2 \sinh ^{-1}(c x)^2+1\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )+6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 c^3 d \sqrt {c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a^2*c*x*(d + c^2*d*x^2) - 12*a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 6*
a*b*d*Sqrt[1 + c^2*x^2]*(Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] - Sinh[2*ArcSinh[c*x]])) - b^2*d*
Sqrt[1 + c^2*x^2]*(4*ArcSinh[c*x]^3 + 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - 3*(1 + 2*ArcSinh[c*x]^2)*Sinh[2*Ar
cSinh[c*x]]))/(24*c^3*d*Sqrt[d + c^2*d*x^2])

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{2}}{\sqrt {c^{2} d x^{2} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{\sqrt {c^{2} d x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.40, size = 530, normalized size = 2.60 \[ \frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{6 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3}}{4 d \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{4 c^{2} d \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{2 c d \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{4 c^{3} d \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{3}}{2 d \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x}{2 c^{2} d \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{d \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{2 c d \sqrt {c^{2} x^{2}+1}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{c^{2} d \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}}{4 c^{3} d \sqrt {c^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x/c^2/d*(c^2*d*x^2+d)^(1/2)-1/2*a^2/c^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)-1/
6*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3/d*arcsinh(c*x)^3+1/4*b^2*(d*(c^2*x^2+1))^(1/2)/d/(c^2*x^2+1)
*x^3+1/4*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/d/(c^2*x^2+1)*x-1/2*b^2*(d*(c^2*x^2+1))^(1/2)/c/d/(c^2*x^2+1)^(1/2)*arc
sinh(c*x)*x^2-1/4*b^2*(d*(c^2*x^2+1))^(1/2)/c^3/d/(c^2*x^2+1)^(1/2)*arcsinh(c*x)+1/2*b^2*(d*(c^2*x^2+1))^(1/2)
/d/(c^2*x^2+1)*arcsinh(c*x)^2*x^3+1/2*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/d/(c^2*x^2+1)*arcsinh(c*x)^2*x-1/2*a*b*(d*
(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3/d*arcsinh(c*x)^2+a*b*(d*(c^2*x^2+1))^(1/2)/d/(c^2*x^2+1)*arcsinh(c*x)
*x^3-1/2*a*b*(d*(c^2*x^2+1))^(1/2)/c/d/(c^2*x^2+1)^(1/2)*x^2+a*b*(d*(c^2*x^2+1))^(1/2)/c^2/d/(c^2*x^2+1)*arcsi
nh(c*x)*x-1/4*a*b*(d*(c^2*x^2+1))^(1/2)/c^3/d/(c^2*x^2+1)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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