3.3.46 \(\int \frac {x (a+b \sinh ^{-1}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [246]

Optimal. Leaf size=145 \[ \frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{6 c^2 d^3} \]

[Out]

1/12*b^2/c^2/d^3/(c^2*x^2+1)+1/6*b*x*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+1)^(3/2)-1/4*(a+b*arcsinh(c*x))^2/c^2/d
^3/(c^2*x^2+1)^2-1/6*b^2*ln(c^2*x^2+1)/c^2/d^3+1/3*b*x*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5798, 5788, 5787, 266, 267} \begin {gather*} \frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {c^2 x^2+1}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b^2}{12 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {b^2 \log \left (c^2 x^2+1\right )}{6 c^2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 266

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

Rule 267

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

Rule 5787

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

Rule 5788

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

Rule 5798

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

Rubi steps

\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}\\ &=\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \int \frac {x}{\left (1+c^2 x^2\right )^2} \, dx}{6 d^3}+\frac {b \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{3 d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{6 c^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 152, normalized size = 1.05 \begin {gather*} \frac {-3 a^2+b^2+b^2 c^2 x^2+6 a b c x \sqrt {1+c^2 x^2}+4 a b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-3 a+b c x \sqrt {1+c^2 x^2} \left (3+2 c^2 x^2\right )\right ) \sinh ^{-1}(c x)-3 b^2 \sinh ^{-1}(c x)^2-2 \left (b+b c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{12 d^3 \left (c+c^3 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(409\) vs. \(2(131)=262\).
time = 4.23, size = 410, normalized size = 2.83

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right )}{3 d^{3}}+\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right ) c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 b^{2} \arcsinh \left (c x \right ) c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{4 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} c^{2} x^{2}}{12 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2}}{12 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 d^{3}}+\frac {2 a b \left (-\frac {\arcsinh \left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(410\)
default	\(\frac {-\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right )}{3 d^{3}}+\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right ) c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 b^{2} \arcsinh \left (c x \right ) c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{4 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} c^{2} x^{2}}{12 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2}}{12 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 d^{3}}+\frac {2 a b \left (-\frac {\arcsinh \left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-1/4*a^2/d^3/(c^2*x^2+1)^2+2/3*b^2/d^3*arcsinh(c*x)+1/3*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)*arcsinh(c*x)*(c^2
*x^2+1)^(1/2)*c^3*x^3-1/3*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)*arcsinh(c*x)*c^4*x^4+1/2*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)
*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x-2/3*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)*arcsinh(c*x)*c^2*x^2-1/4*b^2/d^3/(c^4*x^
4+2*c^2*x^2+1)*arcsinh(c*x)^2+1/12*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)*c^2*x^2-1/3*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)*arc
sinh(c*x)+1/12*b^2/d^3/(c^4*x^4+2*c^2*x^2+1)-1/3*b^2/d^3*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+2*a*b/d^3*(-1/4*arcsi
nh(c*x)/(c^2*x^2+1)^2+1/12/(c^2*x^2+1)^(3/2)*c*x+1/6*c*x/(c^2*x^2+1)^(1/2)))

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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*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/4*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) - 1/4*a^2/(c^6*d^3*x^4 + 2*c^4
*d^3*x^2 + c^2*d^3) + integrate(1/2*((4*a*b*c^2 + b^2*c^2)*x^2 + sqrt(c^2*x^2 + 1)*(4*a*b*c + b^2*c)*x + b^2)*
log(c*x + sqrt(c^2*x^2 + 1))/(c^8*d^3*x^7 + 3*c^6*d^3*x^5 + 3*c^4*d^3*x^3 + c^2*d^3*x + (c^7*d^3*x^6 + 3*c^5*d
^3*x^4 + 3*c^3*d^3*x^2 + c*d^3)*sqrt(c^2*x^2 + 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (131) = 262\).
time = 0.46, size = 273, normalized size = 1.88 \begin {gather*} \frac {4 \, a b c^{4} x^{4} + {\left (8 \, a b + b^{2}\right )} c^{2} x^{2} - 3 \, b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - 3 \, a^{2} + 4 \, a b + b^{2} - 2 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} + 6 \, a b c^{2} x^{2} + {\left (2 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (a b c^{4} x^{4} + 2 \, a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (2 \, a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*a*b*c^4*x^4 + (8*a*b + b^2)*c^2*x^2 - 3*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2 - 3*a^2 + 4*a*b + b^2 - 2*(
b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) + 2*(3*a*b*c^4*x^4 + 6*a*b*c^2*x^2 + (2*b^2*c^3*x^3 + 3*b^
2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 6*(a*b*c^4*x^4 + 2*a*b*c^2*x^2 + a*b)*log(-c*x + sqrt
(c^2*x^2 + 1)) + 2*(2*a*b*c^3*x^3 + 3*a*b*c*x)*sqrt(c^2*x^2 + 1))/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**3,x)

[Out]

(Integral(a**2*x/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(b**2*x*asinh(c*x)**2/(c**6*x**6 +
3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(2*a*b*x*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),
x))/d**3

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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*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x/(c^2*d*x^2 + d)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^3,x)

[Out]

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^3, x)

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