3.2.52 \(\int \frac {a+b \sinh ^{-1}(c x)}{x^2 \sqrt {d+c^2 d x^2}} \, dx\) [152]

Optimal. Leaf size=63 \[ -\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {b c \sqrt {1+c^2 x^2} \log (x)}{\sqrt {d+c^2 d x^2}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 29} \begin {gather*} \frac {b c \sqrt {c^2 x^2+1} \log (x)}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 67, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {b c \sqrt {d \left (1+c^2 x^2\right )} \log (x)}{d \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(d*x)) + (b*c*Sqrt[d*(1 + c^2*x^2)]*Log[x])/(d*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. \(182\) vs. \(2(57)=114\).
time = 1.97, size = 183, normalized size = 2.90

method	result	size
default	\(-\frac {a \sqrt {c^{2} d \,x^{2}+d}}{d x}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\left (c^{2} x^{2}+1\right ) x d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 101, normalized size = 1.60 \begin {gather*} -\frac {{\left (\left (-1\right )^{2 \, c^{2} d x^{2} + 2 \, d} \sqrt {d} \log \left (2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) - \sqrt {d} \log \left (x^{2} + \frac {1}{c^{2}}\right )\right )} b c}{2 \, d} - \frac {\sqrt {c^{2} d x^{2} + d} b \operatorname {arsinh}\left (c x\right )}{d x} - \frac {\sqrt {c^{2} d x^{2} + d} a}{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((-1)^(2*c^2*d*x^2 + 2*d)*sqrt(d)*log(2*c^2*d + 2*d/x^2) - sqrt(d)*log(x^2 + 1/c^2))*b*c/d - sqrt(c^2*d*x
^2 + d)*b*arcsinh(c*x)/(d*x) - sqrt(c^2*d*x^2 + d)*a/(d*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (57) = 114\).
time = 0.38, size = 132, normalized size = 2.10 \begin {gather*} \frac {b c \sqrt {d} x \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} + \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} + d}{c^{2} x^{4} + x^{2}}\right ) - 2 \, \sqrt {c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {c^{2} d x^{2} + d} a}{2 \, d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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