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

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

[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]

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Rubi [A]  time = 0.0891756, antiderivative size = 63, normalized size of antiderivative = 1., 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 = {5723, 29} \[ \frac{b c \sqrt{c^2 x^2+1} \log (x)}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \]

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 5723

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*d^IntPart[p]*(d + e
*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Arc
Sinh[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 29

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

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*}

Mathematica [A]  time = 0.158615, size = 67, normalized size = 1.06 \[ \frac{b c \log (x) \sqrt{d \left (c^2 x^2+1\right )}}{d \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \]

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]  time = 0.118, size = 183, normalized size = 2.9 \begin{align*} -{\frac{a}{dx}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{b{\it Arcsinh} \left ( cx \right ) c}{d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) x{c}^{2}}{d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{ \left ({c}^{2}{x}^{2}+1 \right ) dx}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{bc}{d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}

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)

[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)/d*x*c^2-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/d/x+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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 3.12669, size = 301, normalized size = 4.78 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asinh}{\left (c x \right )}}{x^{2} \sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{c^{2} d x^{2} + d} x^{2}}\,{d x} \end{align*}

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)