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

Optimal. Leaf size=105 \[ \frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.114713, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5737, 29, 5675} \[ \frac{c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5737

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

Rule 29

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

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
]

Rubi steps

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

Mathematica [A]  time = 0.320564, size = 129, normalized size = 1.23 \[ -\frac{a \sqrt{d \left (c^2 x^2+1\right )}}{x}+a c \sqrt{d} \log \left (\sqrt{d} \sqrt{d \left (c^2 x^2+1\right )}+c d x\right )+\frac{b c \sqrt{d \left (c^2 x^2+1\right )} \left (-\frac{2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c x}+2 \log (c x)+\sinh ^{-1}(c x)^2\right )}{2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.128, size = 263, normalized size = 2.5 \begin{align*} -{\frac{a}{dx} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+a{c}^{2}x\sqrt{{c}^{2}d{x}^{2}+d}+{a{c}^{2}d\ln \left ({{c}^{2}dx{\frac{1}{\sqrt{{c}^{2}d}}}}+\sqrt{{c}^{2}d{x}^{2}+d} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}c}{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{b{\it Arcsinh} \left ( cx \right ) c\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}}{{c}^{2}{x}^{2}+1}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{x \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{bc\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))*(c^2*d*x^2+d)^(1/2)/x^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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