3.160 \(\int \frac{a+b \sinh ^{-1}(c x)}{(d+c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt{c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.0382054, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5687, 260} \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5687

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*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
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 260

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]

Rubi steps

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

Mathematica [A]  time = 0.159034, size = 100, normalized size = 1.32 \[ \frac{\sqrt{c^2 d x^2+d} \left (2 a c x \sqrt{c^2 x^2+1}-\left (b c^2 x^2+b\right ) \log \left (c^2 x^2+1\right )+2 b c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)\right )}{2 c d^2 \left (c^2 x^2+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.085, size = 143, normalized size = 1.9 \begin{align*}{\frac{ax}{d}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) }{c{d}^{2}}\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}{{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b}{c{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.24511, size = 88, normalized size = 1.16 \begin{align*} -\frac{b c \sqrt{\frac{1}{c^{4} d}} \log \left (x^{2} + \frac{1}{c^{2}}\right )}{2 \, d} + \frac{b x \operatorname{arsinh}\left (c x\right )}{\sqrt{c^{2} d x^{2} + d} d} + \frac{a x}{\sqrt{c^{2} d x^{2} + d} d} \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)^(3/2),x, algorithm="maxima")

[Out]

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

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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 )}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), 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 )}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{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)**(3/2),x)

[Out]

Integral((a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(3/2), 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}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

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