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

Optimal. Leaf size=55 \[ \frac{b x}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]

[Out]

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

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Rubi [A]  time = 0.0496438, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5717, 191} \[ \frac{b x}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0674566, size = 74, normalized size = 1.35 \[ -\frac{a}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac{b x}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 61, normalized size = 1.1 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{a}{2\,{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{b}{{d}^{2}} \left ( -{\frac{{\it Arcsinh} \left ( cx \right ) }{2\,{c}^{2}{x}^{2}+2}}+{\frac{cx}{2}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, b{\left (\frac{2 \, \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 1}{c^{4} d^{2} x^{2} + c^{2} d^{2}} - 4 \, \int \frac{1}{2 \,{\left (c^{6} d^{2} x^{5} + 2 \, c^{4} d^{2} x^{3} + c^{2} d^{2} x +{\left (c^{5} d^{2} x^{4} + 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac{a}{2 \,{\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.39258, size = 135, normalized size = 2.45 \begin{align*} \frac{a c^{2} x^{2} + \sqrt{c^{2} x^{2} + 1} b c x - b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{2 \,{\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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