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

Optimal. Leaf size=66 \[ c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-\frac{d \left (a+b \sinh ^{-1}(c x)\right )}{x}-b c d \sqrt{c^2 x^2+1}-b c d \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0814023, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {14, 5730, 12, 446, 80, 63, 208} \[ c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-\frac{d \left (a+b \sinh ^{-1}(c x)\right )}{x}-b c d \sqrt{c^2 x^2+1}-b c d \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5730

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0265772, size = 74, normalized size = 1.12 \[ a c^2 d x-\frac{a d}{x}-b c d \sqrt{c^2 x^2+1}-b c d \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+b c^2 d x \sinh ^{-1}(c x)-\frac{b d \sinh ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 69, normalized size = 1.1 \begin{align*} c \left ( da \left ( cx-{\frac{1}{cx}} \right ) +db \left ({\it Arcsinh} \left ( cx \right ) cx-{\frac{{\it Arcsinh} \left ( cx \right ) }{cx}}-\sqrt{{c}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09682, size = 89, normalized size = 1.35 \begin{align*} a c^{2} d x +{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b c d -{\left (c \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arsinh}\left (c x\right )}{x}\right )} b d - \frac{a d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.61708, size = 346, normalized size = 5.24 \begin{align*} \frac{a c^{2} d x^{2} - b c d x \log \left (-c x + \sqrt{c^{2} x^{2} + 1} + 1\right ) + b c d x \log \left (-c x + \sqrt{c^{2} x^{2} + 1} - 1\right ) - \sqrt{c^{2} x^{2} + 1} b c d x -{\left (b c^{2} - b\right )} d x \log \left (-c x + \sqrt{c^{2} x^{2} + 1}\right ) - a d +{\left (b c^{2} d x^{2} -{\left (b c^{2} - b\right )} d x - b d\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}{\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((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x^2,x, algorithm="giac")

[Out]

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