3.79 \(\int \frac{(d+e x^2) (a+b \text{csch}^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=91 \[ -\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d \sqrt{-c^2 x^2-1}}{\sqrt{-c^2 x^2}}-\frac{b e x \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{\sqrt{-c^2 x^2}} \]

[Out]

(b*c*d*Sqrt[-1 - c^2*x^2])/Sqrt[-(c^2*x^2)] - (d*(a + b*ArcCsch[c*x]))/x + e*x*(a + b*ArcCsch[c*x]) - (b*e*x*A
rcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/Sqrt[-(c^2*x^2)]

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Rubi [A]  time = 0.0676791, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 6302, 451, 217, 203} \[ -\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d \sqrt{-c^2 x^2-1}}{\sqrt{-c^2 x^2}}-\frac{b e x \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{\sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*Sqrt[-1 - c^2*x^2])/Sqrt[-(c^2*x^2)] - (d*(a + b*ArcCsch[c*x]))/x + e*x*(a + b*ArcCsch[c*x]) - (b*e*x*A
rcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/Sqrt[-(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 6302

Int[((a_.) + ArcCsch[(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*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 451

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.12998, size = 89, normalized size = 0.98 \[ -\frac{a d}{x}+a e x+b c d \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{b d \text{csch}^{-1}(c x)}{x}+b e x \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.194, size = 107, normalized size = 1.2 \begin{align*} c \left ({\frac{a}{{c}^{2}} \left ( cxe-{\frac{cd}{x}} \right ) }+{\frac{b}{{c}^{2}} \left ({\rm arccsch} \left (cx\right )cxe-{\frac{{\rm arccsch} \left (cx\right )cd}{x}}+{\frac{1}{{c}^{2}{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1} \left ({c}^{2}d\sqrt{{c}^{2}{x}^{2}+1}+e{\it Arcsinh} \left ( cx \right ) cx \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.974879, size = 113, normalized size = 1.24 \begin{align*}{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsch}\left (c x\right )}{x}\right )} b d + a e x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e}{2 \, c} - \frac{a d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.76196, size = 493, normalized size = 5.42 \begin{align*} \frac{b c^{2} d x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + b c^{2} d x + a c e x^{2} - b e x \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - a c d -{\left (b c d - b c e\right )} x \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) +{\left (b c d - b c e\right )} x \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) +{\left (b c e x^{2} - b c d +{\left (b c d - b c e\right )} x\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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