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

Optimal. Leaf size=177 \[ -\frac{b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.195302, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5742, 5760, 4182, 2279, 2391, 8} \[ -\frac{b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5742

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 + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f
, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5760

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{d+c^2 d x^2} \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int 1 \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b \sqrt{d+c^2 d x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.451983, size = 168, normalized size = 0.95 \[ \frac{b \sqrt{c^2 d x^2+d} \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt{c^2 x^2+1}}+a \sqrt{c^2 d x^2+d}-a \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+a \sqrt{d} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.146, size = 331, normalized size = 1.9 \begin{align*} -\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{{c}^{2}d{x}^{2}+d} \right ) } \right ) \sqrt{d}a+a\sqrt{{c}^{2}d{x}^{2}+d}+{\frac{b{\it Arcsinh} \left ( cx \right ){x}^{2}{c}^{2}}{{c}^{2}{x}^{2}+1}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{bcx\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) }{{c}^{2}{x}^{2}+1}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{b{\it Arcsinh} \left ( cx \right ) \sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{b\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{b{\it Arcsinh} \left ( cx \right ) \sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{b\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{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,x)

[Out]

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

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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,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}, 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,x, algorithm="fricas")

[Out]

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

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))/x, 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}\,{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,x, algorithm="giac")

[Out]

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