3.36 \(\int \frac{a+b \sinh ^{-1}(c x)}{x^4 (d+c^2 d x^2)} \, dx\)

Optimal. Leaf size=156 \[ -\frac{i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 d x^2}+\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{6 d} \]

[Out]

-(b*c*Sqrt[1 + c^2*x^2])/(6*d*x^2) - (a + b*ArcSinh[c*x])/(3*d*x^3) + (c^2*(a + b*ArcSinh[c*x]))/(d*x) + (2*c^
3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d + (7*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(6*d) - (I*b*c^3*PolyL
og[2, (-I)*E^ArcSinh[c*x]])/d + (I*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]])/d

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Rubi [A]  time = 0.246556, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5747, 5693, 4180, 2279, 2391, 266, 63, 208, 51} \[ -\frac{i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 d x^2}+\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)),x]

[Out]

-(b*c*Sqrt[1 + c^2*x^2])/(6*d*x^2) - (a + b*ArcSinh[c*x])/(3*d*x^3) + (c^2*(a + b*ArcSinh[c*x]))/(d*x) + (2*c^
3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d + (7*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(6*d) - (I*b*c^3*PolyL
og[2, (-I)*E^ArcSinh[c*x]])/d + (I*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]])/d

Rule 5747

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] + (-Dist[(c^2*(m + 2*p + 3))/(f^2
*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^
2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSin
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && Int
egerQ[m]

Rule 5693

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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 266

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

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]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rubi steps

\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left (d+c^2 d x^2\right )} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}-c^2 \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx+\frac{(b c) \int \frac{1}{x^3 \sqrt{1+c^2 x^2}} \, dx}{3 d}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+c^4 \int \frac{a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{6 d}-\frac{\left (b c^3\right ) \int \frac{1}{x \sqrt{1+c^2 x^2}} \, dx}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{c^3 \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{12 d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{6 d}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{d}-\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{6 d}-\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 d x^2}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{6 d}-\frac{i b c^3 \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{i b c^3 \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.213495, size = 247, normalized size = 1.58 \[ \frac{6 b \left (-c^2\right )^{3/2} x^3 \text{PolyLog}\left (2,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right )-6 b \left (-c^2\right )^{3/2} x^3 \text{PolyLog}\left (2,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+6 a c^2 x^2+6 a c^3 x^3 \tan ^{-1}(c x)-2 a-b c x \sqrt{c^2 x^2+1}+6 b c^2 x^2 \sinh ^{-1}(c x)+7 b c^3 x^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-6 b \left (-c^2\right )^{3/2} x^3 \sinh ^{-1}(c x) \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )+6 b \left (-c^2\right )^{3/2} x^3 \sinh ^{-1}(c x) \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-2 b \sinh ^{-1}(c x)}{6 d x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)),x]

[Out]

(-2*a + 6*a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x] + 6*b*c^2*x^2*ArcSinh[c*x] + 6*a*c^3*x^3*ArcT
an[c*x] + 7*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] - 6*b*(-c^2)^(3/2)*x^3*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x]
)/Sqrt[-c^2]] + 6*b*(-c^2)^(3/2)*x^3*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 6*b*(-c^2)^(3/2)*x^
3*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 6*b*(-c^2)^(3/2)*x^3*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/
(6*d*x^3)

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Maple [A]  time = 0.016, size = 261, normalized size = 1.7 \begin{align*} -{\frac{a}{3\,d{x}^{3}}}+{\frac{{c}^{2}a}{dx}}+{\frac{{c}^{3}a\arctan \left ( cx \right ) }{d}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{3\,d{x}^{3}}}+{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) }{dx}}+{\frac{b{c}^{3}{\it Arcsinh} \left ( cx \right ) \arctan \left ( cx \right ) }{d}}-{\frac{bc}{6\,d{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{7\,b{c}^{3}}{6\,d}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) }+{\frac{b{c}^{3}\arctan \left ( cx \right ) }{d}\ln \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{b{c}^{3}\arctan \left ( cx \right ) }{d}\ln \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i{c}^{3}b}{d}{\it dilog} \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{i{c}^{3}b}{d}{\it dilog} \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d),x)

[Out]

-1/3*a/d/x^3+c^2*a/d/x+c^3*a/d*arctan(c*x)-1/3*b/d*arcsinh(c*x)/x^3+c^2*b/d*arcsinh(c*x)/x+c^3*b/d*arcsinh(c*x
)*arctan(c*x)-1/6*b*c*(c^2*x^2+1)^(1/2)/d/x^2+7/6*c^3*b/d*arctanh(1/(c^2*x^2+1)^(1/2))+c^3*b/d*arctan(c*x)*ln(
1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-c^3*b/d*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I*c^3*b/d*dilog(1+I*(
1+I*c*x)/(c^2*x^2+1)^(1/2))+I*c^3*b/d*dilog(1-I*(1+I*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}{3} \,{\left (\frac{3 \, c^{3} \arctan \left (c x\right )}{d} + \frac{3 \, c^{2} x^{2} - 1}{d x^{3}}\right )} a + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2} d x^{6} + d x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arsinh}\left (c x\right ) + a}{c^{2} d x^{6} + d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arcsinh(c*x) + a)/(c^2*d*x^6 + d*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{2} x^{6} + x^{4}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(c**2*x**6 + x**4), x) + Integral(b*asinh(c*x)/(c**2*x**6 + x**4), x))/d

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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 )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)*x^4), x)