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

Optimal. Leaf size=171 \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{b \sqrt{c^2 x^2+1}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{c^2 x^2+1}} \]

[Out]

b/(2*c^5*d^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[1 + c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcSinh[c*x]))/(2*c^4*d^2) - (
x^3*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*(1 + c^2*x^2)) - (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^5*d^2
) + (((3*I)/2)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*
d^2)

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Rubi [A]  time = 0.240776, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5751, 5767, 5693, 4180, 2279, 2391, 261, 266, 43} \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{b \sqrt{c^2 x^2+1}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b/(2*c^5*d^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[1 + c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcSinh[c*x]))/(2*c^4*d^2) - (
x^3*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*(1 + c^2*x^2)) - (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^5*d^2
) + (((3*I)/2)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*
d^2)

Rule 5751

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

Rule 5767

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(e*(m + 2*p + 1)), x] + (-Dist[(f^2*(m - 1))/(c^2
*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d
+ e*x^2)^FracPart[p])/(c*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(
a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[m
, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx &=-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac{b \int \frac{x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}+\frac{3 \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{(3 b) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{2 c^3 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac{3 \int \frac{a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac{3 b \sqrt{1+c^2 x^2}}{2 c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \left (1+c^2 x\right )^{3/2}}+\frac{1}{c^2 \sqrt{1+c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{1+c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{(3 i b) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^5 d^2}-\frac{(3 i b) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{1+c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{1+c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{3 i b \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end{align*}

Mathematica [A]  time = 0.339399, size = 268, normalized size = 1.57 \[ \frac{3 i b \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-3 i b \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+2 a c^3 x^3-3 a c^2 x^2 \tan ^{-1}(c x)+3 a c x-3 a \tan ^{-1}(c x)-2 b c^2 x^2 \sqrt{c^2 x^2+1}-b \sqrt{c^2 x^2+1}+2 b c^3 x^3 \sinh ^{-1}(c x)-3 i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+3 i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+3 b c x \sinh ^{-1}(c x)-3 i b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+3 i b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{2 c^5 d^2 \left (c^2 x^2+1\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*a*c*x + 2*a*c^3*x^3 - b*Sqrt[1 + c^2*x^2] - 2*b*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c*x*ArcSinh[c*x] + 2*b*c^3*
x^3*ArcSinh[c*x] - 3*a*ArcTan[c*x] - 3*a*c^2*x^2*ArcTan[c*x] - (3*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]]
- (3*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (3*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (3
*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (3*I)*b*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] -
(3*I)*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c^5*d^2*(1 + c^2*x^2))

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Maple [A]  time = 0.016, size = 285, normalized size = 1.7 \begin{align*}{\frac{ax}{{c}^{4}{d}^{2}}}+{\frac{ax}{2\,{c}^{4}{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,a\arctan \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) x}{{c}^{4}{d}^{2}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) x}{2\,{c}^{4}{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,b{\it Arcsinh} \left ( cx \right ) \arctan \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}}-{\frac{b{x}^{2}}{{c}^{3}{d}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b}{2\,{c}^{5}{d}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{3\,b\arctan \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}\ln \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{3\,b\arctan \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}\ln \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{3\,i}{2}}b}{{c}^{5}{d}^{2}}{\it dilog} \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{3\,i}{2}}b}{{c}^{5}{d}^{2}}{\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(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^2,x)

[Out]

1/c^4*a/d^2*x+1/2/c^4*a/d^2*x/(c^2*x^2+1)-3/2/c^5*a/d^2*arctan(c*x)+1/c^4*b/d^2*arcsinh(c*x)*x+1/2/c^4*b/d^2*a
rcsinh(c*x)*x/(c^2*x^2+1)-3/2/c^5*b/d^2*arcsinh(c*x)*arctan(c*x)-1/c^3*b/d^2*x^2/(c^2*x^2+1)^(1/2)-1/2*b/c^5/d
^2/(c^2*x^2+1)^(1/2)-3/2/c^5*b/d^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2/c^5*b/d^2*arctan(c*x)*l
n(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*I/c^5*b/d^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*I/c^5*b/d^2*dilo
g(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}{2} \, a{\left (\frac{x}{c^{6} d^{2} x^{2} + c^{4} d^{2}} + \frac{2 \, x}{c^{4} d^{2}} - \frac{3 \, \arctan \left (c x\right )}{c^{5} d^{2}}\right )} + b \int \frac{x^{4} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

(Integral(a*x**4/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b*x**4*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^{4}}{{\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^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

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