3.220 \(\int \frac{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)}{x} \, dx\)

Optimal. Leaf size=329 \[ \frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{29}{120} a c^2 x \sqrt{a^2 c x^2+c}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{149}{120} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{1}{20} a c x \left (a^2 c x^2+c\right )^{3/2}+\frac{1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]

[Out]

(-29*a*c^2*x*Sqrt[c + a^2*c*x^2])/120 - (a*c*x*(c + a^2*c*x^2)^(3/2))/20 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]
+ (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/5 - (2*c^3*Sqrt[1 + a^2*x^2]*
ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (149*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)
/Sqrt[c + a^2*c*x^2]])/120 + (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c +
a^2*c*x^2] - (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.550225, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4950, 4946, 4958, 4954, 217, 206, 4930, 195} \[ \frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{29}{120} a c^2 x \sqrt{a^2 c x^2+c}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{149}{120} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{1}{20} a c x \left (a^2 c x^2+c\right )^{3/2}+\frac{1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x,x]

[Out]

(-29*a*c^2*x*Sqrt[c + a^2*c*x^2])/120 - (a*c*x*(c + a^2*c*x^2)^(3/2))/20 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]
+ (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/5 - (2*c^3*Sqrt[1 + a^2*x^2]*
ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (149*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)
/Sqrt[c + a^2*c*x^2]])/120 + (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c +
a^2*c*x^2] - (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(
m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcTan[c*x]
))/Sqrt[d + e*x^2], x], x] - Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]

Rule 4958

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 4954

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(-2*(a + b*ArcTan[c
*x])*ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x] + (Simp[(I*b*PolyLog[2, -(Sqrt[1 + I*c*x]/Sqrt[1 -
I*c*x])])/Sqrt[d], x] - Simp[(I*b*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c,
d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

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 206

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

Rule 4930

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{1}{5} (a c) \int \left (c+a^2 c x^2\right )^{3/2} \, dx+c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{1}{20} \left (3 a c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx-\frac{1}{3} \left (a c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx+c^3 \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{29}{120} a c^2 x \sqrt{c+a^2 c x^2}-\frac{1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{1}{40} \left (3 a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{6} \left (a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{29}{120} a c^2 x \sqrt{c+a^2 c x^2}-\frac{1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{1}{40} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )-\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{29}{120} a c^2 x \sqrt{c+a^2 c x^2}-\frac{1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149}{120} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.343376, size = 268, normalized size = 0.81 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (120 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-120 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-6 a^3 x^3 \sqrt{a^2 x^2+1}-35 a x \sqrt{a^2 x^2+1}+24 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+88 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+184 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-29 \sinh ^{-1}(a x)+120 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-120 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+120 \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-120 \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )\right )}{120 \sqrt{a^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(-35*a*x*Sqrt[1 + a^2*x^2] - 6*a^3*x^3*Sqrt[1 + a^2*x^2] - 29*ArcSinh[a*x] + 184*Sqrt
[1 + a^2*x^2]*ArcTan[a*x] + 88*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 24*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x
] + 120*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 120*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 120*Log[Cos[ArcT
an[a*x]/2] - Sin[ArcTan[a*x]/2]] - 120*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (120*I)*PolyLog[2, -E^(I
*ArcTan[a*x])] - (120*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(120*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.342, size = 198, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 24\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-6\,{a}^{3}{x}^{3}+88\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-35\,ax+184\,\arctan \left ( ax \right ) \right ) }{120}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{c}^{2}}{60}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 60\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -149\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -60\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -60\,i{\it dilog} \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x,x)

[Out]

1/120*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(24*arctan(a*x)*x^4*a^4-6*a^3*x^3+88*arctan(a*x)*a^2*x^2-35*a*x+184*arctan
(a*x))-1/60*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(60*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*I*arctan((1+I*
a*x)/(a^2*x^2+1)^(1/2))-60*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))-60*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^
2*x^2+1)^(1/2)

________________________________________________________________________________________

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^2*c*x^2+c)^(5/2)*arctan(a*x)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)/x, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)/x,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)/x, x)