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

Optimal. Leaf size=400 \[ -\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

2/(27*a^6*c*(c + a^2*c*x^2)^(3/2)) - 32/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*a^3*c*(c + a^
2*c*x^2)^(3/2)) - (10*x*ArcTan[a*x])/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a^4*c*(c + a^2*c
*x^2)^(3/2)) + (5*ArcTan[a*x]^2)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^6*c^
3) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2
]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]
) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.81751, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4964, 4930, 4890, 4886, 4894, 4940, 266, 43} \[ -\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(27*a^6*c*(c + a^2*c*x^2)^(3/2)) - 32/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*a^3*c*(c + a^
2*c*x^2)^(3/2)) - (10*x*ArcTan[a*x])/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a^4*c*(c + a^2*c
*x^2)^(3/2)) + (5*ArcTan[a*x]^2)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^6*c^
3) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2
]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]
) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2])

Rule 4964

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

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 4890

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/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 4886

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

Rule 4894

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b
*p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int
[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/m^2, Int[(f*x)^m*(d +
e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p
)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && G
tQ[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^5 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac{\int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a^2}+\frac{\int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac{2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac{\int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{\operatorname{Subst}\left (\int \frac{x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{9 a^2}-\frac{2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^5 c}-\frac{2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}\\ &=-\frac{10}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac{1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{9 a^2}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2}{27 a^6 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{32}{9 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{4 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.31527, size = 229, normalized size = 0.57 \[ \frac{-432 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+432 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-9 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2 \left (-20 \cos \left (2 \tan ^{-1}(a x)\right )+\cos \left (4 \tan ^{-1}(a x)\right )-45\right )+6 \tan ^{-1}(a x) \left (-72 \sqrt{a^2 x^2+1} \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+72 \sqrt{a^2 x^2+1} \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+\left (a^2 x^2+1\right ) \sin \left (4 \tan ^{-1}(a x)\right )-124 a x\right )+8 \left (\cos \left (2 \tan ^{-1}(a x)\right )-95\right )}{216 a^6 c^2 \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*(-95 + Cos[2*ArcTan[a*x]]) - 9*(1 + a^2*x^2)*ArcTan[a*x]^2*(-45 - 20*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]
]) - (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, I*E^(
I*ArcTan[a*x])] + 6*ArcTan[a*x]*(-124*a*x - 72*Sqrt[1 + a^2*x^2]*Log[1 - I*E^(I*ArcTan[a*x])] + 72*Sqrt[1 + a^
2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] + (1 + a^2*x^2)*Sin[4*ArcTan[a*x]]))/(216*a^6*c^2*Sqrt[c + a^2*c*x^2])

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Maple [A]  time = 1.515, size = 454, normalized size = 1.1 \begin{align*}{\frac{ \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 7\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-14+14\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -7+7\,iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{216\,{c}^{3}{a}^{6} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{c}^{3}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{2\,i}{{c}^{3}{a}^{6}} \left ( i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(6*I*arctan(a*x)+9*arctan(a*x)^2-2)*(I*x^3*a^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1
)^2/c^3/a^6+7/8*(arctan(a*x)^2-2+2*I*arctan(a*x))*(1+I*a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/a^6/c^3/(a^2*x^2+1)-7/8*
(c*(a*x-I)*(a*x+I))^(1/2)*(-1+I*a*x)*(arctan(a*x)^2-2-2*I*arctan(a*x))/a^6/c^3/(a^2*x^2+1)-1/216*(c*(a*x-I)*(a
*x+I))^(1/2)*(I*x^3*a^3-3*a^2*x^2-3*I*a*x+1)*(-6*I*arctan(a*x)+9*arctan(a*x)^2-2)/a^6/c^3/(a^4*x^4+2*a^2*x^2+1
)+arctan(a*x)^2*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/a^6-2*I*(I*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*arc
tan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-dilog(1-I*(1+I*a*x)/(a^2*x
^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/a^6/c^3

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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(x^5*arctan(a*x)^2/(a^2*c*x^2+c)^(5/2),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{a^{2} c x^{2} + c} x^{5} \arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^5*arctan(a*x)^2/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \operatorname{atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*atan(a*x)**2/(c*(a**2*x**2 + 1))**(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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