Integrand size = 22, antiderivative size = 300 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=-\frac {1}{2} a c \sqrt {c+a^2 c x^2}-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \]
-a*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))-3*I*a*c^2*arctan(a*x)*arct an((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+ 3/2*I*a*c^2*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2 )/(a^2*c*x^2+c)^(1/2)-3/2*I*a*c^2*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1 /2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/2*a*c*(a^2*c*x^2+c)^(1/2)-c*a rctan(a*x)*(a^2*c*x^2+c)^(1/2)/x+1/2*a^2*c*x*arctan(a*x)*(a^2*c*x^2+c)^(1/ 2)
Time = 1.07 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.73 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (-a x \sqrt {1+a^2 x^2}-2 \sqrt {1+a^2 x^2} \arctan (a x)+a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+3 a x \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-3 a x \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )-2 a x \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )\right )+2 a x \log \left (\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+3 i a x \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-3 i a x \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{2 x \sqrt {1+a^2 x^2}} \]
(c*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) - 2*Sqrt[1 + a^2*x^2]*Arc Tan[a*x] + a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 3*a*x*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - 3*a*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 2*a*x*Log[Cos[ArcTan[a*x]/2]] + 2*a*x*Log[Sin[ArcTan[a*x]/2]] + (3*I)*a*x *PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (3*I)*a*x*PolyLog[2, I*E^(I*ArcTan[a *x])]))/(2*x*Sqrt[1 + a^2*x^2])
Time = 1.52 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.33, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5485, 5413, 5425, 5421, 5485, 5425, 5421, 5479, 243, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \sqrt {a^2 c x^2+c} \arctan (a x)dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle a^2 c \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (a^2 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle c \left (c \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (c \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (c \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\) |
a^2*c*(-1/2*Sqrt[c + a^2*c*x^2]/a + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 + (c*Sqrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(2*Sqrt[c + a^2*c* x^2])) + c*(c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqr t[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]) + (a^2*c*Sqrt[1 + a^2*x^2]*(((-2*I)*Ar cTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I) *Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/ Sqrt[1 - I*a*x]])/a))/Sqrt[c + a^2*c*x^2])
3.3.13.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
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]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[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]))
Time = 0.53 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+3 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) a x -3 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) a x +3 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) a x -3 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) a x -\sqrt {a^{2} x^{2}+1}\, a x -2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a x +2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) a x -2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) c}{2 \sqrt {a^{2} x^{2}+1}\, x}\) | \(263\) |
1/2*(c*(a*x-I)*(I+a*x))^(1/2)*(arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2+3*arc tan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*a*x-3*arctan(a*x)*ln(1+I*(1+I *a*x)/(a^2*x^2+1)^(1/2))*a*x+3*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*a* x-3*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*a*x-(a^2*x^2+1)^(1/2)*a*x-2*l n((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*a*x+2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)*a *x-2*arctan(a*x)*(a^2*x^2+1)^(1/2))*c/(a^2*x^2+1)^(1/2)/x
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x^{2}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^2} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^2} \,d x \]