Integrand size = 24, antiderivative size = 458 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {2 i a c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:
-(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x-2*I*a*c*(a^2*x^2+1)^(1/2)*arctan((1+I *a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2)-4*a*c*(a^2*x^2+ 1)^(1/2)*arctan(a*x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c )^(1/2)+2*I*a*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2* x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-2*I*a*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*po lylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+2*I*a*c*(a^2*x^ 2+1)^(1/2)*polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2) -2*I*a*c*(a^2*x^2+1)^(1/2)*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2 *c*x^2+c)^(1/2)-2*a*c*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1) ^(1/2))/(a^2*c*x^2+c)^(1/2)+2*a*c*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I*a*x)/ (a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)
Time = 0.61 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=-\frac {a \sqrt {c \left (1+a^2 x^2\right )} \left (\frac {\sqrt {1+a^2 x^2} \arctan (a x)^2}{a x}-2 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-\arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+\arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+2 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-2 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+2 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}} \] Input:
Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^2,x]
Output:
-((a*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2)/(a*x) - 2*Ar cTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan [a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTa n[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I *E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 2*PolyLog[3, ( -I)*E^(I*ArcTan[a*x])] - 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2* x^2])
Time = 2.06 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.66, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5485, 5425, 5423, 3042, 4669, 3011, 2720, 5479, 5493, 5489, 7143}
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.
\(\displaystyle \int \frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{x^2} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\) |
\(\Big \downarrow \) 5425 |
\(\displaystyle \frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\) |
\(\Big \downarrow \) 5423 |
\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5479 |
\(\displaystyle c \left (2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle c \left (\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5489 |
\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle c \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\) |
Input:
Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^2,x]
Output:
c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*( -2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(S qrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I *a*x]]))/Sqrt[c + a^2*c*x^2]) + (a*c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I *ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTa n[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[ 2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])])))/Sqrt[c + a^2* c*x^2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt Q[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]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 2.55 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2}}{x}+\frac {a \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {dilog}\left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\) | \(308\) |
Input:
int((a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x^2,x,method=_RETURNVERBOSE)
Output:
-(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)^2/x+a*(c*(a*x-I)*(a*x+I))^(1/2)*(ar ctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^2*ln(1+I*(1+I* a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^ (1/2))+2*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan( a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/ 2))+2*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*polylog(3,I*(1+I*a*x)/(a^2* x^2+1)^(1/2))-2*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/ 2)
\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x^2,x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2/x^2, x)
\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx \] Input:
integrate((a**2*c*x**2+c)**(1/2)*atan(a*x)**2/x**2,x)
Output:
Integral(sqrt(c*(a**2*x**2 + 1))*atan(a*x)**2/x**2, x)
\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2/x^2, x)
Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x^2,x, algorithm="giac")
Output:
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. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c}}{x^2} \,d x \] Input:
int((atan(a*x)^2*(c + a^2*c*x^2)^(1/2))/x^2,x)
Output:
int((atan(a*x)^2*(c + a^2*c*x^2)^(1/2))/x^2, x)
\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx=\sqrt {c}\, \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2}}{x^{2}}d x \right ) \] Input:
int((a^2*c*x^2+c)^(1/2)*atan(a*x)^2/x^2,x)
Output:
sqrt(c)*int((sqrt(a**2*x**2 + 1)*atan(a*x)**2)/x**2,x)