Integrand size = 21, antiderivative size = 626 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a \sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a \sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {3 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {3 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}} \]
-I*c*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)/a /(a^2*c*x^2+c)^(1/2)-6*I*c*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1 /2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+3/2*I*c*arctan(a*x)^2*polylog (2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2) -3/2*I*c*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1 )^(1/2)/a/(a^2*c*x^2+c)^(1/2)+3*I*c*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x) ^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-3*I*c*polylog(2,I*(1+I*a*x )^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-3*c*arcta n(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2* c*x^2+c)^(1/2)+3*c*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a ^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-3*I*c*polylog(4,-I*(1+I*a*x)/(a^2*x^ 2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+3*I*c*polylog(4,I*(1+I *a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-3/2*arcta n(a*x)^2*(a^2*c*x^2+c)^(1/2)/a+1/2*x*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)
Time = 0.64 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.41 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {i \sqrt {c \left (1+a^2 x^2\right )} \left (12 \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)-3 i \sqrt {1+a^2 x^2} \arctan (a x)^2+i a x \sqrt {1+a^2 x^2} \arctan (a x)^3+2 \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \left (2+\arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+3 \left (2+\arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-6 i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+6 i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+6 \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-6 \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )\right )}{2 a \sqrt {1+a^2 x^2}} \]
((-1/2*I)*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a *x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)* PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^ (I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I *ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2])
Time = 1.36 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5415, 5425, 5421, 5423, 3042, 4669, 3011, 7163, 2720, 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.
| \(\displaystyle \int \arctan (a x)^3 \sqrt {a^2 c x^2+c} \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle 3 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} c \int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle \frac {3 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\) |
(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcT an[a*x]^3)/2 + (3*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))/Sqr t[c + a^2*c*x^2] + (c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]* ArcTan[a*x]^3 + 3*(I*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (2 *I)*((-I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + PolyLog[4, (-I) *E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + PolyLog[4, I* E^(I*ArcTan[a*x])]))))/(2*a*Sqrt[c + a^2*c*x^2])
3.5.15.3.1 Defintions of rubi rules used
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_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
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[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[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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 4.34 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (x \arctan \left (a x \right ) a -3\right )}{2 a}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 a \sqrt {a^{2} x^{2}+1}}\) | \(422\) |
1/2/a*(c*(a*x-I)*(I+a*x))^(1/2)*arctan(a*x)^2*(x*arctan(a*x)*a-3)-1/2*(c*( a*x-I)*(I+a*x))^(1/2)*(arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-a rctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan(a*x)^2*polylog (2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x) /(a^2*x^2+1)^(1/2))+6*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*ar ctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*ln(1-I*( 1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1 )^(1/2))+6*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(4,I*(1+ I*a*x)/(a^2*x^2+1)^(1/2))-6*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*d ilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a/(a^2*x^2+1)^(1/2)
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3} \,d x } \]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]