Integrand size = 22, antiderivative size = 281 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=-\frac {1}{6} a c x \sqrt {c+a^2 c x^2}+c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7}{6} c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \] Output:
-1/6*a*c*x*(a^2*c*x^2+c)^(1/2)+c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)+1/3*(a^2* c*x^2+c)^(3/2)*arctan(a*x)-2*c^2*(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)-7/6*c^(3/2)*arctanh(a*c^ (1/2)*x/(a^2*c*x^2+c)^(1/2))+I*c^2*(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)-I*c^2*(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)
Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (-a x \sqrt {1+a^2 x^2}+8 \sqrt {1+a^2 x^2} \arctan (a x)+2 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+6 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-6 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+\log \left (-a x+\sqrt {1+a^2 x^2}\right )+6 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-6 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+6 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-6 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{6 \sqrt {1+a^2 x^2}} \] Input:
Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]
Output:
(c*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) + 8*Sqrt[1 + a^2*x^2]*Arc Tan[a*x] + 2*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 6*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 6*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + Log[-(a*x ) + Sqrt[1 + a^2*x^2]] + 6*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 6*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (6*I)*PolyLog[2, -E^(I*Ar cTan[a*x])] - (6*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(6*Sqrt[1 + a^2*x^2])
Time = 1.01 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5485, 5465, 211, 224, 219, 5481, 224, 219, 5493, 5489}
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.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\int \sqrt {a^2 c x^2+c}dx}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\) |
\(\Big \downarrow \) 219 |
\(\displaystyle c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\) |
\(\Big \downarrow \) 5481 |
\(\displaystyle c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\) |
\(\Big \downarrow \) 5489 |
\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\) |
Input:
Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]
Output:
a^2*c*(((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c* x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a))/(3*a )) + c*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sq rt[c + a^2*c*x^2]] + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[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])
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
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] - Simp[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]
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] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq rt[d + e*x^2]), x], x] - Simp[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]
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]
Time = 1.48 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, a x -8 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}+6 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {dilog}\left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-14 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c}{6 \sqrt {a^{2} x^{2}+1}}\) | \(185\) |
Input:
int((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x,method=_RETURNVERBOSE)
Output:
-1/6/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)*(-2*arctan(a*x)*(a^2*x^2+ 1)^(1/2)*a^2*x^2+(a^2*x^2+1)^(1/2)*a*x-8*arctan(a*x)*(a^2*x^2+1)^(1/2)+6*a rctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*dilog(1+(1+I*a*x)/(a^2*x^ 2+1)^(1/2))-14*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*dilog((1+I*a*x)/( a^2*x^2+1)^(1/2)))*c
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}{x}\, dx \] Input:
integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)/x,x)
Output:
Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x, algorithm="maxima")
Output:
1/3*(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 1/6*(a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*(a*c*x*cos(1/2*arctan2(4*a*x, -a^2*x^2 + 3)) + 2*c* sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) + 1/12*(c*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) + 2, a*x + (a ^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x^2 - 3))) + c*a rctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3 )) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2 *x^2 - 3))) + 12*c*integrate(sqrt(a^2*x^2 + 1)*arctan(a*x)/x, x))*sqrt(c)
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,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. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x} \,d x \] Input:
int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x,x)
Output:
int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\frac {\sqrt {c}\, c \left (2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}+2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-\sqrt {a^{2} x^{2}+1}\, a x +6 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )}{x}d x \right )-\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )\right )}{6} \] Input:
int((a^2*c*x^2+c)^(3/2)*atan(a*x)/x,x)
Output:
(sqrt(c)*c*(2*sqrt(a**2*x**2 + 1)*atan(a*x)*a**2*x**2 + 2*sqrt(a**2*x**2 + 1)*atan(a*x) - sqrt(a**2*x**2 + 1)*a*x + 6*int((sqrt(a**2*x**2 + 1)*atan( a*x))/x,x) - log(sqrt(a**2*x**2 + 1) + a*x)))/6