Integrand size = 22, antiderivative size = 107 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c^2}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 c^{3/2}} \] Output:
-x/a^3/c/(a^2*c*x^2+c)^(1/2)+arctan(a*x)/a^4/c/(a^2*c*x^2+c)^(1/2)+(a^2*c* x^2+c)^(1/2)*arctan(a*x)/a^4/c^2-arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/ a^4/c^(3/2)
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {-a x \sqrt {c+a^2 c x^2}+\left (2+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)-\sqrt {c} \left (1+a^2 x^2\right ) \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{a^4 c^2 \left (1+a^2 x^2\right )} \] Input:
Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]
Output:
(-(a*x*Sqrt[c + a^2*c*x^2]) + (2 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x ] - Sqrt[c]*(1 + a^2*x^2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^4*c ^2*(1 + a^2*x^2))
Time = 0.47 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5499, 5465, 208, 224, 219}
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.
\(\displaystyle \int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5499 |
\(\displaystyle \frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a^2 c}-\frac {\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a^2 c}-\frac {\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}\) |
Input:
Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]
Output:
-((x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt[c + a^2*c*x^2]))/ a^2) + ((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/ Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c]))/(a^2*c)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
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_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan [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]
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.04
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) a^{2} x^{2}+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) a^{2} x^{2}-\sqrt {a^{2} x^{2}+1}\, a x +2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )\right ) \sqrt {a^{2} x^{2}+1}\, \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{4} c^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(218\) |
Input:
int(x^3*arctan(a*x)/(a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
(arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)*a ^2*x^2+ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-I)*a^2*x^2-(a^2*x^2+1)^(1/2)*a*x+2*a rctan(a*x)*(a^2*x^2+1)^(1/2)-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)+ln((1+I*a*x )/(a^2*x^2+1)^(1/2)-I))*(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/a^4/c^ 2/(a^4*x^4+2*a^2*x^2+1)
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {{\left (a^{2} x^{2} + 1\right )} \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - {\left (a^{2} x^{2} + 2\right )} \arctan \left (a x\right )\right )}}{2 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")
Output:
1/2*((a^2*x^2 + 1)*sqrt(c)*log(-2*a^2*c*x^2 + 2*sqrt(a^2*c*x^2 + c)*a*sqrt (c)*x - c) - 2*sqrt(a^2*c*x^2 + c)*(a*x - (a^2*x^2 + 2)*arctan(a*x)))/(a^6 *c^2*x^2 + a^4*c^2)
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**3*atan(a*x)/(a**2*c*x**2+c)**(3/2),x)
Output:
Integral(x**3*atan(a*x)/(c*(a**2*x**2 + 1))**(3/2), x)
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate(x^3*arctan(a*x)/(a^2*c*x^2 + c)^(3/2), x)
Exception generated. \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(3/2),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. \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((x^3*atan(a*x))/(c + a^2*c*x^2)^(3/2),x)
Output:
int((x^3*atan(a*x))/(c + a^2*c*x^2)^(3/2), x)
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}+2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-\sqrt {a^{2} x^{2}+1}\, a x -\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right ) a^{2} x^{2}-\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )-a^{2} x^{2}-1\right )}{a^{4} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:
int(x^3*atan(a*x)/(a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*(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 - log(sqrt(a**2*x**2 + 1) + a*x)*a**2* x**2 - log(sqrt(a**2*x**2 + 1) + a*x) - a**2*x**2 - 1))/(a**4*c**2*(a**2*x **2 + 1))