Integrand size = 22, antiderivative size = 120 \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {x \sqrt {c+a^2 c x^2}}{6 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^2 c}+\frac {5 \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^4 \sqrt {c}} \] Output:
-1/6*x*(a^2*c*x^2+c)^(1/2)/a^3/c-2/3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^4/c +1/3*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2/c+5/6*arctanh(a*c^(1/2)*x/(a^ 2*c*x^2+c)^(1/2))/a^4/c^(1/2)
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {-a x \sqrt {c+a^2 c x^2}+2 \left (-2+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)+5 \sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{6 a^4 c} \] Input:
Integrate[(x^3*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
(-(a*x*Sqrt[c + a^2*c*x^2]) + 2*(-2 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[ a*x] + 5*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(6*a^4*c)
Time = 0.52 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5487, 262, 224, 219, 5465, 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.
\(\displaystyle \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}} \, dx\) |
\(\Big \downarrow \) 5487 |
\(\displaystyle -\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle -\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\) |
Input:
Int[(x^3*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2]) /(2*a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c]))/( 3*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])))/(3*a^2)
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b* ArcTan[c*x])^p/(c^2*d*m)), x] + (-Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1)*(( a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[f^2*((m - 1)/(c^ 2*m)) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\left (2 x^{2} a^{2} \arctan \left (a x \right )-a x -4 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,a^{4}}-\frac {5 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}+\frac {5 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}\) | \(165\) |
Input:
int(x^3*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*(2*x^2*a^2*arctan(a*x)-a*x-4*arctan(a*x))*(c*(a*x-I)*(a*x+I))^(1/2)/c/ a^4-5/6*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-I)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x ^2+1)^(1/2)/a^4/c+5/6*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)*(c*(a*x-I)*(a*x+I) )^(1/2)/(a^2*x^2+1)^(1/2)/a^4/c
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - 2 \, {\left (a^{2} x^{2} - 2\right )} \arctan \left (a x\right )\right )} - 5 \, \sqrt {c} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{12 \, a^{4} c} \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-1/12*(2*sqrt(a^2*c*x^2 + c)*(a*x - 2*(a^2*x^2 - 2)*arctan(a*x)) - 5*sqrt( c)*log(-2*a^2*c*x^2 - 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c))/(a^4*c)
\[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \] Input:
integrate(x**3*atan(a*x)/(a**2*c*x**2+c)**(1/2),x)
Output:
Integral(x**3*atan(a*x)/sqrt(c*(a**2*x**2 + 1)), x)
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74 \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {a {\left (\frac {\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}}{a^{2}} - \frac {4 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} - 2 \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \arctan \left (a x\right )}{6 \, \sqrt {c}} \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")
Output:
-1/6*(a*((sqrt(a^2*x^2 + 1)*x/a^2 - arcsinh(a*x)/a^3)/a^2 - 4*arcsinh(a*x) /a^5) - 2*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arctan(a*x ))/sqrt(c)
Exception generated. \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*arctan(a*x)/(a^2*c*x^2+c)^(1/2),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. \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \] Input:
int((x^3*atan(a*x))/(c + a^2*c*x^2)^(1/2),x)
Output:
int((x^3*atan(a*x))/(c + a^2*c*x^2)^(1/2), x)
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}-4 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-\sqrt {a^{2} x^{2}+1}\, a x +5 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )\right )}{6 a^{4} c} \] Input:
int(x^3*atan(a*x)/(a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt(a**2*x**2 + 1)*atan(a*x)*a**2*x**2 - 4*sqrt(a**2*x**2 + 1 )*atan(a*x) - sqrt(a**2*x**2 + 1)*a*x + 5*log(sqrt(a**2*x**2 + 1) + a*x))) /(6*a**4*c)