Integrand size = 23, antiderivative size = 135 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b \left (c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d^2 \sqrt {c^2 d-e}} \]
-b*c*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)+b*(c^2*d-2*e)*arctanh(c*(e*x ^2+d)^(1/2)/(c^2*d-e)^(1/2))/d^2/(c^2*d-e)^(1/2)+(-a-b*arctan(c*x))/d/x/(e *x^2+d)^(1/2)-2*e*x*(a+b*arctan(c*x))/d^2/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.27 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-\frac {2 a \left (d+2 e x^2\right )}{x \sqrt {d+e x^2}}-\frac {2 b \left (d+2 e x^2\right ) \arctan (c x)}{x \sqrt {d+e x^2}}+2 b c \sqrt {d} \log (x)-2 b c \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+\frac {b \left (c^2 d-2 e\right ) \log \left (-\frac {4 c d^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-2 e\right ) \sqrt {c^2 d-e} (i+c x)}\right )}{\sqrt {c^2 d-e}}+\frac {b \left (c^2 d-2 e\right ) \log \left (-\frac {4 c d^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-2 e\right ) \sqrt {c^2 d-e} (-i+c x)}\right )}{\sqrt {c^2 d-e}}}{2 d^2} \]
((-2*a*(d + 2*e*x^2))/(x*Sqrt[d + e*x^2]) - (2*b*(d + 2*e*x^2)*ArcTan[c*x] )/(x*Sqrt[d + e*x^2]) + 2*b*c*Sqrt[d]*Log[x] - 2*b*c*Sqrt[d]*Log[d + Sqrt[ d]*Sqrt[d + e*x^2]] + (b*(c^2*d - 2*e)*Log[(-4*c*d^2*(c*d - I*e*x + Sqrt[c ^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - 2*e)*Sqrt[c^2*d - e]*(I + c*x))])/ Sqrt[c^2*d - e] + (b*(c^2*d - 2*e)*Log[(-4*c*d^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - 2*e)*Sqrt[c^2*d - e]*(-I + c*x))])/Sqr t[c^2*d - e])/(2*d^2)
Time = 0.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5511, 25, 27, 435, 174, 73, 221}
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.
\(\displaystyle \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5511 |
\(\displaystyle -b c \int -\frac {2 e x^2+d}{d^2 x \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle b c \int \frac {2 e x^2+d}{d^2 x \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c \int \frac {2 e x^2+d}{x \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx}{d^2}-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {b c \int \frac {2 e x^2+d}{x^2 \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2}{2 d^2}-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {b c \left (d \int \frac {1}{x^2 \sqrt {e x^2+d}}dx^2-\left (c^2 d-2 e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2\right )}{2 d^2}-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b c \left (\frac {2 d \int \frac {1}{\frac {x^4}{e}-\frac {d}{e}}d\sqrt {e x^2+d}}{e}-\frac {2 \left (c^2 d-2 e\right ) \int \frac {1}{\frac {c^2 x^4}{e}-\frac {c^2 d}{e}+1}d\sqrt {e x^2+d}}{e}\right )}{2 d^2}-\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 e x (a+b \arctan (c x))}{d^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{d x \sqrt {d+e x^2}}+\frac {b c \left (\frac {2 \left (c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{c \sqrt {c^2 d-e}}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right )}{2 d^2}\) |
-((a + b*ArcTan[c*x])/(d*x*Sqrt[d + e*x^2])) - (2*e*x*(a + b*ArcTan[c*x])) /(d^2*Sqrt[d + e*x^2]) + (b*c*(-2*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + (2*(c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(c*Sqrt[ c^2*d - e])))/(2*d^2)
3.13.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim p[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2 *x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILt Q[(m - 1)/2, 0]))
\[\int \frac {a +b \arctan \left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (119) = 238\).
Time = 0.42 (sec) , antiderivative size = 1317, normalized size of antiderivative = 9.76 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/4*(((b*c^2*d*e - 2*b*e^2)*x^3 + (b*c^2*d^2 - 2*b*d*e)*x)*sqrt(c^2*d - e)*log((c^4*e^2*x^4 + 8*c^4*d^2 - 8*c^2*d*e + 2*(4*c^4*d*e - 3*c^2*e^2)*x^ 2 - 4*(c^3*e*x^2 + 2*c^3*d - c*e)*sqrt(c^2*d - e)*sqrt(e*x^2 + d) + e^2)/( c^4*x^4 + 2*c^2*x^2 + 1)) - 2*((b*c^3*d*e - b*c*e^2)*x^3 + (b*c^3*d^2 - b* c*d*e)*x)*sqrt(d)*log(-(e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) + 4* (a*c^2*d^2 - a*d*e + 2*(a*c^2*d*e - a*e^2)*x^2 + (b*c^2*d^2 - b*d*e + 2*(b *c^2*d*e - b*e^2)*x^2)*arctan(c*x))*sqrt(e*x^2 + d))/((c^2*d^3*e - d^2*e^2 )*x^3 + (c^2*d^4 - d^3*e)*x), 1/2*(((b*c^2*d*e - 2*b*e^2)*x^3 + (b*c^2*d^2 - 2*b*d*e)*x)*sqrt(-c^2*d + e)*arctan(-1/2*(c^2*e*x^2 + 2*c^2*d - e)*sqrt (-c^2*d + e)*sqrt(e*x^2 + d)/(c^3*d^2 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + ((b*c^3*d*e - b*c*e^2)*x^3 + (b*c^3*d^2 - b*c*d*e)*x)*sqrt(d)*log(-(e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) - 2*(a*c^2*d^2 - a*d*e + 2*(a*c^2* d*e - a*e^2)*x^2 + (b*c^2*d^2 - b*d*e + 2*(b*c^2*d*e - b*e^2)*x^2)*arctan( c*x))*sqrt(e*x^2 + d))/((c^2*d^3*e - d^2*e^2)*x^3 + (c^2*d^4 - d^3*e)*x), 1/4*(4*((b*c^3*d*e - b*c*e^2)*x^3 + (b*c^3*d^2 - b*c*d*e)*x)*sqrt(-d)*arct an(sqrt(-d)/sqrt(e*x^2 + d)) - ((b*c^2*d*e - 2*b*e^2)*x^3 + (b*c^2*d^2 - 2 *b*d*e)*x)*sqrt(c^2*d - e)*log((c^4*e^2*x^4 + 8*c^4*d^2 - 8*c^2*d*e + 2*(4 *c^4*d*e - 3*c^2*e^2)*x^2 - 4*(c^3*e*x^2 + 2*c^3*d - c*e)*sqrt(c^2*d - e)* sqrt(e*x^2 + d) + e^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) - 4*(a*c^2*d^2 - a*d*e + 2*(a*c^2*d*e - a*e^2)*x^2 + (b*c^2*d^2 - b*d*e + 2*(b*c^2*d*e - b*e^2)...
\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]