Integrand size = 23, antiderivative size = 179 \[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {b c \sqrt {d+e x^2}}{6 d x^2}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}+\frac {b c \left (2 c^2 d+3 e\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{6 d^{3/2}}-\frac {b \sqrt {c^2 d-e} \left (c^2 d+2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2} \]
1/6*b*c*(2*c^2*d+3*e)*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)-1/3*b*(c^2* d+2*e)*arctanh(c*(e*x^2+d)^(1/2)/(c^2*d-e)^(1/2))*(c^2*d-e)^(1/2)/d^2-1/6* b*c*(e*x^2+d)^(1/2)/d/x^2-1/3*(a+b*arctan(c*x))*(e*x^2+d)^(1/2)/d/x^3+2/3* e*(a+b*arctan(c*x))*(e*x^2+d)^(1/2)/d^2/x
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.08 \[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {\frac {\sqrt {d+e x^2} \left (b c d x+2 a \left (d-2 e x^2\right )\right )}{x^3}+\frac {2 b \left (d-2 e x^2\right ) \sqrt {d+e x^2} \arctan (c x)}{x^3}+b c \sqrt {d} \left (2 c^2 d+3 e\right ) \log (x)-b c \sqrt {d} \left (2 c^2 d+3 e\right ) \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+\frac {b \left (c^4 d^2+c^2 d e-2 e^2\right ) \log \left (\frac {12 c d^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} \left (c^4 d^2+c^2 d e-2 e^2\right ) (i+c x)}\right )}{\sqrt {c^2 d-e}}+\frac {b \left (c^4 d^2+c^2 d e-2 e^2\right ) \log \left (\frac {12 c d^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} \left (c^4 d^2+c^2 d e-2 e^2\right ) (-i+c x)}\right )}{\sqrt {c^2 d-e}}}{6 d^2} \]
-1/6*((Sqrt[d + e*x^2]*(b*c*d*x + 2*a*(d - 2*e*x^2)))/x^3 + (2*b*(d - 2*e* x^2)*Sqrt[d + e*x^2]*ArcTan[c*x])/x^3 + b*c*Sqrt[d]*(2*c^2*d + 3*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 3*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^4* d^2 + c^2*d*e - 2*e^2)*Log[(12*c*d^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + c^2*d*e - 2*e^2)*(I + c*x))])/Sq rt[c^2*d - e] + (b*(c^4*d^2 + c^2*d*e - 2*e^2)*Log[(12*c*d^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + c^2*d*e - 2*e^2)*(-I + c*x))])/Sqrt[c^2*d - e])/d^2
Time = 0.42 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5511, 27, 435, 166, 27, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int -\frac {\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{3 d^2 x^3 \left (c^2 x^2+1\right )}dx+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{x^3 \left (c^2 x^2+1\right )}dx}{3 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle \frac {b c \int \frac {\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{x^4 \left (c^2 x^2+1\right )}dx^2}{6 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {b c \left (\int -\frac {e \left (d c^2+4 e\right ) x^2+d \left (2 d c^2+3 e\right )}{2 x^2 \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2-\frac {d \sqrt {d+e x^2}}{x^2}\right )}{6 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \left (-\frac {1}{2} \int \frac {e \left (d c^2+4 e\right ) x^2+d \left (2 d c^2+3 e\right )}{x^2 \left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2-\frac {d \sqrt {d+e x^2}}{x^2}\right )}{6 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {b c \left (\frac {1}{2} \left (2 \left (c^2 d-e\right ) \left (c^2 d+2 e\right ) \int \frac {1}{\left (c^2 x^2+1\right ) \sqrt {e x^2+d}}dx^2-d \left (2 c^2 d+3 e\right ) \int \frac {1}{x^2 \sqrt {e x^2+d}}dx^2\right )-\frac {d \sqrt {d+e x^2}}{x^2}\right )}{6 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b c \left (\frac {1}{2} \left (\frac {4 \left (c^2 d-e\right ) \left (c^2 d+2 e\right ) \int \frac {1}{\frac {c^2 x^4}{e}-\frac {c^2 d}{e}+1}d\sqrt {e x^2+d}}{e}-\frac {2 d \left (2 c^2 d+3 e\right ) \int \frac {1}{\frac {x^4}{e}-\frac {d}{e}}d\sqrt {e x^2+d}}{e}\right )-\frac {d \sqrt {d+e x^2}}{x^2}\right )}{6 d^2}+\frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 e \sqrt {d+e x^2} (a+b \arctan (c x))}{3 d^2 x}-\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{3 d x^3}+\frac {b c \left (\frac {1}{2} \left (2 \sqrt {d} \left (2 c^2 d+3 e\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-\frac {4 \sqrt {c^2 d-e} \left (c^2 d+2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{c}\right )-\frac {d \sqrt {d+e x^2}}{x^2}\right )}{6 d^2}\) |
-1/3*(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(d*x^3) + (2*e*Sqrt[d + e*x^2]* (a + b*ArcTan[c*x]))/(3*d^2*x) + (b*c*(-((d*Sqrt[d + e*x^2])/x^2) + (2*Sqr t[d]*(2*c^2*d + 3*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] - (4*Sqrt[c^2*d - e] *(c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/c)/2))/(6*d^2 )
3.13.8.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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]))
\[\int \frac {a +b \arctan \left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}d x\]
Time = 0.39 (sec) , antiderivative size = 868, normalized size of antiderivative = 4.85 \[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=\left [\frac {{\left (b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} d - e} x^{3} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} - 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + {\left (2 \, b c^{3} d + 3 \, b c e\right )} \sqrt {d} x^{3} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (b c d x - 4 \, a e x^{2} + 2 \, a d - 2 \, {\left (2 \, b e x^{2} - b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, d^{2} x^{3}}, -\frac {2 \, {\left (b c^{2} d + 2 \, b e\right )} \sqrt {-c^{2} d + e} x^{3} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - {\left (2 \, b c^{3} d + 3 \, b c e\right )} \sqrt {d} x^{3} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (b c d x - 4 \, a e x^{2} + 2 \, a d - 2 \, {\left (2 \, b e x^{2} - b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, d^{2} x^{3}}, -\frac {2 \, {\left (2 \, b c^{3} d + 3 \, b c e\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) - {\left (b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} d - e} x^{3} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} - 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 2 \, {\left (b c d x - 4 \, a e x^{2} + 2 \, a d - 2 \, {\left (2 \, b e x^{2} - b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, d^{2} x^{3}}, -\frac {{\left (b c^{2} d + 2 \, b e\right )} \sqrt {-c^{2} d + e} x^{3} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b c^{3} d + 3 \, b c e\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (b c d x - 4 \, a e x^{2} + 2 \, a d - 2 \, {\left (2 \, b e x^{2} - b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{6 \, d^{2} x^{3}}\right ] \]
[1/12*((b*c^2*d + 2*b*e)*sqrt(c^2*d - e)*x^3*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 + 3*b*c*e)*sqrt(d)*x^3*log(-(e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(d) + 2* d)/x^2) - 2*(b*c*d*x - 4*a*e*x^2 + 2*a*d - 2*(2*b*e*x^2 - b*d)*arctan(c*x) )*sqrt(e*x^2 + d))/(d^2*x^3), -1/12*(2*(b*c^2*d + 2*b*e)*sqrt(-c^2*d + e)* x^3*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)) - (2*b*c^3*d + 3*b*c*e)*sqrt(d )*x^3*log(-(e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) + 2*(b*c*d*x - 4 *a*e*x^2 + 2*a*d - 2*(2*b*e*x^2 - b*d)*arctan(c*x))*sqrt(e*x^2 + d))/(d^2* x^3), -1/12*(2*(2*b*c^3*d + 3*b*c*e)*sqrt(-d)*x^3*arctan(sqrt(-d)/sqrt(e*x ^2 + d)) - (b*c^2*d + 2*b*e)*sqrt(c^2*d - e)*x^3*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*d*x - 4*a*e*x^2 + 2*a*d - 2*(2*b*e*x^2 - b*d)*arctan(c*x))*sqrt(e*x ^2 + d))/(d^2*x^3), -1/6*((b*c^2*d + 2*b*e)*sqrt(-c^2*d + e)*x^3*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)) + (2*b*c^3*d + 3*b*c*e)*sqrt(-d)*x^3*arctan (sqrt(-d)/sqrt(e*x^2 + d)) + (b*c*d*x - 4*a*e*x^2 + 2*a*d - 2*(2*b*e*x^2 - b*d)*arctan(c*x))*sqrt(e*x^2 + d))/(d^2*x^3)]
\[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]