Integrand size = 16, antiderivative size = 146 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {b c^2 (c d-e) (c d+e) \arctan (c x)}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {a+b \arctan (c x)}{2 e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c^3 d \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2} \]
-1/2*b*c/(c^2*d^2+e^2)/(e*x+d)+1/2*b*c^2*(c*d-e)*(c*d+e)*arctan(c*x)/e/(c^ 2*d^2+e^2)^2+1/2*(-a-b*arctan(c*x))/e/(e*x+d)^2+b*c^3*d*ln(e*x+d)/(c^2*d^2 +e^2)^2-1/2*b*c^3*d*ln(c^2*x^2+1)/(c^2*d^2+e^2)^2
Time = 0.36 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.32 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=-\frac {2 (a+b \arctan (c x))+\frac {b c (d+e x) \left (2 e \left (c^2 d^2+e^2\right )-\left (c^2 d \left (\sqrt {-c^2} d-2 e\right )-\sqrt {-c^2} e^2\right ) (d+e x) \log \left (1-\sqrt {-c^2} x\right )-\left (\sqrt {-c^2} e^2-c^2 d \left (\sqrt {-c^2} d+2 e\right )\right ) (d+e x) \log \left (1+\sqrt {-c^2} x\right )-4 c^2 d e (d+e x) \log (d+e x)\right )}{\left (c^2 d^2+e^2\right )^2}}{4 e (d+e x)^2} \]
-1/4*(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(2*e*(c^2*d^2 + e^2) - (c^2*d *(Sqrt[-c^2]*d - 2*e) - Sqrt[-c^2]*e^2)*(d + e*x)*Log[1 - Sqrt[-c^2]*x] - (Sqrt[-c^2]*e^2 - c^2*d*(Sqrt[-c^2]*d + 2*e))*(d + e*x)*Log[1 + Sqrt[-c^2] *x] - 4*c^2*d*e*(d + e*x)*Log[d + e*x]))/(c^2*d^2 + e^2)^2)/(e*(d + e*x)^2 )
Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5387, 480, 657, 2009}
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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5387 |
| \(\displaystyle \frac {b c \int \frac {1}{(d+e x)^2 \left (c^2 x^2+1\right )}dx}{2 e}-\frac {a+b \arctan (c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 480 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \int \frac {d-e x}{(d+e x) \left (c^2 x^2+1\right )}dx}{c^2 d^2+e^2}-\frac {e}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \arctan (c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \int \left (\frac {2 d e^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {d^2 c^2-2 d e x c^2-e^2}{\left (c^2 d^2+e^2\right ) \left (c^2 x^2+1\right )}\right )dx}{c^2 d^2+e^2}-\frac {e}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \arctan (c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \left (\frac {\arctan (c x) (c d-e) (c d+e)}{c \left (c^2 d^2+e^2\right )}-\frac {d e \log \left (c^2 x^2+1\right )}{c^2 d^2+e^2}+\frac {2 d e \log (d+e x)}{c^2 d^2+e^2}\right )}{c^2 d^2+e^2}-\frac {e}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \arctan (c x)}{2 e (d+e x)^2}\) |
-1/2*(a + b*ArcTan[c*x])/(e*(d + e*x)^2) + (b*c*(-(e/((c^2*d^2 + e^2)*(d + e*x))) + (c^2*(((c*d - e)*(c*d + e)*ArcTan[c*x])/(c*(c^2*d^2 + e^2)) + (2 *d*e*Log[d + e*x])/(c^2*d^2 + e^2) - (d*e*Log[1 + c^2*x^2])/(c^2*d^2 + e^2 )))/(c^2*d^2 + e^2)))/(2*e)
3.1.7.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c + d*x)^(n + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[b/(b*c^2 + a*d^2) I nt[(c + d*x)^(n + 1)*((c - d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])/(e*(q + 1))), x] - Simp[b*( c/(e*(q + 1))) Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b , c, d, e, q}, x] && NeQ[q, -1]
Time = 1.62 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}\right )}{2 e}\right )}{c}\) | \(153\) |
| derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(157\) |
| default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(157\) |
| parallelrisch | \(-\frac {b c d \,e^{2}+e^{3} a -x^{2} a \,c^{4} d^{2} e -2 b \,d^{3} \arctan \left (c x \right ) x \,c^{4}-x^{2} b \,c^{3} d \,e^{2}-x b \,c^{3} d^{2} e +3 \arctan \left (c x \right ) b \,c^{2} d^{2} e -2 x a \,c^{4} d^{3}+\ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d^{3}+x b c \,e^{3}+a \,c^{2} d^{2} e -x^{2} \arctan \left (c x \right ) b \,c^{4} d^{2} e +\arctan \left (c x \right ) b \,e^{3}-x^{2} a \,c^{2} e^{3}-2 \ln \left (e x +d \right ) b \,c^{3} d^{3}+x^{2} \arctan \left (c x \right ) b \,c^{2} e^{3}-2 x a \,c^{2} d \,e^{2}+2 x \arctan \left (c x \right ) b \,c^{2} d \,e^{2}+\ln \left (c^{2} x^{2}+1\right ) x^{2} b \,c^{3} d \,e^{2}-2 \ln \left (e x +d \right ) x^{2} b \,c^{3} d \,e^{2}+2 \ln \left (c^{2} x^{2}+1\right ) x b \,c^{3} d^{2} e -4 \ln \left (e x +d \right ) x b \,c^{3} d^{2} e}{2 \left (e x +d \right )^{2} \left (c^{4} d^{4}+2 c^{2} d^{2} e^{2}+e^{4}\right )}\) | \(329\) |
| risch | \(\text {Expression too large to display}\) | \(2102\) |
-1/2*a/(e*x+d)^2/e+b/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arctan(c*x)+1/2*c^3/e*(-e /(c^2*d^2+e^2)/(c*e*x+c*d)+2*e*c*d/(c^2*d^2+e^2)^2*ln(c*e*x+c*d)+1/(c^2*d^ 2+e^2)^2*(-c*d*e*ln(c^2*x^2+1)+(c^2*d^2-e^2)*arctan(c*x))))
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (138) = 276\).
Time = 0.35 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.14 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=-\frac {a c^{4} d^{4} + b c^{3} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + b c d e^{3} + a e^{4} + {\left (b c^{3} d^{2} e^{2} + b c e^{4}\right )} x + {\left (3 \, b c^{2} d^{2} e^{2} + b e^{4} - {\left (b c^{4} d^{2} e^{2} - b c^{2} e^{4}\right )} x^{2} - 2 \, {\left (b c^{4} d^{3} e - b c^{2} d e^{3}\right )} x\right )} \arctan \left (c x\right ) + {\left (b c^{3} d e^{3} x^{2} + 2 \, b c^{3} d^{2} e^{2} x + b c^{3} d^{3} e\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{3} d e^{3} x^{2} + 2 \, b c^{3} d^{2} e^{2} x + b c^{3} d^{3} e\right )} \log \left (e x + d\right )}{2 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}} \]
-1/2*(a*c^4*d^4 + b*c^3*d^3*e + 2*a*c^2*d^2*e^2 + b*c*d*e^3 + a*e^4 + (b*c ^3*d^2*e^2 + b*c*e^4)*x + (3*b*c^2*d^2*e^2 + b*e^4 - (b*c^4*d^2*e^2 - b*c^ 2*e^4)*x^2 - 2*(b*c^4*d^3*e - b*c^2*d*e^3)*x)*arctan(c*x) + (b*c^3*d*e^3*x ^2 + 2*b*c^3*d^2*e^2*x + b*c^3*d^3*e)*log(c^2*x^2 + 1) - 2*(b*c^3*d*e^3*x^ 2 + 2*b*c^3*d^2*e^2*x + b*c^3*d^3*e)*log(e*x + d))/(c^4*d^6*e + 2*c^2*d^4* e^3 + d^2*e^5 + (c^4*d^4*e^3 + 2*c^2*d^2*e^5 + e^7)*x^2 + 2*(c^4*d^5*e^2 + 2*c^2*d^3*e^4 + d*e^6)*x)
Result contains complex when optimal does not.
Time = 3.27 (sec) , antiderivative size = 2866, normalized size of antiderivative = 19.63 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=\text {Too large to display} \]
Piecewise((a*x/d**3, Eq(c, 0) & Eq(e, 0)), ((a*x + b*x*atan(c*x) - b*log(x **2 + c**(-2))/(2*c))/d**3, Eq(e, 0)), (-a/(2*d**2*e + 4*d*e**2*x + 2*e**3 *x**2), Eq(c, 0)), (-4*a*d**2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x** 2) + 3*I*b*d**2*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2 ) + 2*I*b*d**2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 2*I*b*d*e* x*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) + I*b*d*e*x/ (8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - I*b*e**2*x**2*atanh(e*x/d )/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2), Eq(c, -I*e/d)), (-4*a*d* *2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 3*I*b*d**2*atanh(e*x/d )/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 2*I*b*d**2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) + 2*I*b*d*e*x*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - I*b*d*e*x/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) + I*b*e**2*x**2*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2* x + 8*d**2*e**3*x**2), Eq(c, I*e/d)), (-a*c**4*d**4/(2*c**4*d**6*e + 4*c** 4*d**5*e**2*x + 2*c**4*d**4*e**3*x**2 + 4*c**2*d**4*e**3 + 8*c**2*d**3*e** 4*x + 4*c**2*d**2*e**5*x**2 + 2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2* a*c**2*d**2*e**2/(2*c**4*d**6*e + 4*c**4*d**5*e**2*x + 2*c**4*d**4*e**3*x* *2 + 4*c**2*d**4*e**3 + 8*c**2*d**3*e**4*x + 4*c**2*d**2*e**5*x**2 + 2*d** 2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - a*e**4/(2*c**4*d**6*e + 4*c**4*d**5*e **2*x + 2*c**4*d**4*e**3*x**2 + 4*c**2*d**4*e**3 + 8*c**2*d**3*e**4*x +...
Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=-\frac {1}{2} \, {\left ({\left (\frac {c^{2} d \log \left (c^{2} x^{2} + 1\right )}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {2 \, c^{2} d \log \left (e x + d\right )}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {{\left (c^{4} d^{2} - c^{2} e^{2}\right )} \arctan \left (c x\right )}{{\left (c^{4} d^{4} e + 2 \, c^{2} d^{2} e^{3} + e^{5}\right )} c} + \frac {1}{c^{2} d^{3} + d e^{2} + {\left (c^{2} d^{2} e + e^{3}\right )} x}\right )} c + \frac {\arctan \left (c x\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right )} b - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
-1/2*((c^2*d*log(c^2*x^2 + 1)/(c^4*d^4 + 2*c^2*d^2*e^2 + e^4) - 2*c^2*d*lo g(e*x + d)/(c^4*d^4 + 2*c^2*d^2*e^2 + e^4) - (c^4*d^2 - c^2*e^2)*arctan(c* x)/((c^4*d^4*e + 2*c^2*d^2*e^3 + e^5)*c) + 1/(c^2*d^3 + d*e^2 + (c^2*d^2*e + e^3)*x))*c + arctan(c*x)/(e^3*x^2 + 2*d*e^2*x + d^2*e))*b - 1/2*a/(e^3* x^2 + 2*d*e^2*x + d^2*e)
\[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x + d\right )}^{3}} \,d x } \]
Time = 4.81 (sec) , antiderivative size = 591, normalized size of antiderivative = 4.05 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^3} \, dx=\frac {\frac {x\,\left (a\,c^2\,d^2+\frac {b\,c\,d\,e}{2}+a\,e^2\right )}{d\,\left (c^2\,d^2+e^2\right )}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{2\,e}+\frac {x^2\,\left (\frac {a\,c^2\,d^2\,e}{2}+\frac {b\,c\,d\,e^2}{2}+\frac {a\,e^3}{2}\right )}{d^2\,\left (c^2\,d^2+e^2\right )}+\frac {x^4\,\left (\frac {a\,c^4\,d^2\,e}{2}+\frac {b\,c^3\,d\,e^2}{2}+\frac {a\,c^2\,e^3}{2}\right )}{d^2\,\left (c^2\,d^2+e^2\right )}+\frac {x^3\,\left (a\,c^4\,d^2+\frac {b\,c^3\,d\,e}{2}+a\,c^2\,e^2\right )}{d\,\left (c^2\,d^2+e^2\right )}-\frac {b\,c^2\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e}}{c^2\,d^2\,x^2+2\,c^2\,d\,e\,x^3+c^2\,e^2\,x^4+d^2+2\,d\,e\,x+e^2\,x^2}+\frac {b\,c^3\,d\,\ln \left (d+e\,x\right )}{c^4\,d^4+2\,c^2\,d^2\,e^2+e^4}-\frac {b\,c^3\,d\,\ln \left (c^2\,x^2+1\right )}{2\,\left (c^4\,d^4+2\,c^2\,d^2\,e^2+e^4\right )}+\frac {\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,{\left (c^2\right )}^{7/2}\,\left (c^4\,d^4+8\,c^2\,d^2\,e^2+2\,e^4\right )\,\left (3\,c^6\,d^4+26\,c^4\,d^2\,e^2+4\,c^2\,e^4\right )\,\left (27\,b\,c^{10}\,d^{10}+23\,b\,c^8\,d^8\,e^2-34\,b\,c^6\,d^6\,e^4-26\,b\,c^4\,d^4\,e^6+7\,b\,c^2\,d^2\,e^8+3\,b\,e^{10}\right )}{2\,c\,\left (81\,c^{26}\,d^{20}\,e+1662\,c^{24}\,d^{18}\,e^3+11515\,c^{22}\,d^{16}\,e^5+32306\,c^{20}\,d^{14}\,e^7+43705\,c^{18}\,d^{12}\,e^9+28142\,c^{16}\,d^{10}\,e^{11}+4857\,c^{14}\,d^8\,e^{13}-3650\,c^{12}\,d^6\,e^{15}-2054\,c^{10}\,d^4\,e^{17}-380\,c^8\,d^2\,e^{19}-24\,c^6\,e^{21}\right )} \]
((x*(a*e^2 + a*c^2*d^2 + (b*c*d*e)/2))/(d*(e^2 + c^2*d^2)) - (b*atan(c*x)) /(2*e) + (x^2*((a*e^3)/2 + (b*c*d*e^2)/2 + (a*c^2*d^2*e)/2))/(d^2*(e^2 + c ^2*d^2)) + (x^4*((a*c^2*e^3)/2 + (a*c^4*d^2*e)/2 + (b*c^3*d*e^2)/2))/(d^2* (e^2 + c^2*d^2)) + (x^3*(a*c^4*d^2 + a*c^2*e^2 + (b*c^3*d*e)/2))/(d*(e^2 + c^2*d^2)) - (b*c^2*x^2*atan(c*x))/(2*e))/(d^2 + e^2*x^2 + 2*d*e*x + c^2*d ^2*x^2 + c^2*e^2*x^4 + 2*c^2*d*e*x^3) + (b*c^3*d*log(d + e*x))/(e^4 + c^4* d^4 + 2*c^2*d^2*e^2) - (b*c^3*d*log(c^2*x^2 + 1))/(2*(e^4 + c^4*d^4 + 2*c^ 2*d^2*e^2)) + (atan((c^2*x)/(c^2)^(1/2))*(c^2)^(7/2)*(2*e^4 + c^4*d^4 + 8* c^2*d^2*e^2)*(3*c^6*d^4 + 4*c^2*e^4 + 26*c^4*d^2*e^2)*(3*b*e^10 + 27*b*c^1 0*d^10 + 7*b*c^2*d^2*e^8 - 26*b*c^4*d^4*e^6 - 34*b*c^6*d^6*e^4 + 23*b*c^8* d^8*e^2))/(2*c*(81*c^26*d^20*e - 24*c^6*e^21 - 380*c^8*d^2*e^19 - 2054*c^1 0*d^4*e^17 - 3650*c^12*d^6*e^15 + 4857*c^14*d^8*e^13 + 28142*c^16*d^10*e^1 1 + 43705*c^18*d^12*e^9 + 32306*c^20*d^14*e^7 + 11515*c^22*d^16*e^5 + 1662 *c^24*d^18*e^3))