Integrand size = 10, antiderivative size = 106 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=-\frac {\left (1-6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {\left (1-6 a^2+a^4\right ) \arctan (a+b x)}{4 b^4}+\frac {a \left (1-a^2\right ) \log \left (1+(a+b x)^2\right )}{2 b^4} \]
-1/4*(-6*a^2+1)*x/b^3-1/2*a*(b*x+a)^2/b^4+1/12*(b*x+a)^3/b^4+1/4*x^4*arcco t(b*x+a)+1/4*(a^4-6*a^2+1)*arctan(b*x+a)/b^4+1/2*a*(-a^2+1)*ln(1+(b*x+a)^2 )/b^4
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\frac {6 \left (-1+6 a^2\right ) b x-12 a (a+b x)^2+2 (a+b x)^3+6 b^4 x^4 \cot ^{-1}(a+b x)-3 i (-i+a)^4 \log (i-a-b x)+3 i (i+a)^4 \log (i+a+b x)}{24 b^4} \]
(6*(-1 + 6*a^2)*b*x - 12*a*(a + b*x)^2 + 2*(a + b*x)^3 + 6*b^4*x^4*ArcCot[ a + b*x] - (3*I)*(-I + a)^4*Log[I - a - b*x] + (3*I)*(I + a)^4*Log[I + a + b*x])/(24*b^4)
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5571, 25, 27, 5388, 478, 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.
\(\displaystyle \int x^3 \cot ^{-1}(a+b x) \, dx\) |
\(\Big \downarrow \) 5571 |
\(\displaystyle \frac {\int x^3 \cot ^{-1}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -x^3 \cot ^{-1}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -b^3 x^3 \cot ^{-1}(a+b x)d(a+b x)}{b^4}\) |
\(\Big \downarrow \) 5388 |
\(\displaystyle -\frac {-\frac {1}{4} \int \frac {b^4 x^4}{(a+b x)^2+1}d(a+b x)-\frac {1}{4} b^4 x^4 \cot ^{-1}(a+b x)}{b^4}\) |
\(\Big \downarrow \) 478 |
\(\displaystyle -\frac {-\frac {1}{4} \int \left (6 a^2-4 (a+b x) a+(a+b x)^2+\frac {a^4-6 a^2+4 \left (1-a^2\right ) (a+b x) a+1}{(a+b x)^2+1}-1\right )d(a+b x)-\frac {1}{4} b^4 x^4 \cot ^{-1}(a+b x)}{b^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{4} \left (\left (1-6 a^2\right ) (a+b x)-2 a \left (1-a^2\right ) \log \left ((a+b x)^2+1\right )-\left (a^4-6 a^2+1\right ) \arctan (a+b x)-\frac {1}{3} (a+b x)^3+2 a (a+b x)^2\right )-\frac {1}{4} b^4 x^4 \cot ^{-1}(a+b x)}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCot[a + b*x]) + ((1 - 6*a^2)*(a + b*x) + 2*a*(a + b*x) ^2 - (a + b*x)^3/3 - (1 - 6*a^2 + a^4)*ArcTan[a + b*x] - 2*a*(1 - a^2)*Log [1 + (a + b*x)^2])/4)/b^4)
3.1.99.3.1 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[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand Integrand[(c + d*x)^n/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ [n, 1]
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCot[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]
Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && I GtQ[p, 0]
Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(-\frac {-3 \,\operatorname {arccot}\left (b x +a \right ) x^{4} b^{4}-b^{3} x^{3}+3 a \,b^{2} x^{2}+3 \,\operatorname {arccot}\left (b x +a \right ) a^{4}+6 a^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )-9 a^{2} b x -18 \,\operatorname {arccot}\left (b x +a \right ) a^{2}+15 a^{3}-6 a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )+3 b x +3 \,\operatorname {arccot}\left (b x +a \right )-9 a}{12 b^{4}}\) | \(131\) |
parts | \(\frac {x^{4} \operatorname {arccot}\left (b x +a \right )}{4}+\frac {b \left (\frac {\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x -x}{b^{4}}+\frac {\frac {\left (-4 a^{3} b +4 a b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (-3 a^{4}-2 a^{2}+1-\frac {\left (-4 a^{3} b +4 a b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}}{b^{4}}\right )}{4}\) | \(135\) |
derivativedivides | \(\frac {\frac {\operatorname {arccot}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccot}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccot}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccot}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccot}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}-\frac {b x}{4}-\frac {a}{4}+\frac {\left (-4 a^{3}+4 a \right ) \ln \left (1+\left (b x +a \right )^{2}\right )}{8}+\frac {\left (a^{4}-6 a^{2}+1\right ) \arctan \left (b x +a \right )}{4}}{b^{4}}\) | \(157\) |
default | \(\frac {\frac {\operatorname {arccot}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccot}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccot}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccot}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccot}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}-\frac {b x}{4}-\frac {a}{4}+\frac {\left (-4 a^{3}+4 a \right ) \ln \left (1+\left (b x +a \right )^{2}\right )}{8}+\frac {\left (a^{4}-6 a^{2}+1\right ) \arctan \left (b x +a \right )}{4}}{b^{4}}\) | \(157\) |
risch | \(\frac {i x^{4} \ln \left (1+i \left (b x +a \right )\right )}{8}-\frac {i x^{4} \ln \left (1-i \left (b x +a \right )\right )}{8}+\frac {\pi \,x^{4}}{8}+\frac {x^{3}}{12 b}+\frac {a^{4} \arctan \left (b x +a \right )}{4 b^{4}}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{4}}+\frac {3 a^{2} x}{4 b^{3}}-\frac {3 a^{2} \arctan \left (b x +a \right )}{2 b^{4}}+\frac {a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{4}}-\frac {x}{4 b^{3}}+\frac {\arctan \left (b x +a \right )}{4 b^{4}}\) | \(163\) |
-1/12*(-3*arccot(b*x+a)*x^4*b^4-b^3*x^3+3*a*b^2*x^2+3*arccot(b*x+a)*a^4+6* a^3*ln(b^2*x^2+2*a*b*x+a^2+1)-9*a^2*b*x-18*arccot(b*x+a)*a^2+15*a^3-6*a*ln (b^2*x^2+2*a*b*x+a^2+1)+3*b*x+3*arccot(b*x+a)-9*a)/b^4
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\frac {3 \, b^{4} x^{4} \operatorname {arccot}\left (b x + a\right ) + b^{3} x^{3} - 3 \, a b^{2} x^{2} + 3 \, {\left (3 \, a^{2} - 1\right )} b x + 3 \, {\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (b x + a\right ) - 6 \, {\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{12 \, b^{4}} \]
1/12*(3*b^4*x^4*arccot(b*x + a) + b^3*x^3 - 3*a*b^2*x^2 + 3*(3*a^2 - 1)*b* x + 3*(a^4 - 6*a^2 + 1)*arctan(b*x + a) - 6*(a^3 - a)*log(b^2*x^2 + 2*a*b* x + a^2 + 1))/b^4
Time = 0.38 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.46 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {acot}{\left (a + b x \right )}}{4 b^{4}} - \frac {a^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} + \frac {3 a^{2} x}{4 b^{3}} + \frac {3 a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 b^{4}} - \frac {a x^{2}}{4 b^{2}} + \frac {a \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} + \frac {x^{4} \operatorname {acot}{\left (a + b x \right )}}{4} + \frac {x^{3}}{12 b} - \frac {x}{4 b^{3}} - \frac {\operatorname {acot}{\left (a + b x \right )}}{4 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acot}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*acot(a + b*x)/(4*b**4) - a**3*log(a**2 + 2*a*b*x + b**2*x **2 + 1)/(2*b**4) + 3*a**2*x/(4*b**3) + 3*a**2*acot(a + b*x)/(2*b**4) - a* x**2/(4*b**2) + a*log(a**2 + 2*a*b*x + b**2*x**2 + 1)/(2*b**4) + x**4*acot (a + b*x)/4 + x**3/(12*b) - x/(4*b**3) - acot(a + b*x)/(4*b**4), Ne(b, 0)) , (x**4*acot(a)/4, True))
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arccot}\left (b x + a\right ) + \frac {1}{12} \, b {\left (\frac {b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} - 1\right )} x}{b^{4}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b^{5}} - \frac {6 \, {\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{5}}\right )} \]
1/4*x^4*arccot(b*x + a) + 1/12*b*((b^2*x^3 - 3*a*b*x^2 + 3*(3*a^2 - 1)*x)/ b^4 + 3*(a^4 - 6*a^2 + 1)*arctan((b^2*x + a*b)/b)/b^5 - 6*(a^3 - a)*log(b^ 2*x^2 + 2*a*b*x + a^2 + 1)/b^5)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (92) = 184\).
Time = 0.73 (sec) , antiderivative size = 617, normalized size of antiderivative = 5.82 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\text {Too large to display} \]
1/192*(96*a^3*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^5 + 72*a^2* arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^6 + 24*a*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^7 + 3*arctan(1/(b*x + a))*tan(1/2*arctan( 1/(b*x + a)))^8 + 96*a^3*log(16*tan(1/2*arctan(1/(b*x + a)))^2/(tan(1/2*ar ctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arct an(1/(b*x + a)))^4 - 96*a^3*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a) ))^3 + 144*a^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^4 - 144*a^ 2*tan(1/2*arctan(1/(b*x + a)))^5 - 72*a*arctan(1/(b*x + a))*tan(1/2*arctan (1/(b*x + a)))^5 - 24*a*tan(1/2*arctan(1/(b*x + a)))^6 - 12*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^6 - 2*tan(1/2*arctan(1/(b*x + a)))^7 - 96*a*log(16*tan(1/2*arctan(1/(b*x + a)))^2/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^4 + 72*a^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^2 + 144*a^2*tan(1 /2*arctan(1/(b*x + a)))^3 + 72*a*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^3 - 48*a*tan(1/2*arctan(1/(b*x + a)))^4 - 30*arctan(1/(b*x + a))*t an(1/2*arctan(1/(b*x + a)))^4 + 30*tan(1/2*arctan(1/(b*x + a)))^5 - 24*a*a rctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a))) - 24*a*tan(1/2*arctan(1/(b *x + a)))^2 - 12*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^2 - 30*t an(1/2*arctan(1/(b*x + a)))^3 + 3*arctan(1/(b*x + a)) + 2*tan(1/2*arctan(1 /(b*x + a))))/(b^4*tan(1/2*arctan(1/(b*x + a)))^4)
Time = 0.92 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.25 \[ \int x^3 \cot ^{-1}(a+b x) \, dx=\frac {\mathrm {atan}\left (a+b\,x\right )}{4\,b^4}+\frac {x^4\,\mathrm {acot}\left (a+b\,x\right )}{4}-\frac {x}{4\,b^3}+\frac {x^3}{12\,b}-\frac {a^3\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,b^4}-\frac {3\,a^2\,\mathrm {atan}\left (a+b\,x\right )}{2\,b^4}+\frac {a^4\,\mathrm {atan}\left (a+b\,x\right )}{4\,b^4}-\frac {a\,x^2}{4\,b^2}+\frac {3\,a^2\,x}{4\,b^3}+\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,b^4} \]