3.1.8 \(\int \frac {a+b \text {ArcTan}(c x)}{(d+e x)^4} \, dx\) [8]

Optimal. Leaf size=206 \[ -\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {b c^4 d \left (c^2 d^2-3 e^2\right ) \text {ArcTan}(c x)}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {a+b \text {ArcTan}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 \left (c^2 d^2+e^2\right )^3} \]

[Out]

-1/6*b*c/(c^2*d^2+e^2)/(e*x+d)^2-2/3*b*c^3*d/(c^2*d^2+e^2)^2/(e*x+d)+1/3*b*c^4*d*(c^2*d^2-3*e^2)*arctan(c*x)/e
/(c^2*d^2+e^2)^3+1/3*(-a-b*arctan(c*x))/e/(e*x+d)^3+1/3*b*c^3*(3*c^2*d^2-e^2)*ln(e*x+d)/(c^2*d^2+e^2)^3-1/6*b*
c^3*(3*c^2*d^2-e^2)*ln(c^2*x^2+1)/(c^2*d^2+e^2)^3

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Rubi [A]
time = 0.13, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4972, 724, 815, 649, 209, 266} \begin {gather*} -\frac {a+b \text {ArcTan}(c x)}{3 e (d+e x)^3}+\frac {b c^4 d \text {ArcTan}(c x) \left (c^2 d^2-3 e^2\right )}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 \left (c^2 d^2+e^2\right )^3}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x)^4,x]

[Out]

-1/6*(b*c)/((c^2*d^2 + e^2)*(d + e*x)^2) - (2*b*c^3*d)/(3*(c^2*d^2 + e^2)^2*(d + e*x)) + (b*c^4*d*(c^2*d^2 - 3
*e^2)*ArcTan[c*x])/(3*e*(c^2*d^2 + e^2)^3) - (a + b*ArcTan[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(3*c^2*d^2 - e^2)*
Log[d + e*x])/(3*(c^2*d^2 + e^2)^3) - (b*c^3*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)^3)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 724

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a
*e^2))), x] + Dist[c/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*((d - e*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 4972

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] - Dist[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]

Rubi steps

\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \left (1+c^2 x^2\right )} \, dx}{3 e}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \frac {d-e x}{(d+e x)^2 \left (1+c^2 x^2\right )} \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \left (\frac {2 d e^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {e^2 \left (-3 c^2 d^2+e^2\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac {\left (b c^3\right ) \int \frac {c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac {\left (b c^5 d \left (c^2 d^2-3 e^2\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {\left (b c^5 \left (3 c^2 d^2-e^2\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 \left (c^2 d^2+e^2\right )^3}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 \left (c^2 d^2+e^2\right )^3}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 254, normalized size = 1.23 \begin {gather*} -\frac {2 (a+b \text {ArcTan}(c x))+\frac {b c (d+e x) \left (e \left (c^2 d^2+e^2\right )^2+4 c^2 d e \left (c^2 d^2+e^2\right ) (d+e x)-c^2 \left (c^2 d^2 \left (\sqrt {-c^2} d-3 e\right )+e^2 \left (-3 \sqrt {-c^2} d+e\right )\right ) (d+e x)^2 \log \left (1-\sqrt {-c^2} x\right )-c^2 \left (e^2 \left (3 \sqrt {-c^2} d+e\right )-c^2 d^2 \left (\sqrt {-c^2} d+3 e\right )\right ) (d+e x)^2 \log \left (1+\sqrt {-c^2} x\right )-2 c^2 e \left (3 c^2 d^2-e^2\right ) (d+e x)^2 \log (d+e x)\right )}{\left (c^2 d^2+e^2\right )^3}}{6 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x])/(d + e*x)^4,x]

[Out]

-1/6*(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(e*(c^2*d^2 + e^2)^2 + 4*c^2*d*e*(c^2*d^2 + e^2)*(d + e*x) - c^2*
(c^2*d^2*(Sqrt[-c^2]*d - 3*e) + e^2*(-3*Sqrt[-c^2]*d + e))*(d + e*x)^2*Log[1 - Sqrt[-c^2]*x] - c^2*(e^2*(3*Sqr
t[-c^2]*d + e) - c^2*d^2*(Sqrt[-c^2]*d + 3*e))*(d + e*x)^2*Log[1 + Sqrt[-c^2]*x] - 2*c^2*e*(3*c^2*d^2 - e^2)*(
d + e*x)^2*Log[d + e*x]))/(c^2*d^2 + e^2)^3)/(e*(d + e*x)^3)

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Maple [A]
time = 0.20, size = 286, normalized size = 1.39

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \arctan \left (c x \right )}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{6} \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {b \,c^{4} e^{2} \ln \left (c^{2} x^{2}+1\right )}{6 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {b \,c^{7} \arctan \left (c x \right ) d^{3}}{3 e \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{5} e \arctan \left (c x \right ) d}{\left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{4}}{6 \left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )^{2}}+\frac {b \,c^{6} \ln \left (c e x +c d \right ) d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{4} e^{2} \ln \left (c e x +c d \right )}{3 \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {2 b \,c^{5} d}{3 \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c e x +c d \right )}}{c}\)	\(286\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \arctan \left (c x \right )}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{6} \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {b \,c^{4} e^{2} \ln \left (c^{2} x^{2}+1\right )}{6 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {b \,c^{7} \arctan \left (c x \right ) d^{3}}{3 e \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{5} e \arctan \left (c x \right ) d}{\left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{4}}{6 \left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )^{2}}+\frac {b \,c^{6} \ln \left (c e x +c d \right ) d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {b \,c^{4} e^{2} \ln \left (c e x +c d \right )}{3 \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {2 b \,c^{5} d}{3 \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c e x +c d \right )}}{c}\)	\(286\)
risch	\(\text {Expression too large to display}\)	\(6058\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/3*a*c^4/(c*e*x+c*d)^3/e-1/3*b*c^4/(c*e*x+c*d)^3/e*arctan(c*x)-1/2*b*c^6/(c^2*d^2+e^2)^3*ln(c^2*x^2+1)*
d^2+1/6*b*c^4*e^2/(c^2*d^2+e^2)^3*ln(c^2*x^2+1)+1/3*b*c^7/e/(c^2*d^2+e^2)^3*arctan(c*x)*d^3-b*c^5*e/(c^2*d^2+e
^2)^3*arctan(c*x)*d-1/6*b*c^4/(c^2*d^2+e^2)/(c*e*x+c*d)^2+b*c^6/(c^2*d^2+e^2)^3*ln(c*e*x+c*d)*d^2-1/3*b*c^4*e^
2/(c^2*d^2+e^2)^3*ln(c*e*x+c*d)-2/3*b*c^5*d/(c^2*d^2+e^2)^2/(c*e*x+c*d))

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Maxima [A]
time = 0.49, size = 354, normalized size = 1.72 \begin {gather*} -\frac {1}{6} \, {\left (c {\left (\frac {{\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} - \frac {2 \, {\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (x e + d\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} + \frac {4 \, c^{2} d x e + 5 \, c^{2} d^{2} + e^{2}}{c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} + 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + d^{2} e^{4} + 2 \, {\left (c^{4} d^{5} e + 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x} - \frac {2 \, {\left (c^{6} d^{3} - 3 \, c^{4} d e^{2}\right )} \arctan \left (c x\right )}{{\left (c^{6} d^{6} e + 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} + e^{7}\right )} c}\right )} + \frac {2 \, \arctan \left (c x\right )}{x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e}\right )} b - \frac {a}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/6*(c*((3*c^4*d^2 - c^2*e^2)*log(c^2*x^2 + 1)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) - 2*(3*c^4*d^2
- c^2*e^2)*log(x*e + d)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) + (4*c^2*d*x*e + 5*c^2*d^2 + e^2)/(c^
4*d^6 + 2*c^2*d^4*e^2 + (c^4*d^4*e^2 + 2*c^2*d^2*e^4 + e^6)*x^2 + d^2*e^4 + 2*(c^4*d^5*e + 2*c^2*d^3*e^3 + d*e
^5)*x) - 2*(c^6*d^3 - 3*c^4*d*e^2)*arctan(c*x)/((c^6*d^6*e + 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 + e^7)*c)) + 2*arct
an(c*x)/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e))*b - 1/3*a/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (188) = 376\).
time = 1.99, size = 620, normalized size = 3.01 \begin {gather*} -\frac {2 \, a c^{6} d^{6} + 5 \, b c^{5} d^{5} e - 2 \, {\left (3 \, b c^{6} d^{5} x e - 3 \, b c^{4} d x^{3} e^{5} - b e^{6} - 3 \, {\left (3 \, b c^{4} d^{2} x^{2} + b c^{2} d^{2}\right )} e^{4} + {\left (b c^{6} d^{3} x^{3} - 9 \, b c^{4} d^{3} x\right )} e^{3} + 3 \, {\left (b c^{6} d^{4} x^{2} - 2 \, b c^{4} d^{4}\right )} e^{2}\right )} \arctan \left (c x\right ) + {\left (b c x + 2 \, a\right )} e^{6} + {\left (4 \, b c^{3} d x^{2} + b c d\right )} e^{5} + 2 \, {\left (5 \, b c^{3} d^{2} x + 3 \, a c^{2} d^{2}\right )} e^{4} + 2 \, {\left (2 \, b c^{5} d^{3} x^{2} + 3 \, b c^{3} d^{3}\right )} e^{3} + 3 \, {\left (3 \, b c^{5} d^{4} x + 2 \, a c^{4} d^{4}\right )} e^{2} + {\left (9 \, b c^{5} d^{4} x e^{2} + 3 \, b c^{5} d^{5} e - b c^{3} x^{3} e^{6} - 3 \, b c^{3} d x^{2} e^{5} + 3 \, {\left (b c^{5} d^{2} x^{3} - b c^{3} d^{2} x\right )} e^{4} + {\left (9 \, b c^{5} d^{3} x^{2} - b c^{3} d^{3}\right )} e^{3}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (9 \, b c^{5} d^{4} x e^{2} + 3 \, b c^{5} d^{5} e - b c^{3} x^{3} e^{6} - 3 \, b c^{3} d x^{2} e^{5} + 3 \, {\left (b c^{5} d^{2} x^{3} - b c^{3} d^{2} x\right )} e^{4} + {\left (9 \, b c^{5} d^{3} x^{2} - b c^{3} d^{3}\right )} e^{3}\right )} \log \left (x e + d\right )}{6 \, {\left (3 \, c^{6} d^{8} x e^{2} + c^{6} d^{9} e + x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, {\left (c^{2} d^{2} x^{3} + d^{2} x\right )} e^{8} + {\left (9 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{7} + 3 \, {\left (c^{4} d^{4} x^{3} + 3 \, c^{2} d^{4} x\right )} e^{6} + 3 \, {\left (3 \, c^{4} d^{5} x^{2} + c^{2} d^{5}\right )} e^{5} + {\left (c^{6} d^{6} x^{3} + 9 \, c^{4} d^{6} x\right )} e^{4} + 3 \, {\left (c^{6} d^{7} x^{2} + c^{4} d^{7}\right )} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(2*a*c^6*d^6 + 5*b*c^5*d^5*e - 2*(3*b*c^6*d^5*x*e - 3*b*c^4*d*x^3*e^5 - b*e^6 - 3*(3*b*c^4*d^2*x^2 + b*c^
2*d^2)*e^4 + (b*c^6*d^3*x^3 - 9*b*c^4*d^3*x)*e^3 + 3*(b*c^6*d^4*x^2 - 2*b*c^4*d^4)*e^2)*arctan(c*x) + (b*c*x +
2*a)*e^6 + (4*b*c^3*d*x^2 + b*c*d)*e^5 + 2*(5*b*c^3*d^2*x + 3*a*c^2*d^2)*e^4 + 2*(2*b*c^5*d^3*x^2 + 3*b*c^3*d
^3)*e^3 + 3*(3*b*c^5*d^4*x + 2*a*c^4*d^4)*e^2 + (9*b*c^5*d^4*x*e^2 + 3*b*c^5*d^5*e - b*c^3*x^3*e^6 - 3*b*c^3*d
*x^2*e^5 + 3*(b*c^5*d^2*x^3 - b*c^3*d^2*x)*e^4 + (9*b*c^5*d^3*x^2 - b*c^3*d^3)*e^3)*log(c^2*x^2 + 1) - 2*(9*b*
c^5*d^4*x*e^2 + 3*b*c^5*d^5*e - b*c^3*x^3*e^6 - 3*b*c^3*d*x^2*e^5 + 3*(b*c^5*d^2*x^3 - b*c^3*d^2*x)*e^4 + (9*b
*c^5*d^3*x^2 - b*c^3*d^3)*e^3)*log(x*e + d))/(3*c^6*d^8*x*e^2 + c^6*d^9*e + x^3*e^10 + 3*d*x^2*e^9 + 3*(c^2*d^
2*x^3 + d^2*x)*e^8 + (9*c^2*d^3*x^2 + d^3)*e^7 + 3*(c^4*d^4*x^3 + 3*c^2*d^4*x)*e^6 + 3*(3*c^4*d^5*x^2 + c^2*d^
5)*e^5 + (c^6*d^6*x^3 + 9*c^4*d^6*x)*e^4 + 3*(c^6*d^7*x^2 + c^4*d^7)*e^3)

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Sympy [C] Result contains complex when optimal does not.
time = 7.26, size = 9202, normalized size = 44.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))/(e*x+d)**4,x)

[Out]

Piecewise((a*x/d**4, Eq(c, 0) & Eq(e, 0)), (-a/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3), Eq(c,
0)), ((a*x + b*x*atan(c*x) - b*log(x**2 + c**(-2))/(2*c))/d**4, Eq(e, 0)), (-24*a*d**3/(72*d**6*e + 216*d**5*
e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 21*I*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216
*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 10*I*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3
*e**4*x**3) - 9*I*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**
3) + 9*I*b*d**2*e*x/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*I*b*d*e**2*x**2
*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 3*I*b*d*e**2*x**2/(72*d
**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 3*I*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e +
216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c, -I*e/d)), (-24*a*d**3/(72*d**6*e + 216*d**5*e
**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 21*I*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*
d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 10*I*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*
e**4*x**3) + 9*I*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3
) - 9*I*b*d**2*e*x/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*I*b*d*e**2*x**2*
atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 3*I*b*d*e**2*x**2/(72*d*
*6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 3*I*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 2
16*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c, I*e/d)), (-2*a*c**6*d**6/(6*c**6*d**9*e + 18*c
**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 5
4*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x
**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6*a*c**4*d**4*e
**2/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3
+ 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**
6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**1
0*x**3) - 6*a*c**2*d**2*e**4/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*
x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**
5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x
+ 18*d*e**9*x**2 + 6*e**10*x**3) - 2*a*e**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*
c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**
3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 +
18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 6*b*c**6*d**5*e*x*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e*
*2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5
*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c*
*2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 6*b*c**6*d**4*e**2*x**2*at
an(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e
**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4
*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*
e**10*x**3) + 2*b*c**6*d**3*e**3*x**3*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2
+ 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6
*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**
7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 3*b*c**5*d**5*e*log(x**2 + c**(-2))/(6*c**6*d**9*e + 18*
c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x +
54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*
x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 6*b*c**5*d**5*
e*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4
*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c*
*2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x*
*2 + 6*e**10*x**3) - 5*b*c**5*d**5*e/(6*c**6*d*...

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))/(d + e*x)^4,x)

[Out]

int((a + b*atan(c*x))/(d + e*x)^4, x)

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