3.12.50 \(\int (c+d x^2)^4 \text {ArcTan}(a x) \, dx\) [1150]

Optimal. Leaf size=244 \[ -\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \text {ArcTan}(a x)+\frac {4}{3} c^3 d x^3 \text {ArcTan}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {ArcTan}(a x)+\frac {4}{7} c d^3 x^7 \text {ArcTan}(a x)+\frac {1}{9} d^4 x^9 \text {ArcTan}(a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \]

[Out]

-1/630*d*(420*a^6*c^3-378*a^4*c^2*d+180*a^2*c*d^2-35*d^3)*x^2/a^7-1/1260*d^2*(378*a^4*c^2-180*a^2*c*d+35*d^2)*
x^4/a^5-1/378*(36*a^2*c-7*d)*d^3*x^6/a^3-1/72*d^4*x^8/a+c^4*x*arctan(a*x)+4/3*c^3*d*x^3*arctan(a*x)+6/5*c^2*d^
2*x^5*arctan(a*x)+4/7*c*d^3*x^7*arctan(a*x)+1/9*d^4*x^9*arctan(a*x)-1/630*(315*a^8*c^4-420*a^6*c^3*d+378*a^4*c
^2*d^2-180*a^2*c*d^3+35*d^4)*ln(a^2*x^2+1)/a^9

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Rubi [A]
time = 0.13, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {200, 5032, 1824, 266} \begin {gather*} -\frac {d^3 x^6 \left (36 a^2 c-7 d\right )}{378 a^3}-\frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{1260 a^5}-\frac {d x^2 \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right )}{630 a^7}-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{630 a^9}+c^4 x \text {ArcTan}(a x)+\frac {4}{3} c^3 d x^3 \text {ArcTan}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {ArcTan}(a x)+\frac {4}{7} c d^3 x^7 \text {ArcTan}(a x)+\frac {1}{9} d^4 x^9 \text {ArcTan}(a x)-\frac {d^4 x^8}{72 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4*ArcTan[a*x],x]

[Out]

-1/630*(d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/a^7 - (d^2*(378*a^4*c^2 - 180*a^2*c*d +
35*d^2)*x^4)/(1260*a^5) - ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) - (d^4*x^8)/(72*a) + c^4*x*ArcTan[a*x] + (4*c^3
*d*x^3*ArcTan[a*x])/3 + (6*c^2*d^2*x^5*ArcTan[a*x])/5 + (4*c*d^3*x^7*ArcTan[a*x])/7 + (d^4*x^9*ArcTan[a*x])/9
- ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(630*a^9)

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 5032

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rubi steps

\begin {align*} \int \left (c+d x^2\right )^4 \tan ^{-1}(a x) \, dx &=c^4 x \tan ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tan ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tan ^{-1}(a x)-a \int \frac {c^4 x+\frac {4}{3} c^3 d x^3+\frac {6}{5} c^2 d^2 x^5+\frac {4}{7} c d^3 x^7+\frac {d^4 x^9}{9}}{1+a^2 x^2} \, dx\\ &=c^4 x \tan ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tan ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tan ^{-1}(a x)-a \int \left (\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x}{315 a^8}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^3}{315 a^6}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^5}{63 a^4}+\frac {d^4 x^7}{9 a^2}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) x}{315 a^8 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \tan ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tan ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tan ^{-1}(a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \int \frac {x}{1+a^2 x^2} \, dx}{315 a^7}\\ &=-\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \tan ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tan ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tan ^{-1}(a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 212, normalized size = 0.87 \begin {gather*} -\frac {a^2 d x^2 \left (-420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )-4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )-24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \text {ArcTan}(a x)+12 \left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{7560 a^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^4*ArcTan[a*x],x]

[Out]

-1/7560*(a^2*d*x^2*(-420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) - 4*a^4*d*(1134*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 3
*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) - 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^
2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcTan[a*x] + 12*(315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*
c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/a^9

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Maple [A]
time = 0.28, size = 254, normalized size = 1.04

method	result	size
derivativedivides	\(\frac {\arctan \left (a x \right ) c^{4} a x +\frac {4 a \arctan \left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \arctan \left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \arctan \left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \arctan \left (a x \right ) d^{4} x^{9}}{9}-\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\)	\(254\)
default	\(\frac {\arctan \left (a x \right ) c^{4} a x +\frac {4 a \arctan \left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \arctan \left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \arctan \left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \arctan \left (a x \right ) d^{4} x^{9}}{9}-\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\)	\(254\)
meijerg	\(\frac {d^{4} \left (\frac {x^{2} a^{2} \left (-15 a^{6} x^{6}+20 x^{4} a^{4}-30 a^{2} x^{2}+60\right )}{270}+\frac {4 x^{10} a^{10} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{9 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{9}\right )}{4 a^{9}}+\frac {d^{3} c \left (-\frac {a^{2} x^{2} \left (4 x^{4} a^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 a^{8} x^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{a^{7}}+\frac {3 c^{2} d^{2} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{2 a^{5}}+\frac {d \,c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 x^{4} a^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{a^{3}}+\frac {c^{4} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\)	\(331\)
risch	\(\frac {i \ln \left (-i a x +1\right ) c^{4} x}{2}+\frac {2 i \ln \left (-i a x +1\right ) d \,c^{3} x^{3}}{3}+\frac {3 i \ln \left (-i a x +1\right ) c^{2} d^{2} x^{5}}{5}-\frac {d^{4} x^{8}}{72 a}+\frac {2 i \ln \left (-i a x +1\right ) d^{3} c \,x^{7}}{7}-\frac {2 c \,d^{3} x^{6}}{21 a}+\frac {i \left (-\frac {1}{9} d^{4} x^{9}-\frac {4}{7} d^{3} c \,x^{7}-\frac {6}{5} c^{2} d^{2} x^{5}-\frac {4}{3} d \,c^{3} x^{3}-c^{4} x \right ) \ln \left (i a x +1\right )}{2}-\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {d^{4} x^{6}}{54 a^{3}}+\frac {i \ln \left (-i a x +1\right ) d^{4} x^{9}}{18}-\frac {2 c^{3} d \,x^{2}}{3 a}+\frac {c \,d^{3} x^{4}}{7 a^{3}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}-\frac {d^{4} x^{4}}{36 a^{5}}+\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}-\frac {2 c \,d^{3} x^{2}}{7 a^{5}}-\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}+\frac {d^{4} x^{2}}{18 a^{7}}+\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^4*arctan(a*x),x,method=_RETURNVERBOSE)

[Out]

1/a*(arctan(a*x)*c^4*a*x+4/3*a*arctan(a*x)*c^3*d*x^3+6/5*a*arctan(a*x)*c^2*d^2*x^5+4/7*a*arctan(a*x)*c*d^3*x^7
+1/9*a*arctan(a*x)*d^4*x^9-1/315/a^8*(210*c^3*a^8*d*x^2+189/2*c^2*a^8*d^2*x^4-189*c^2*a^6*d^2*x^2+30*c*a^8*d^3
*x^6-45*a^6*c*d^3*x^4+35/8*d^4*a^8*x^8+90*a^4*c*d^3*x^2-35/6*d^4*a^6*x^6+35/4*d^4*a^4*x^4-35/2*d^4*a^2*x^2+1/2
*(315*a^8*c^4-420*a^6*c^3*d+378*a^4*c^2*d^2-180*a^2*c*d^3+35*d^4)*ln(a^2*x^2+1)))

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Maxima [A]
time = 0.26, size = 226, normalized size = 0.93 \begin {gather*} -\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} - 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} - 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d - 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} - 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \arctan \left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arctan(a*x),x, algorithm="maxima")

[Out]

-1/7560*a*((105*a^6*d^4*x^8 + 20*(36*a^6*c*d^3 - 7*a^4*d^4)*x^6 + 6*(378*a^6*c^2*d^2 - 180*a^4*c*d^3 + 35*a^2*
d^4)*x^4 + 12*(420*a^6*c^3*d - 378*a^4*c^2*d^2 + 180*a^2*c*d^3 - 35*d^4)*x^2)/a^8 + 12*(315*a^8*c^4 - 420*a^6*
c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*log(a^2*x^2 + 1)/a^10) + 1/315*(35*d^4*x^9 + 180*c*d^3*x^7 +
378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^4*x)*arctan(a*x)

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Fricas [A]
time = 4.29, size = 237, normalized size = 0.97 \begin {gather*} -\frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} - 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} - 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d - 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} - 35 \, a^{2} d^{4}\right )} x^{2} - 24 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \arctan \left (a x\right ) + 12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{7560 \, a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arctan(a*x),x, algorithm="fricas")

[Out]

-1/7560*(105*a^8*d^4*x^8 + 20*(36*a^8*c*d^3 - 7*a^6*d^4)*x^6 + 6*(378*a^8*c^2*d^2 - 180*a^6*c*d^3 + 35*a^4*d^4
)*x^4 + 12*(420*a^8*c^3*d - 378*a^6*c^2*d^2 + 180*a^4*c*d^3 - 35*a^2*d^4)*x^2 - 24*(35*a^9*d^4*x^9 + 180*a^9*c
*d^3*x^7 + 378*a^9*c^2*d^2*x^5 + 420*a^9*c^3*d*x^3 + 315*a^9*c^4*x)*arctan(a*x) + 12*(315*a^8*c^4 - 420*a^6*c^
3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*log(a^2*x^2 + 1))/a^9

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Sympy [A]
time = 0.68, size = 314, normalized size = 1.29 \begin {gather*} \begin {cases} c^{4} x \operatorname {atan}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {atan}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {atan}{\left (a x \right )}}{9} - \frac {c^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} - \frac {2 c^{3} d x^{2}}{3 a} - \frac {3 c^{2} d^{2} x^{4}}{10 a} - \frac {2 c d^{3} x^{6}}{21 a} - \frac {d^{4} x^{8}}{72 a} + \frac {2 c^{3} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} + \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} + \frac {c d^{3} x^{4}}{7 a^{3}} + \frac {d^{4} x^{6}}{54 a^{3}} - \frac {3 c^{2} d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5 a^{5}} - \frac {2 c d^{3} x^{2}}{7 a^{5}} - \frac {d^{4} x^{4}}{36 a^{5}} + \frac {2 c d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{7 a^{7}} + \frac {d^{4} x^{2}}{18 a^{7}} - \frac {d^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{18 a^{9}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**4*atan(a*x),x)

[Out]

Piecewise((c**4*x*atan(a*x) + 4*c**3*d*x**3*atan(a*x)/3 + 6*c**2*d**2*x**5*atan(a*x)/5 + 4*c*d**3*x**7*atan(a*
x)/7 + d**4*x**9*atan(a*x)/9 - c**4*log(x**2 + a**(-2))/(2*a) - 2*c**3*d*x**2/(3*a) - 3*c**2*d**2*x**4/(10*a)
- 2*c*d**3*x**6/(21*a) - d**4*x**8/(72*a) + 2*c**3*d*log(x**2 + a**(-2))/(3*a**3) + 3*c**2*d**2*x**2/(5*a**3)
+ c*d**3*x**4/(7*a**3) + d**4*x**6/(54*a**3) - 3*c**2*d**2*log(x**2 + a**(-2))/(5*a**5) - 2*c*d**3*x**2/(7*a**
5) - d**4*x**4/(36*a**5) + 2*c*d**3*log(x**2 + a**(-2))/(7*a**7) + d**4*x**2/(18*a**7) - d**4*log(x**2 + a**(-
2))/(18*a**9), Ne(a, 0)), (0, True))

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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((d*x^2+c)^4*arctan(a*x),x, algorithm="giac")

[Out]

sage0*x

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Mupad [B]
time = 0.20, size = 233, normalized size = 0.95 \begin {gather*} \mathrm {atan}\left (a\,x\right )\,\left (c^4\,x+\frac {4\,c^3\,d\,x^3}{3}+\frac {6\,c^2\,d^2\,x^5}{5}+\frac {4\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{9}\right )+x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}-\frac {2\,c^3\,d}{3\,a}\right )+x^6\,\left (\frac {d^4}{54\,a^3}-\frac {2\,c\,d^3}{21\,a}\right )-x^4\,\left (\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )-\frac {\ln \left (a^2\,x^2+1\right )\,\left (315\,a^8\,c^4-420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2-180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}-\frac {d^4\,x^8}{72\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)*(c + d*x^2)^4,x)

[Out]

atan(a*x)*(c^4*x + (d^4*x^9)/9 + (4*c^3*d*x^3)/3 + (4*c*d^3*x^7)/7 + (6*c^2*d^2*x^5)/5) + x^2*(((d^4/(9*a^3) -
(4*c*d^3)/(7*a))/a^2 + (6*c^2*d^2)/(5*a))/(2*a^2) - (2*c^3*d)/(3*a)) + x^6*(d^4/(54*a^3) - (2*c*d^3)/(21*a))
- x^4*((d^4/(9*a^3) - (4*c*d^3)/(7*a))/(4*a^2) + (3*c^2*d^2)/(10*a)) - (log(a^2*x^2 + 1)*(35*d^4 + 315*a^8*c^4
- 180*a^2*c*d^3 - 420*a^6*c^3*d + 378*a^4*c^2*d^2))/(630*a^9) - (d^4*x^8)/(72*a)

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