3.1.60 \(\int x^7 (a+b \text {ArcTan}(c x^2)) \, dx\) [60]

Optimal. Leaf size=54 \[ \frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}-\frac {b \text {ArcTan}\left (c x^2\right )}{8 c^4}+\frac {1}{8} x^8 \left (a+b \text {ArcTan}\left (c x^2\right )\right ) \]

[Out]

1/8*b*x^2/c^3-1/24*b*x^6/c-1/8*b*arctan(c*x^2)/c^4+1/8*x^8*(a+b*arctan(c*x^2))

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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4946, 281, 308, 209} \begin {gather*} \frac {1}{8} x^8 \left (a+b \text {ArcTan}\left (c x^2\right )\right )-\frac {b \text {ArcTan}\left (c x^2\right )}{8 c^4}+\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*ArcTan[c*x^2]),x]

[Out]

(b*x^2)/(8*c^3) - (b*x^6)/(24*c) - (b*ArcTan[c*x^2])/(8*c^4) + (x^8*(a + b*ArcTan[c*x^2]))/8

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^7 \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{4} (b c) \int \frac {x^9}{1+c^2 x^4} \, dx\\ &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{8} (b c) \text {Subst}\left (\int \frac {x^4}{1+c^2 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{8} (b c) \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )}{8 c^3}\\ &=\frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}-\frac {b \tan ^{-1}\left (c x^2\right )}{8 c^4}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.09 \begin {gather*} \frac {b x^2}{8 c^3}-\frac {b x^6}{24 c}+\frac {a x^8}{8}-\frac {b \text {ArcTan}\left (c x^2\right )}{8 c^4}+\frac {1}{8} b x^8 \text {ArcTan}\left (c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*ArcTan[c*x^2]),x]

[Out]

(b*x^2)/(8*c^3) - (b*x^6)/(24*c) + (a*x^8)/8 - (b*ArcTan[c*x^2])/(8*c^4) + (b*x^8*ArcTan[c*x^2])/8

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Maple [A]
time = 0.21, size = 50, normalized size = 0.93

method	result	size
default	\(\frac {x^{8} a}{8}+\frac {b \,x^{8} \arctan \left (c \,x^{2}\right )}{8}-\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}-\frac {b \arctan \left (c \,x^{2}\right )}{8 c^{4}}\)	\(50\)
risch	\(-\frac {i x^{8} b \ln \left (i c \,x^{2}+1\right )}{16}+\frac {i x^{8} b \ln \left (-i c \,x^{2}+1\right )}{16}+\frac {x^{8} a}{8}-\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}-\frac {b \arctan \left (c \,x^{2}\right )}{8 c^{4}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a+b*arctan(c*x^2)),x,method=_RETURNVERBOSE)

[Out]

1/8*x^8*a+1/8*b*x^8*arctan(c*x^2)-1/24*b*x^6/c+1/8*b*x^2/c^3-1/8*b*arctan(c*x^2)/c^4

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Maxima [A]
time = 0.47, size = 54, normalized size = 1.00 \begin {gather*} \frac {1}{8} \, a x^{8} + \frac {1}{24} \, {\left (3 \, x^{8} \arctan \left (c x^{2}\right ) - c {\left (\frac {c^{2} x^{6} - 3 \, x^{2}}{c^{4}} + \frac {3 \, \arctan \left (c x^{2}\right )}{c^{5}}\right )}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2)),x, algorithm="maxima")

[Out]

1/8*a*x^8 + 1/24*(3*x^8*arctan(c*x^2) - c*((c^2*x^6 - 3*x^2)/c^4 + 3*arctan(c*x^2)/c^5))*b

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Fricas [A]
time = 2.64, size = 51, normalized size = 0.94 \begin {gather*} \frac {3 \, a c^{4} x^{8} - b c^{3} x^{6} + 3 \, b c x^{2} + 3 \, {\left (b c^{4} x^{8} - b\right )} \arctan \left (c x^{2}\right )}{24 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2)),x, algorithm="fricas")

[Out]

1/24*(3*a*c^4*x^8 - b*c^3*x^6 + 3*b*c*x^2 + 3*(b*c^4*x^8 - b)*arctan(c*x^2))/c^4

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Sympy [A]
time = 28.52, size = 58, normalized size = 1.07 \begin {gather*} \begin {cases} \frac {a x^{8}}{8} + \frac {b x^{8} \operatorname {atan}{\left (c x^{2} \right )}}{8} - \frac {b x^{6}}{24 c} + \frac {b x^{2}}{8 c^{3}} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{8 c^{4}} & \text {for}\: c \neq 0 \\\frac {a x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(a+b*atan(c*x**2)),x)

[Out]

Piecewise((a*x**8/8 + b*x**8*atan(c*x**2)/8 - b*x**6/(24*c) + b*x**2/(8*c**3) - b*atan(c*x**2)/(8*c**4), Ne(c,
0)), (a*x**8/8, True))

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Giac [A]
time = 0.43, size = 60, normalized size = 1.11 \begin {gather*} \frac {3 \, a c x^{8} + {\left (3 \, c x^{8} \arctan \left (c x^{2}\right ) - \frac {3 \, \arctan \left (c x^{2}\right )}{c^{3}} - \frac {c^{9} x^{6} - 3 \, c^{7} x^{2}}{c^{9}}\right )} b}{24 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2)),x, algorithm="giac")

[Out]

1/24*(3*a*c*x^8 + (3*c*x^8*arctan(c*x^2) - 3*arctan(c*x^2)/c^3 - (c^9*x^6 - 3*c^7*x^2)/c^9)*b)/c

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Mupad [B]
time = 0.36, size = 49, normalized size = 0.91 \begin {gather*} \frac {a\,x^8}{8}+\frac {b\,x^2}{8\,c^3}-\frac {b\,x^6}{24\,c}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{8\,c^4}+\frac {b\,x^8\,\mathrm {atan}\left (c\,x^2\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*atan(c*x^2)),x)

[Out]

(a*x^8)/8 + (b*x^2)/(8*c^3) - (b*x^6)/(24*c) - (b*atan(c*x^2))/(8*c^4) + (b*x^8*atan(c*x^2))/8

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