3.12.84 \(\int e^{4 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [1184]

Optimal. Leaf size=35 \[ \frac {c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a} \]

[Out]

1/3*c^3*(a*x+1)^6/a-1/7*c^3*(a*x+1)^7/a

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Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6275, 45} \begin {gather*} \frac {c^3 (a x+1)^6}{3 a}-\frac {c^3 (a x+1)^7}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x) (1+a x)^5 \, dx\\ &=c^3 \int \left (2 (1+a x)^5-(1+a x)^6\right ) \, dx\\ &=\frac {c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.66 \begin {gather*} -\frac {c^3 (1+a x)^6 (-4+3 a x)}{21 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

-1/21*(c^3*(1 + a*x)^6*(-4 + 3*a*x))/a

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Maple [A]
time = 0.08, size = 45, normalized size = 1.29

method	result	size
gosper	\(-\frac {c^{3} x \left (3 x^{6} a^{6}+14 x^{5} a^{5}+21 a^{4} x^{4}-35 a^{2} x^{2}-42 a x -21\right )}{21}\)	\(45\)
default	\(c^{3} \left (-\frac {1}{7} a^{6} x^{7}-\frac {2}{3} a^{5} x^{6}-a^{4} x^{5}+\frac {5}{3} a^{2} x^{3}+2 x^{2} a +x \right )\)	\(45\)
risch	\(-\frac {1}{7} a^{6} c^{3} x^{7}-\frac {2}{3} a^{5} c^{3} x^{6}-a^{4} c^{3} x^{5}+\frac {5}{3} a^{2} c^{3} x^{3}+2 a \,c^{3} x^{2}+c^{3} x\)	\(60\)
norman	\(\frac {-2 a \,c^{3} x^{2}-c^{3} x -\frac {2}{3} a^{2} c^{3} x^{3}+2 a^{3} c^{3} x^{4}+\frac {8}{3} a^{4} c^{3} x^{5}+\frac {2}{3} a^{5} c^{3} x^{6}-\frac {6}{7} a^{6} c^{3} x^{7}-\frac {2}{3} a^{7} c^{3} x^{8}-\frac {1}{7} a^{8} c^{3} x^{9}}{a^{2} x^{2}-1}\)	\(106\)
meijerg	\(\frac {c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {11}{2}} \left (-110 a^{8} x^{8}-198 x^{6} a^{6}-462 a^{4} x^{4}-2310 a^{2} x^{2}+3465\right )}{385 a^{10} \left (-a^{2} x^{2}+1\right )}-\frac {9 \left (-a^{2}\right )^{\frac {11}{2}} \arctanh \left (a x \right )}{a^{11}}\right )}{2 \sqrt {-a^{2}}}-\frac {3 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {9}{2}} \left (-18 x^{6} a^{6}-42 a^{4} x^{4}-210 a^{2} x^{2}+315\right )}{45 a^{8} \left (-a^{2} x^{2}+1\right )}-\frac {7 \left (-a^{2}\right )^{\frac {9}{2}} \arctanh \left (a x \right )}{a^{9}}\right )}{2 \sqrt {-a^{2}}}-\frac {7 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-70 a^{2} x^{2}+105\right )}{21 a^{6} \left (-a^{2} x^{2}+1\right )}-\frac {5 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{\sqrt {-a^{2}}}-\frac {7 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{\sqrt {-a^{2}}}+\frac {2 c^{3} \left (-\frac {a^{2} x^{2} \left (-5 x^{6} a^{6}-10 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{15 \left (-a^{2} x^{2}+1\right )}-4 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {4 c^{3} \left (\frac {a^{2} x^{2} \left (-2 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{-4 a^{2} x^{2}+4}+3 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {4 c^{3} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 a \,c^{3} x^{2}}{-a^{2} x^{2}+1}+\frac {c^{3} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}\)	\(597\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/7*a^6*x^7-2/3*a^5*x^6-a^4*x^5+5/3*a^2*x^3+2*x^2*a+x)

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Maxima [A]
time = 0.27, size = 59, normalized size = 1.69 \begin {gather*} -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x

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Fricas [A]
time = 0.33, size = 59, normalized size = 1.69 \begin {gather*} -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).
time = 0.04, size = 63, normalized size = 1.80 \begin {gather*} - \frac {a^{6} c^{3} x^{7}}{7} - \frac {2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac {5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**3,x)

[Out]

-a**6*c**3*x**7/7 - 2*a**5*c**3*x**6/3 - a**4*c**3*x**5 + 5*a**2*c**3*x**3/3 + 2*a*c**3*x**2 + c**3*x

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Giac [A]
time = 0.43, size = 59, normalized size = 1.69 \begin {gather*} -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x

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Mupad [B]
time = 0.03, size = 59, normalized size = 1.69 \begin {gather*} -\frac {a^6\,c^3\,x^7}{7}-\frac {2\,a^5\,c^3\,x^6}{3}-a^4\,c^3\,x^5+\frac {5\,a^2\,c^3\,x^3}{3}+2\,a\,c^3\,x^2+c^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a^2*c*x^2)^3*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)

[Out]

c^3*x + 2*a*c^3*x^2 + (5*a^2*c^3*x^3)/3 - a^4*c^3*x^5 - (2*a^5*c^3*x^6)/3 - (a^6*c^3*x^7)/7

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