3.12.83 \(\int e^{4 \tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\) [1183]

Optimal. Leaf size=52 \[ \frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a} \]

[Out]

4/7*c^4*(a*x+1)^7/a-1/2*c^4*(a*x+1)^8/a+1/9*c^4*(a*x+1)^9/a

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Rubi [A]
time = 0.04, antiderivative size = 52, 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^4 (a x+1)^9}{9 a}-\frac {c^4 (a x+1)^8}{2 a}+\frac {4 c^4 (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)^4,x]

[Out]

(4*c^4*(1 + a*x)^7)/(7*a) - (c^4*(1 + a*x)^8)/(2*a) + (c^4*(1 + a*x)^9)/(9*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 )^4 \, dx &=c^4 \int (1-a x)^2 (1+a x)^6 \, dx\\ &=c^4 \int \left (4 (1+a x)^6-4 (1+a x)^7+(1+a x)^8\right ) \, dx\\ &=\frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 31, normalized size = 0.60 \begin {gather*} \frac {c^4 (1+a x)^7 \left (23-35 a x+14 a^2 x^2\right )}{126 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(1 + a*x)^7*(23 - 35*a*x + 14*a^2*x^2))/(126*a)

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Maple [A]
time = 0.09, size = 69, normalized size = 1.33

method	result	size
gosper	\(\frac {c^{4} x \left (14 a^{8} x^{8}+63 a^{7} x^{7}+72 x^{6} a^{6}-84 x^{5} a^{5}-252 a^{4} x^{4}-126 a^{3} x^{3}+168 a^{2} x^{2}+252 a x +126\right )}{126}\)	\(69\)
default	\(c^{4} \left (\frac {1}{9} a^{8} x^{9}+\frac {1}{2} a^{7} x^{8}+\frac {4}{7} a^{6} x^{7}-\frac {2}{3} a^{5} x^{6}-2 a^{4} x^{5}-a^{3} x^{4}+\frac {4}{3} a^{2} x^{3}+2 x^{2} a +x \right )\)	\(69\)
risch	\(\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{2} a^{7} c^{4} x^{8}+\frac {4}{7} c^{4} a^{6} x^{7}-\frac {2}{3} a^{5} c^{4} x^{6}-2 a^{4} c^{4} x^{5}-a^{3} c^{4} x^{4}+\frac {4}{3} a^{2} c^{4} x^{3}+2 a \,c^{4} x^{2}+c^{4} x\)	\(93\)
norman	\(\frac {-2 a \,c^{4} x^{2}-c^{4} x -\frac {1}{3} a^{2} c^{4} x^{3}+3 a^{3} c^{4} x^{4}+\frac {10}{3} a^{4} c^{4} x^{5}-\frac {1}{3} a^{5} c^{4} x^{6}-\frac {7}{6} a^{7} c^{4} x^{8}+\frac {29}{63} a^{8} c^{4} x^{9}+\frac {1}{2} a^{9} c^{4} x^{10}+\frac {1}{9} a^{10} c^{4} x^{11}-\frac {18}{7} c^{4} a^{6} x^{7}}{a^{2} x^{2}-1}\)	\(128\)
meijerg	\(\frac {c^{4} \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}}}-\frac {c^{4} \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 )}{\sqrt {-a^{2}}}-\frac {c^{4} \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 )}{\sqrt {-a^{2}}}-\frac {14 c^{4} \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 {2 a \,c^{4} x^{2}}{-a^{2} x^{2}+1}+\frac {c^{4} \left (\frac {x \left (-a^{2}\right )^{\frac {13}{2}} \left (-910 a^{10} x^{10}-1430 a^{8} x^{8}-2574 x^{6} a^{6}-6006 a^{4} x^{4}-30030 a^{2} x^{2}+45045\right )}{4095 a^{12} \left (-a^{2} x^{2}+1\right )}-\frac {11 \left (-a^{2}\right )^{\frac {13}{2}} \arctanh \left (a x \right )}{a^{13}}\right )}{2 \sqrt {-a^{2}}}+\frac {4 c^{4} \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 {2 c^{4} \left (\frac {x^{2} a^{2} \left (-3 a^{8} x^{8}-5 x^{6} a^{6}-10 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{-12 a^{2} x^{2}+12}+5 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {17 c^{4} \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 {17 c^{4} \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 )}{2 \sqrt {-a^{2}}}+\frac {6 c^{4} \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^{4} \left (-\frac {x^{2} a^{2} \left (-3 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {6 c^{4} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}\)	\(821\)

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)^4,x,method=_RETURNVERBOSE)

[Out]

c^4*(1/9*a^8*x^9+1/2*a^7*x^8+4/7*a^6*x^7-2/3*a^5*x^6-2*a^4*x^5-a^3*x^4+4/3*a^2*x^3+2*x^2*a+x)

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Maxima [A]
time = 0.26, size = 92, normalized size = 1.77 \begin {gather*} \frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} 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)^4,x, algorithm="maxima")

[Out]

1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2*a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*
c^4*x^3 + 2*a*c^4*x^2 + c^4*x

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Fricas [A]
time = 0.32, size = 92, normalized size = 1.77 \begin {gather*} \frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} 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)^4,x, algorithm="fricas")

[Out]

1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2*a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*
c^4*x^3 + 2*a*c^4*x^2 + c^4*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).
time = 0.04, size = 100, normalized size = 1.92 \begin {gather*} \frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{2} + \frac {4 a^{6} c^{4} x^{7}}{7} - \frac {2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4 a^{2} c^{4} x^{3}}{3} + 2 a c^{4} x^{2} + c^{4} 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)**4,x)

[Out]

a**8*c**4*x**9/9 + a**7*c**4*x**8/2 + 4*a**6*c**4*x**7/7 - 2*a**5*c**4*x**6/3 - 2*a**4*c**4*x**5 - a**3*c**4*x
**4 + 4*a**2*c**4*x**3/3 + 2*a*c**4*x**2 + c**4*x

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Giac [A]
time = 0.42, size = 92, normalized size = 1.77 \begin {gather*} \frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} 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)^4,x, algorithm="giac")

[Out]

1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2*a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*
c^4*x^3 + 2*a*c^4*x^2 + c^4*x

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Mupad [B]
time = 0.05, size = 92, normalized size = 1.77 \begin {gather*} \frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{2}+\frac {4\,a^6\,c^4\,x^7}{7}-\frac {2\,a^5\,c^4\,x^6}{3}-2\,a^4\,c^4\,x^5-a^3\,c^4\,x^4+\frac {4\,a^2\,c^4\,x^3}{3}+2\,a\,c^4\,x^2+c^4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 2*a*c^4*x^2 + (4*a^2*c^4*x^3)/3 - a^3*c^4*x^4 - 2*a^4*c^4*x^5 - (2*a^5*c^4*x^6)/3 + (4*a^6*c^4*x^7)/7
+ (a^7*c^4*x^8)/2 + (a^8*c^4*x^9)/9

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