3.7.54 \(\int e^{4 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\) [654]

Optimal. Leaf size=100 \[ -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a} \]

[Out]

-1/7*c^4/a^8/x^7-2/3*c^4/a^7/x^6-4/5*c^4/a^6/x^5+c^4/a^5/x^4+10/3*c^4/a^4/x^3+2*c^4/a^3/x^2-4*c^4/a^2/x+c^4*x+
4*c^4*ln(x)/a

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Rubi [A]
time = 0.09, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6292, 6285, 90} \begin {gather*} -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+\frac {4 c^4 \log (x)}{a}+c^4 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*c^4/(a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4
)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6285

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

Rule 6292

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

Rubi steps

\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \frac {(1-a x)^2 (1+a x)^6}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \left (a^8+\frac {1}{x^8}+\frac {4 a}{x^7}+\frac {4 a^2}{x^6}-\frac {4 a^3}{x^5}-\frac {10 a^4}{x^4}-\frac {4 a^5}{x^3}+\frac {4 a^6}{x^2}+\frac {4 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 100, normalized size = 1.00 \begin {gather*} -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*c^4/(a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4
)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a

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Maple [A]
time = 1.41, size = 71, normalized size = 0.71

method	result	size
default	\(\frac {c^{4} \left (a^{8} x +\frac {10 a^{4}}{3 x^{3}}+\frac {a^{3}}{x^{4}}+\frac {2 a^{5}}{x^{2}}-\frac {4 a^{6}}{x}-\frac {4 a^{2}}{5 x^{5}}-\frac {1}{7 x^{7}}-\frac {2 a}{3 x^{6}}+4 a^{7} \ln \left (x \right )\right )}{a^{8}}\)	\(71\)
risch	\(c^{4} x +\frac {-4 a^{6} c^{4} x^{6}+2 a^{5} c^{4} x^{5}+\frac {10}{3} a^{4} c^{4} x^{4}+a^{3} c^{4} x^{3}-\frac {4}{5} a^{2} c^{4} x^{2}-\frac {2}{3} a \,c^{4} x -\frac {1}{7} c^{4}}{x^{7} a^{8}}+\frac {4 c^{4} \ln \left (x \right )}{a}\)	\(91\)
norman	\(\frac {2 a^{8} c^{4} x^{9}+a^{9} c^{4} x^{10}+\frac {c^{4}}{7 a}+\frac {2 c^{4} x}{3}-\frac {5 a^{2} c^{4} x^{3}}{3}-a^{4} c^{4} x^{5}-5 a^{7} c^{4} x^{8}+\frac {23 c^{4} a \,x^{2}}{35}+\frac {22 c^{4} a^{5} x^{6}}{3}-\frac {62 c^{4} x^{4} a^{3}}{15}}{\left (a^{2} x^{2}-1\right ) a^{7} x^{7}}+\frac {4 c^{4} \ln \left (x \right )}{a}\)	\(130\)
meijerg	\(\frac {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 {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 {17 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 {17 c^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {4 c^{4} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}+\frac {2 c^{4} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {6 a \,c^{4} x^{2}}{-a^{2} x^{2}+1}+\frac {4 c^{4} \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}-\frac {14 c^{4} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}+\frac {c^{4} \left (-\frac {2 \left (-315 a^{8} x^{8}+210 a^{6} x^{6}+42 a^{4} x^{4}+18 a^{2} x^{2}+10\right )}{7 x^{7} \left (-a^{2}\right )^{\frac {7}{2}} \left (-10 a^{2} x^{2}+10\right )}+\frac {9 \arctanh \left (a x \right ) a^{7}}{\left (-a^{2}\right )^{\frac {7}{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {6 c^{4} \left (\frac {4 a^{2} x^{2}}{-4 a^{2} x^{2}+4}-3 \ln \left (-a^{2} x^{2}+1\right )+1+6 \ln \left (x \right )+3 \ln \left (-a^{2}\right )-\frac {1}{2 a^{4} x^{4}}-\frac {2}{a^{2} x^{2}}\right )}{a}-\frac {2 c^{4} \left (-\frac {5 a^{2} x^{2}}{-5 a^{2} x^{2}+5}+4 \ln \left (-a^{2} x^{2}+1\right )-1-8 \ln \left (x \right )-4 \ln \left (-a^{2}\right )+\frac {1}{3 a^{6} x^{6}}+\frac {1}{a^{4} x^{4}}+\frac {3}{a^{2} x^{2}}\right )}{a}-\frac {c^{4} \left (-\frac {2 \left (-105 a^{6} x^{6}+70 a^{4} x^{4}+14 a^{2} x^{2}+6\right )}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}} \left (-6 a^{2} x^{2}+6\right )}+\frac {7 \arctanh \left (a x \right ) a^{5}}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{\sqrt {-a^{2}}}\)	\(791\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4/a^8*(a^8*x+10/3*a^4/x^3+a^3/x^4+2*a^5/x^2-4*a^6/x-4/5*a^2/x^5-1/7/x^7-2/3*a/x^6+4*a^7*ln(x))

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Maxima [A]
time = 0.25, size = 92, normalized size = 0.92 \begin {gather*} c^{4} x + \frac {4 \, c^{4} \log \left (x\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 4*c^4*log(x)/a - 1/105*(420*a^6*c^4*x^6 - 210*a^5*c^4*x^5 - 350*a^4*c^4*x^4 - 105*a^3*c^4*x^3 + 84*a^2
*c^4*x^2 + 70*a*c^4*x + 15*c^4)/(a^8*x^7)

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Fricas [A]
time = 0.35, size = 100, normalized size = 1.00 \begin {gather*} \frac {105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*a^8*c^4*x^8 + 420*a^7*c^4*x^7*log(x) - 420*a^6*c^4*x^6 + 210*a^5*c^4*x^5 + 350*a^4*c^4*x^4 + 105*a^
3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x - 15*c^4)/(a^8*x^7)

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Sympy [A]
time = 0.29, size = 100, normalized size = 1.00 \begin {gather*} \frac {a^{8} c^{4} x + 4 a^{7} c^{4} \log {\left (x \right )} + \frac {- 420 a^{6} c^{4} x^{6} + 210 a^{5} c^{4} x^{5} + 350 a^{4} c^{4} x^{4} + 105 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x - 15 c^{4}}{105 x^{7}}}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**8*c**4*x + 4*a**7*c**4*log(x) + (-420*a**6*c**4*x**6 + 210*a**5*c**4*x**5 + 350*a**4*c**4*x**4 + 105*a**3*
c**4*x**3 - 84*a**2*c**4*x**2 - 70*a*c**4*x - 15*c**4)/(105*x**7))/a**8

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Giac [A]
time = 0.40, size = 93, normalized size = 0.93 \begin {gather*} c^{4} x + \frac {4 \, c^{4} \log \left ({\left | x \right |}\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 4*c^4*log(abs(x))/a - 1/105*(420*a^6*c^4*x^6 - 210*a^5*c^4*x^5 - 350*a^4*c^4*x^4 - 105*a^3*c^4*x^3 + 8
4*a^2*c^4*x^2 + 70*a*c^4*x + 15*c^4)/(a^8*x^7)

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Mupad [B]
time = 0.86, size = 72, normalized size = 0.72 \begin {gather*} \frac {c^4\,\left (a^3\,x^3-\frac {4\,a^2\,x^2}{5}-\frac {2\,a\,x}{3}+\frac {10\,a^4\,x^4}{3}+2\,a^5\,x^5-4\,a^6\,x^6+a^8\,x^8+4\,a^7\,x^7\,\ln \left (x\right )-\frac {1}{7}\right )}{a^8\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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