3.7.71 \(\int e^{-2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^3 \, dx\) [671]

Optimal. Leaf size=78 \[ \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 78, 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^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 (-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 78, normalized size = 1.00 \begin {gather*} \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.92, size = 57, normalized size = 0.73

method	result	size
default	\(\frac {c^{3} \left (-a^{6} x -\frac {a^{2}}{3 x^{3}}-\frac {a}{2 x^{4}}+\frac {2 a^{3}}{x^{2}}-\frac {a^{4}}{x}+\frac {1}{5 x^{5}}+2 a^{5} \ln \left (x \right )\right )}{a^{6}}\)	\(57\)
risch	\(-x \,c^{3}+\frac {-a^{4} c^{3} x^{4}+2 a^{3} c^{3} x^{3}-\frac {1}{3} a^{2} c^{3} x^{2}-\frac {1}{2} a \,c^{3} x +\frac {1}{5} c^{3}}{a^{6} x^{5}}+\frac {2 c^{3} \ln \left (x \right )}{a}\)	\(71\)
norman	\(\frac {a^{3} c^{3} x^{4}+\frac {c^{3}}{5 a}-\frac {3 x \,c^{3}}{10}-\frac {5 a \,c^{3} x^{2}}{6}+\frac {5 a^{2} c^{3} x^{3}}{3}-c^{3} a^{6} x^{7}}{a^{5} x^{5} \left (a x +1\right )}+\frac {2 c^{3} \ln \left (x \right )}{a}\)	\(82\)
meijerg	\(-\frac {c^{3} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {4 c^{3} x}{a x +1}-\frac {6 c^{3} \left (\frac {3 a x}{3 a x +3}+2 \ln \left (a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {4 c^{3} \left (\frac {5 a x}{5 a x +5}+4 \ln \left (a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (a \right )-\frac {1}{3 a^{3} x^{3}}+\frac {1}{a^{2} x^{2}}-\frac {3}{a x}\right )}{a}-\frac {c^{3} \left (\frac {7 a x}{7 a x +7}+6 \ln \left (a x +1\right )-1-6 \ln \left (x \right )-6 \ln \left (a \right )-\frac {1}{5 a^{5} x^{5}}+\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{3} x^{3}}+\frac {2}{a^{2} x^{2}}-\frac {5}{a x}\right )}{a}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 71, normalized size = 0.91 \begin {gather*} -c^{3} x + \frac {2 \, c^{3} \log \left (x\right )}{a} - \frac {30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*x + 2*c^3*log(x)/a - 1/30*(30*a^4*c^3*x^4 - 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 + 15*a*c^3*x - 6*c^3)/(a^6*x^
5)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 78, normalized size = 1.00 \begin {gather*} -\frac {30 \, a^{6} c^{3} x^{6} - 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(30*a^6*c^3*x^6 - 60*a^5*c^3*x^5*log(x) + 30*a^4*c^3*x^4 - 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 + 15*a*c^3*x
- 6*c^3)/(a^6*x^5)

________________________________________________________________________________________

Sympy [A]
time = 0.17, size = 76, normalized size = 0.97 \begin {gather*} \frac {- a^{6} c^{3} x + 2 a^{5} c^{3} \log {\left (x \right )} - \frac {30 a^{4} c^{3} x^{4} - 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} + 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**6*c**3*x + 2*a**5*c**3*log(x) - (30*a**4*c**3*x**4 - 60*a**3*c**3*x**3 + 10*a**2*c**3*x**2 + 15*a*c**3*x
- 6*c**3)/(30*x**5))/a**6

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 136, normalized size = 1.74 \begin {gather*} -\frac {2 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (30 \, c^{3} - \frac {71 \, c^{3}}{a x + 1} - \frac {65 \, c^{3}}{{\left (a x + 1\right )}^{2}} + \frac {310 \, c^{3}}{{\left (a x + 1\right )}^{3}} - \frac {270 \, c^{3}}{{\left (a x + 1\right )}^{4}} + \frac {60 \, c^{3}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}}{30 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^3*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a + 2*c^3*log(abs(-1/(a*x + 1) + 1))/a + 1/30*(30*c^3 - 71*c^3/(
a*x + 1) - 65*c^3/(a*x + 1)^2 + 310*c^3/(a*x + 1)^3 - 270*c^3/(a*x + 1)^4 + 60*c^3/(a*x + 1)^5)*(a*x + 1)/(a*(
1/(a*x + 1) - 1)^5)

________________________________________________________________________________________

Mupad [B]
time = 0.85, size = 57, normalized size = 0.73 \begin {gather*} -\frac {c^3\,\left (\frac {a\,x}{2}+\frac {a^2\,x^2}{3}-2\,a^3\,x^3+a^4\,x^4+a^6\,x^6-2\,a^5\,x^5\,\ln \left (x\right )-\frac {1}{5}\right )}{a^6\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________