3.7.36 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^5 \, dx\) [636]

Optimal. Leaf size=128 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a} \]

[Out]

1/9*c^5/a^10/x^9+1/4*c^5/a^9/x^8-3/7*c^5/a^8/x^7-4/3*c^5/a^7/x^6+2/5*c^5/a^6/x^5+3*c^5/a^5/x^4+2/3*c^5/a^4/x^3
-4*c^5/a^3/x^2-3*c^5/a^2/x-c^5*x-2*c^5*ln(x)/a

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Rubi [A]
time = 0.11, antiderivative size = 128, 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^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-\frac {2 c^5 \log (x)}{a}+c^5 (-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^5,x]

[Out]

c^5/(9*a^10*x^9) + c^5/(4*a^9*x^8) - (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) + (2*c^5)/(5*a^6*x^5) + (3*c^5)
/(a^5*x^4) + (2*c^5)/(3*a^4*x^3) - (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) - c^5*x - (2*c^5*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 )^5 \, dx &=-\frac {c^5 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}}\\ &=-\frac {c^5 \int \frac {(1-a x)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}}\\ &=-\frac {c^5 \int \left (a^{10}+\frac {1}{x^{10}}+\frac {2 a}{x^9}-\frac {3 a^2}{x^8}-\frac {8 a^3}{x^7}+\frac {2 a^4}{x^6}+\frac {12 a^5}{x^5}+\frac {2 a^6}{x^4}-\frac {8 a^7}{x^3}-\frac {3 a^8}{x^2}+\frac {2 a^9}{x}\right ) \, dx}{a^{10}}\\ &=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 128, normalized size = 1.00 \begin {gather*} \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^5,x]

[Out]

c^5/(9*a^10*x^9) + c^5/(4*a^9*x^8) - (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) + (2*c^5)/(5*a^6*x^5) + (3*c^5)
/(a^5*x^4) + (2*c^5)/(3*a^4*x^3) - (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) - c^5*x - (2*c^5*Log[x])/a

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Maple [A]
time = 1.59, size = 89, normalized size = 0.70

method	result	size
default	\(\frac {c^{5} \left (-a^{10} x +\frac {2 a^{6}}{3 x^{3}}+\frac {3 a^{5}}{x^{4}}-\frac {4 a^{7}}{x^{2}}-\frac {3 a^{8}}{x}+\frac {2 a^{4}}{5 x^{5}}-\frac {3 a^{2}}{7 x^{7}}-\frac {4 a^{3}}{3 x^{6}}-2 a^{9} \ln \left (x \right )+\frac {1}{9 x^{9}}+\frac {a}{4 x^{8}}\right )}{a^{10}}\)	\(89\)
risch	\(-x \,c^{5}+\frac {-3 a^{8} c^{5} x^{8}-4 a^{7} c^{5} x^{7}+\frac {2}{3} a^{6} c^{5} x^{6}+3 c^{5} a^{5} x^{5}+\frac {2}{5} a^{4} c^{5} x^{4}-\frac {4}{3} a^{3} c^{5} x^{3}-\frac {3}{7} a^{2} c^{5} x^{2}+\frac {1}{4} a \,c^{5} x +\frac {1}{9} c^{5}}{x^{9} a^{10}}-\frac {2 c^{5} \ln \left (x \right )}{a}\)	\(115\)
norman	\(\frac {\frac {c^{5}}{9 a}+\frac {x \,c^{5}}{4}-\frac {3 a \,c^{5} x^{2}}{7}-\frac {4 a^{2} c^{5} x^{3}}{3}+\frac {2 a^{3} c^{5} x^{4}}{5}-4 a^{6} c^{5} x^{7}-a^{9} c^{5} x^{10}+3 c^{5} a^{4} x^{5}+\frac {2 c^{5} a^{5} x^{6}}{3}-3 c^{5} a^{7} x^{8}}{a^{9} x^{9}}-\frac {2 c^{5} \ln \left (x \right )}{a}\)	\(120\)
meijerg	\(-\frac {c^{5} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {4 c^{5} \arctanh \left (a x \right )}{a}-\frac {5 c^{5} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {5 c^{5} \left (-\frac {2 a^{4}}{x \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2 a^{2}}{3 x^{3} \left (-a^{2}\right )^{\frac {5}{2}}}-\frac {2}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}}}+\frac {2 \arctanh \left (a x \right ) a^{5}}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{5} \left (-\frac {2 a^{6}}{x \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2 a^{4}}{3 x^{3} \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2 a^{2}}{5 x^{5} \left (-a^{2}\right )^{\frac {7}{2}}}-\frac {2}{7 x^{7} \left (-a^{2}\right )^{\frac {7}{2}}}+\frac {2 \arctanh \left (a x \right ) a^{7}}{\left (-a^{2}\right )^{\frac {7}{2}}}\right )}{\sqrt {-a^{2}}}-\frac {c^{5} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {5 c^{5} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}-\frac {10 c^{5} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}-\frac {10 c^{5} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{2} x^{2}}\right )}{a}-\frac {5 c^{5} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{3 a^{6} x^{6}}+\frac {1}{2 a^{4} x^{4}}+\frac {1}{a^{2} x^{2}}\right )}{a}-\frac {c^{5} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )-\frac {1}{4 a^{8} x^{8}}-\frac {1}{3 a^{6} x^{6}}-\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{2} x^{2}}\right )}{a}+\frac {c^{5} \left (-\frac {2 a^{8}}{x \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{6}}{3 x^{3} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{4}}{5 x^{5} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2 a^{2}}{7 x^{7} \left (-a^{2}\right )^{\frac {9}{2}}}-\frac {2}{9 x^{9} \left (-a^{2}\right )^{\frac {9}{2}}}+\frac {2 a^{9} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {9}{2}}}\right )}{2 \sqrt {-a^{2}}}\)	\(610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x,method=_RETURNVERBOSE)

[Out]

c^5/a^10*(-a^10*x+2/3*a^6/x^3+3*a^5/x^4-4*a^7/x^2-3*a^8/x+2/5*a^4/x^5-3/7*a^2/x^7-4/3*a^3/x^6-2*a^9*ln(x)+1/9/
x^9+1/4*a/x^8)

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Maxima [A]
time = 0.26, size = 115, normalized size = 0.90 \begin {gather*} -c^{5} x - \frac {2 \, c^{5} \log \left (x\right )}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="maxima")

[Out]

-c^5*x - 2*c^5*log(x)/a - 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 5
04*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)

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Fricas [A]
time = 0.35, size = 122, normalized size = 0.95 \begin {gather*} -\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="fricas")

[Out]

-1/1260*(1260*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) + 3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6
- 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)

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Sympy [A]
time = 0.40, size = 126, normalized size = 0.98 \begin {gather*} \frac {- a^{10} c^{5} x - 2 a^{9} c^{5} \log {\left (x \right )} - \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**5,x)

[Out]

(-a**10*c**5*x - 2*a**9*c**5*log(x) - (3780*a**8*c**5*x**8 + 5040*a**7*c**5*x**7 - 840*a**6*c**5*x**6 - 3780*a
**5*c**5*x**5 - 504*a**4*c**5*x**4 + 1680*a**3*c**5*x**3 + 540*a**2*c**5*x**2 - 315*a*c**5*x - 140*c**5)/(1260
*x**9))/a**10

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Giac [A]
time = 0.41, size = 116, normalized size = 0.91 \begin {gather*} -c^{5} x - \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm="giac")

[Out]

-c^5*x - 2*c^5*log(abs(x))/a - 1/1260*(3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^
5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)

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Mupad [B]
time = 0.89, size = 90, normalized size = 0.70 \begin {gather*} -\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \left (x\right )-\frac {1}{9}\right )}{a^{10}\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a^2*x^2))^5*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

-(c^5*((3*a^2*x^2)/7 - (a*x)/4 + (4*a^3*x^3)/3 - (2*a^4*x^4)/5 - 3*a^5*x^5 - (2*a^6*x^6)/3 + 4*a^7*x^7 + 3*a^8
*x^8 + a^10*x^10 + 2*a^9*x^9*log(x) - 1/9))/(a^10*x^9)

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