3.6.9 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^5} \, dx\) [509]

Optimal. Leaf size=125 \[ -\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \text {ArcSin}(a x)}{a c^5} \]

[Out]

-1/5*(a*x+1)^2/a/c^5/(-a^2*x^2+1)^(5/2)+22/15*(a*x+1)/a/c^5/(-a^2*x^2+1)^(3/2)+2*arcsin(a*x)/a/c^5-2/15*(23*a*
x+30)/a/c^5/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/c^5

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Rubi [A]
time = 0.23, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6266, 6263, 866, 1649, 1828, 655, 222} \begin {gather*} -\frac {(a x+1)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (a x+1)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}-\frac {2 (23 a x+30)}{15 a c^5 \sqrt {1-a^2 x^2}}+\frac {2 \text {ArcSin}(a x)}{a c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^5),x]

[Out]

-1/5*(1 + a*x)^2/(a*c^5*(1 - a^2*x^2)^(5/2)) + (22*(1 + a*x))/(15*a*c^5*(1 - a^2*x^2)^(3/2)) - (2*(30 + 23*a*x
))/(15*a*c^5*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^5) + (2*ArcSin[a*x])/(a*c^5)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 6263

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

Rule 6266

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

Rubi steps

\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx &=-\frac {a^5 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^5}{(1-a x)^5} \, dx}{c^5}\\ &=-\frac {a^5 \int \frac {x^5}{(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^5}\\ &=-\frac {a^5 \int \frac {x^5 (1+a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 \int \frac {(1+a x) \left (\frac {2}{a^5}+\frac {5 x}{a^4}+\frac {5 x^2}{a^3}+\frac {5 x^3}{a^2}+\frac {5 x^4}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^5 \int \frac {\frac {16}{a^5}+\frac {45 x}{a^4}+\frac {30 x^2}{a^3}+\frac {15 x^3}{a^2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}+\frac {a^5 \int \frac {\frac {30}{a^5}+\frac {15 x}{a^4}}{\sqrt {1-a^2 x^2}} \, dx}{15 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \sin ^{-1}(a x)}{a c^5}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 76, normalized size = 0.61 \begin {gather*} \frac {\frac {\sqrt {1-a^2 x^2} \left (56-82 a x-32 a^2 x^2+76 a^3 x^3-15 a^4 x^4\right )}{(-1+a x)^3 (1+a x)}+30 \text {ArcSin}(a x)}{15 a c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^5),x]

[Out]

((Sqrt[1 - a^2*x^2]*(56 - 82*a*x - 32*a^2*x^2 + 76*a^3*x^3 - 15*a^4*x^4))/((-1 + a*x)^3*(1 + a*x)) + 30*ArcSin
[a*x])/(15*a*c^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1290\) vs. \(2(111)=222\).
time = 1.17, size = 1291, normalized size = 10.33

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{5}}-\frac {\left (-\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {383 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{120 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}\right ) a^{5}}{c^{5}}\)	\(227\)
default	\(\text {Expression too large to display}\)	\(1291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x,method=_RETURNVERBOSE)

[Out]

a^5/c^5*(1/40/a^11/(x-1/a)^5*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+5/32/a^7*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(
x-1/a))^(5/2)-3*a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a
*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))+1/2/a^8*(1/a/(x-1/
a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+2*a*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-a^2*
(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/
2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))))+5/128/a^6*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3
/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(
-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))+1/32/a^8*(-1/a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1
/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2
*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(
1/2))))))-5/64/a^7*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/
2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-
a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))+7/16/a^9*(1/3/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+1/3*a*(1/a/(
x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+2*a*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-
a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)
^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))))-5/128/a^6*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a
))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2
)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^5), x)

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Fricas [A]
time = 0.37, size = 151, normalized size = 1.21 \begin {gather*} -\frac {56 \, a^{4} x^{4} - 112 \, a^{3} x^{3} + 112 \, a x + 60 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {-a^{2} x^{2} + 1} - 56}{15 \, {\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="fricas")

[Out]

-1/15*(56*a^4*x^4 - 112*a^3*x^3 + 112*a*x + 60*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) -
1)/(a*x)) + (15*a^4*x^4 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt(-a^2*x^2 + 1) - 56)/(a^5*c^5*x^4 - 2*a^4
*c^5*x^3 + 2*a^2*c^5*x - a*c^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{5} \left (\int \frac {x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{7} \sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\right )\, dx\right )}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**5,x)

[Out]

a**5*(Integral(x**5*sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6 + 6*a**5*x**5 - 6*a**3*x**3 +
2*a**2*x**2 + 2*a*x - 1), x) + Integral(-a**2*x**7*sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6
+ 6*a**5*x**5 - 6*a**3*x**3 + 2*a**2*x**2 + 2*a*x - 1), x))/c**5

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 0.91, size = 275, normalized size = 2.20 \begin {gather*} \frac {41\,a\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^5}+\frac {\sqrt {1-a^2\,x^2}}{8\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {383\,\sqrt {1-a^2\,x^2}}{120\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {\sqrt {1-a^2\,x^2}}{10\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^5*(a*x + 1)^3),x)

[Out]

(41*a*(1 - a^2*x^2)^(1/2))/(60*(a^2*c^5 - 2*a^3*c^5*x + a^4*c^5*x^2)) + (2*asinh(x*(-a^2)^(1/2)))/(c^5*(-a^2)^
(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c^5) + (1 - a^2*x^2)^(1/2)/(8*(-a^2)^(1/2)*(c^5*x*(-a^2)^(1/2) + (c^5*(-a^2)^(
1/2))/a)) - (383*(1 - a^2*x^2)^(1/2))/(120*(-a^2)^(1/2)*(c^5*x*(-a^2)^(1/2) - (c^5*(-a^2)^(1/2))/a)) - (1 - a^
2*x^2)^(1/2)/(10*(-a^2)^(1/2)*(3*c^5*x*(-a^2)^(1/2) - (c^5*(-a^2)^(1/2))/a + a^2*c^5*x^3*(-a^2)^(1/2) - 3*a*c^
5*x^2*(-a^2)^(1/2)))

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