3.12.100 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^5} \, dx\) [1200]

Optimal. Leaf size=119 \[ -\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}} \]

[Out]

1/9*(a*x-1)/a/c^5/(-a^2*x^2+1)^(9/2)+8/63*x/c^5/(-a^2*x^2+1)^(7/2)+16/105*x/c^5/(-a^2*x^2+1)^(5/2)+64/315*x/c^
5/(-a^2*x^2+1)^(3/2)+128/315*x/c^5/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6274, 653, 198, 197} \begin {gather*} \frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]

[Out]

-1/9*(1 - a*x)/(a*c^5*(1 - a^2*x^2)^(9/2)) + (8*x)/(63*c^5*(1 - a^2*x^2)^(7/2)) + (16*x)/(105*c^5*(1 - a^2*x^2
)^(5/2)) + (64*x)/(315*c^5*(1 - a^2*x^2)^(3/2)) + (128*x)/(315*c^5*Sqrt[1 - a^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 6274

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

Rubi steps

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{21 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{105 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{315 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 91, normalized size = 0.76 \begin {gather*} -\frac {35-280 a x-280 a^2 x^2+560 a^3 x^3+560 a^4 x^4-448 a^5 x^5-448 a^6 x^6+128 a^7 x^7+128 a^8 x^8}{315 a c^5 (1-a x)^{7/2} (1+a x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]

[Out]

-1/315*(35 - 280*a*x - 280*a^2*x^2 + 560*a^3*x^3 + 560*a^4*x^4 - 448*a^5*x^5 - 448*a^6*x^6 + 128*a^7*x^7 + 128
*a^8*x^8)/(a*c^5*(1 - a*x)^(7/2)*(1 + a*x)^(9/2))

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.06, size = 1002, normalized size = 8.42

method	result	size
gosper	\(-\frac {128 a^{8} x^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 x^{5} a^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35}{315 \left (-a^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a x +1\right ) c^{5} a}\)	\(90\)
trager	\(-\frac {\left (128 a^{8} x^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 x^{5} a^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35\right ) \sqrt {-a^{2} x^{2}+1}}{315 c^{5} \left (a x +1\right )^{5} \left (a x -1\right )^{4} a}\)	\(97\)
default	\(\text {Expression too large to display}\)	\(1002\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^5*(-63/512/a*((-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(
x+1/a))^(1/2)))-3/64/a^4*(1/5/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)-1/15/(x-1/a)^3*(-a^2*(x-1/a)^2-2*
(x-1/a)*a)^(3/2))-15/128/a^4*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^3*(-a^2*(x+1/a)
^2+2*a*(x+1/a))^(3/2))-35/256/a^2*(-1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-a*((-a^2*(x+1/a)^2+2*a*(x
+1/a))^(1/2)+a/(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^5*(-1/7/a/(x+1/a)
^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+2/7*a*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^
3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)))+1/64/a^5*(1/7/a/(x-1/a)^5*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)-2/7*a*(1/5
/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)-1/15/(x-1/a)^3*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)))-1/32/a^6*(
-1/9/a/(x+1/a)^6*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+1/3*a*(-1/7/a/(x+1/a)^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)
+2/7*a*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2))
))+63/512/a*((-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)-a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*(x-1/a)*a
)^(1/2)))+7/256/a^4/(x-1/a)^3*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)+35/768/a^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1
/a))^(3/2)-7/64/a^2*(1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)+a*((-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)-a/
(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*(x-1/a)*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)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (98) = 196\).
time = 0.41, size = 249, normalized size = 2.09 \begin {gather*} -\frac {35 \, a^{9} x^{9} + 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} - 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} + 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} - 140 \, a^{2} x^{2} + 35 \, a x + {\left (128 \, a^{8} x^{8} + 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} - 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} + 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} - 280 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} + 35}{315 \, {\left (a^{10} c^{5} x^{9} + a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} - 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} - 4 \, a^{3} c^{5} x^{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)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="fricas")

[Out]

-1/315*(35*a^9*x^9 + 35*a^8*x^8 - 140*a^7*x^7 - 140*a^6*x^6 + 210*a^5*x^5 + 210*a^4*x^4 - 140*a^3*x^3 - 140*a^
2*x^2 + 35*a*x + (128*a^8*x^8 + 128*a^7*x^7 - 448*a^6*x^6 - 448*a^5*x^5 + 560*a^4*x^4 + 560*a^3*x^3 - 280*a^2*
x^2 - 280*a*x + 35)*sqrt(-a^2*x^2 + 1) + 35)/(a^10*c^5*x^9 + a^9*c^5*x^8 - 4*a^8*c^5*x^7 - 4*a^7*c^5*x^6 + 6*a
^6*c^5*x^5 + 6*a^5*c^5*x^4 - 4*a^4*c^5*x^3 - 4*a^3*c^5*x^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 {\int \frac {1}{a^{9} x^{9} \sqrt {- a^{2} x^{2} + 1} + a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.33, size = 177, normalized size = 1.49 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {53\,x}{252\,c^5}-\frac {5}{36\,a\,c^5}\right )}{{\left (a\,x-1\right )}^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}}{144\,a\,c^5\,{\left (a\,x+1\right )}^5}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {733\,x}{5040\,c^5}+\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {128\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )}+\frac {64\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1 - a^2*x^2)^(1/2)*((53*x)/(252*c^5) - 5/(36*a*c^5)))/((a*x - 1)^4*(a*x + 1)^4) - (1 - a^2*x^2)^(1/2)/(144*a
*c^5*(a*x + 1)^5) - ((1 - a^2*x^2)^(1/2)*((733*x)/(5040*c^5) + 5/(144*a*c^5)))/((a*x - 1)^3*(a*x + 1)^3) - (12
8*x*(1 - a^2*x^2)^(1/2))/(315*c^5*(a*x - 1)*(a*x + 1)) + (64*x*(1 - a^2*x^2)^(1/2))/(315*c^5*(a*x - 1)^2*(a*x
+ 1)^2)

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