3.13.10 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [1210]

Optimal. Leaf size=234 \[ -\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}} \]

[Out]

-8/3*c^4*(-a*x+1)^6*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+32/7*c^4*(-a*x+1)^7*(-a^2*c*x^2+c)^(1/2)/a/(-a^2
*x^2+1)^(1/2)-3*c^4*(-a*x+1)^8*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+8/9*c^4*(-a*x+1)^9*(-a^2*c*x^2+c)^(1/
2)/a/(-a^2*x^2+1)^(1/2)-1/10*c^4*(-a*x+1)^10*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6278, 6275, 45} \begin {gather*} -\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(9/2)/E^ArcTanh[a*x],x]

[Out]

(-8*c^4*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) + (32*c^4*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7
*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) + (8*c^4*(1 - a*x)^9*Sqr
t[c - a^2*c*x^2])/(9*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 - a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6275

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

Rule 6278

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

Rubi steps

\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^5 (1+a x)^4 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int \left (16 (1-a x)^5-32 (1-a x)^6+24 (1-a x)^7-8 (1-a x)^8+(1-a x)^9\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 76, normalized size = 0.32 \begin {gather*} -\frac {c^4 (-1+a x)^6 \sqrt {c-a^2 c x^2} \left (193+528 a x+588 a^2 x^2+308 a^3 x^3+63 a^4 x^4\right )}{630 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)^(9/2)/E^ArcTanh[a*x],x]

[Out]

-1/630*(c^4*(-1 + a*x)^6*Sqrt[c - a^2*c*x^2]*(193 + 528*a*x + 588*a^2*x^2 + 308*a^3*x^3 + 63*a^4*x^4))/(a*Sqrt
[1 - a^2*x^2])

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Maple [A]
time = 0.07, size = 114, normalized size = 0.49

method	result	size
gosper	\(\frac {x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 x^{5} a^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {-a^{2} x^{2}+1}}{630 \left (a x -1\right )^{5} \left (a x +1\right )^{5}}\)	\(113\)
default	\(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, c^{4} x \left (63 a^{9} x^{9}-70 a^{8} x^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 x^{5} a^{5}-756 a^{4} x^{4}-630 a^{3} x^{3}+840 a^{2} x^{2}+315 a x -630\right )}{630 a^{2} x^{2}-630}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/630*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)*c^4*x*(63*a^9*x^9-70*a^8*x^8-315*a^7*x^7+360*a^6*x^6+630*a^5*x
^5-756*a^4*x^4-630*a^3*x^3+840*a^2*x^2+315*a*x-630)/(a^2*x^2-1)

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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((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(9/2)*sqrt(-a^2*x^2 + 1)/(a*x + 1), x)

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Fricas [A]
time = 0.34, size = 142, normalized size = 0.61 \begin {gather*} \frac {{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{630 \, {\left (a^{2} x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/630*(63*a^9*c^4*x^10 - 70*a^8*c^4*x^9 - 315*a^7*c^4*x^8 + 360*a^6*c^4*x^7 + 630*a^5*c^4*x^6 - 756*a^4*c^4*x^
5 - 630*a^3*c^4*x^4 + 840*a^2*c^4*x^3 + 315*a*c^4*x^2 - 630*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^
2*x^2 - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}{a x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(9/2)/(a*x + 1), x)

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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((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(9/2)*sqrt(-a^2*x^2 + 1)/(a*x + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a^2*c*x^2)^(9/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)

[Out]

int(((c - a^2*c*x^2)^(9/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1), x)

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