3.7.14 \(\int e^{\coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [614]

Optimal. Leaf size=229 \[ \frac {8 (1+a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1+a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1+a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1+a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1+a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 229, 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 = {6327, 6328, 45} \begin {gather*} \frac {(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 6327

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

Rule 6328

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 79, normalized size = 0.34 \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^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))/(630*a^2*Sqrt[1
- 1/(a^2*x^2)]*x)

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Maple [A]
time = 0.13, size = 113, normalized size = 0.49

method	result	size
default	\(\frac {\left (63 a^{9} x^{9}+70 a^{8} x^{8}-315 a^{7} x^{7}-360 a^{6} x^{6}+630 a^{5} x^{5}+756 a^{4} x^{4}-630 a^{3} x^{3}-840 a^{2} x^{2}+315 a x +630\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{630 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(113\)
gosper	\(\frac {x \left (63 a^{9} x^{9}+70 a^{8} x^{8}-315 a^{7} x^{7}-360 a^{6} x^{6}+630 a^{5} x^{5}+756 a^{4} x^{4}-630 a^{3} x^{3}-840 a^{2} x^{2}+315 a x +630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{630 \left (a x -1\right )^{4} \left (a x +1\right )^{5} \sqrt {\frac {a x -1}{a x +1}}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(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+6
30)*x*c^4*(-c*(a^2*x^2-1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(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*x+1))^(1/2)*(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 117, normalized size = 0.51 \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}}{630 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

[Out]

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

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