3.615 \(\int e^{\coth ^{-1}(a x)} (c-a^2 c x^2)^{7/2} \, dx\)

Optimal. Leaf size=183 \[ \frac {(a x+1)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {6 (a x+1)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {8 (a x+1)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]

[Out]

-8/5*(a*x+1)^5*(-a^2*c*x^2+c)^(7/2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7+2*(a*x+1)^6*(-a^2*c*x^2+c)^(7/2)/a^8/(1-1/a^2/
x^2)^(7/2)/x^7-6/7*(a*x+1)^7*(-a^2*c*x^2+c)^(7/2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7+1/8*(a*x+1)^8*(-a^2*c*x^2+c)^(7/
2)/a^8/(1-1/a^2/x^2)^(7/2)/x^7

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Rubi [A]  time = 0.18, antiderivative size = 183, 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 = {6192, 6193, 43} \[ \frac {(a x+1)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {6 (a x+1)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {8 (a x+1)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2),x]

[Out]

(-8*(1 + a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + (2*(1 + a*x)^6*(c - a^2*c*x^2)^(7
/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 + a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*
x^7) + ((1 + a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)

Rule 43

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 6192

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 6193

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u*(-1
+ a*x)^(p - n/2)*(1 + a*x)^(p + n/2))/x^(2*p), 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 )^{7/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (c-a^2 c x^2\right )^{7/2} \int (-1+a x)^3 (1+a x)^4 \, dx}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (c-a^2 c x^2\right )^{7/2} \int \left (-8 (1+a x)^4+12 (1+a x)^5-6 (1+a x)^6+(1+a x)^7\right ) \, dx}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {8 (1+a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1+a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {6 (1+a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1+a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 71, normalized size = 0.39 \[ -\frac {c^3 (a x+1)^5 \left (35 a^3 x^3-135 a^2 x^2+185 a x-93\right ) \sqrt {c-a^2 c x^2}}{280 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2),x]

[Out]

-1/280*(c^3*(1 + a*x)^5*Sqrt[c - a^2*c*x^2]*(-93 + 185*a*x - 135*a^2*x^2 + 35*a^3*x^3))/(a^2*Sqrt[1 - 1/(a^2*x
^2)]*x)

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fricas [A]  time = 0.54, size = 95, normalized size = 0.52 \[ -\frac {{\left (35 \, a^{7} c^{3} x^{8} + 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} + 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} - 280 \, c^{3} x\right )} \sqrt {-a^{2} c}}{280 \, a} \]

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)^(7/2),x, algorithm="fricas")

[Out]

-1/280*(35*a^7*c^3*x^8 + 40*a^6*c^3*x^7 - 140*a^5*c^3*x^6 - 168*a^4*c^3*x^5 + 210*a^3*c^3*x^4 + 280*a^2*c^3*x^
3 - 140*a*c^3*x^2 - 280*c^3*x)*sqrt(-a^2*c)/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

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)^(7/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(7/2)/sqrt((a*x - 1)/(a*x + 1)), x)

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maple [A]  time = 0.04, size = 100, normalized size = 0.55 \[ \frac {x \left (35 a^{7} x^{7}+40 x^{6} a^{6}-140 x^{5} a^{5}-168 x^{4} a^{4}+210 x^{3} a^{3}+280 a^{2} x^{2}-140 a x -280\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{280 \left (a x -1\right )^{3} \left (a x +1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}} \]

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)^(7/2),x)

[Out]

1/280*x*(35*a^7*x^7+40*a^6*x^6-140*a^5*x^5-168*a^4*x^4+210*a^3*x^3+280*a^2*x^2-140*a*x-280)*(-a^2*c*x^2+c)^(7/
2)/(a*x-1)^3/(a*x+1)^4/((a*x-1)/(a*x+1))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

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)^(7/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(7/2)/sqrt((a*x - 1)/(a*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)^(7/2)/((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

int((c - a^2*c*x^2)^(7/2)/((a*x - 1)/(a*x + 1))^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(7/2),x)

[Out]

Timed out

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