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

Optimal. Leaf size=153 \[ -\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

[Out]

(-45*c^3*x*Sqrt[c - a^2*c*x^2])/128 - (15*c^2*x*(c - a^2*c*x^2)^(3/2))/64 - (3*c*x*(c - a^2*c*x^2)^(5/2))/16 +
(9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) - (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)
/Sqrt[c - a^2*c*x^2]])/(128*a)

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Rubi [A]  time = 0.152873, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6167, 6141, 671, 641, 195, 217, 203} \[ -\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*c^3*x*Sqrt[c - a^2*c*x^2])/128 - (15*c^2*x*(c - a^2*c*x^2)^(3/2))/64 - (3*c*x*(c - a^2*c*x^2)^(5/2))/16 +
(9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) - (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)
/Sqrt[c - a^2*c*x^2]])/(128*a)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6141

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^(n/2), Int[(c + d*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] || GtQ[c, 0]) && IGt
Q[n/2, 0]

Rule 671

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

Rule 641

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

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

Rule 203

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

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{5/2} \, dx\right )\\ &=\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{8} (9 c) \int (1+a x) \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{8} (9 c) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{16} \left (15 c^2\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{64} \left (45 c^3\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{128} \left (45 c^4\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{1}{128} \left (45 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}\\ \end{align*}

Mathematica [A]  time = 0.131442, size = 151, normalized size = 0.99 \[ \frac{c^3 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (112 a^8 x^8+144 a^7 x^7-424 a^6 x^6-600 a^5 x^5+558 a^4 x^4+978 a^3 x^3-187 a^2 x^2-837 a x+256\right )+630 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{896 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(256 - 837*a*x - 187*a^2*x^2 + 978*a^3*x^3 + 558*a^4*x^4 - 600*a^5*x^5
- 424*a^6*x^6 + 144*a^7*x^7 + 112*a^8*x^8) + 630*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(896*a*Sqrt[1
- a*x]*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.052, size = 296, normalized size = 1.9 \begin{align*}{\frac{x}{8} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{7\,cx}{48} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{35\,x{c}^{2}}{192} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{c}^{3}x}{128}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{35\,{c}^{4}}{128}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{7\,a} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{cx}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{5\,x{c}^{2}}{12} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{c}^{3}x}{8}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}-{\frac{5\,{c}^{4}}{8}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*(-a^2*c*x^2+c)^(7/2)+7/48*c*x*(-a^2*c*x^2+c)^(5/2)+35/192*c^2*x*(-a^2*c*x^2+c)^(3/2)+35/128*c^3*x*(-a^2*
c*x^2+c)^(1/2)+35/128*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/7/a*(-c*a^2*(x-1/a)^2-2
*a*c*(x-1/a))^(7/2)-1/3*c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(5/2)*x-5/12*c^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(
3/2)*x-5/8*c^3*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x-5/8*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-c*a^2*(
x-1/a)^2-2*a*c*(x-1/a))^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68114, size = 653, normalized size = 4.27 \begin{align*} \left [\frac{315 \, \sqrt{-c} c^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt{-a^{2} c x^{2} + c}}{1792 \, a}, \frac{315 \, c^{\frac{7}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt{-a^{2} c x^{2} + c}}{896 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(112*a^7*c^3*x^7 + 25
6*a^6*c^3*x^6 - 168*a^5*c^3*x^5 - 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c^3*x - 256*c^3)
*sqrt(-a^2*c*x^2 + c))/a, 1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (112*a
^7*c^3*x^7 + 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 - 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c
^3*x - 256*c^3)*sqrt(-a^2*c*x^2 + c))/a]

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Sympy [C]  time = 34.3969, size = 1091, normalized size = 7.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**6*c**3*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1))
- I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c
)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x*
*9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**
2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqr
t(c)*asin(a*x)/(128*a**7), True)) - 2*a**5*c**3*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x
**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0))
, (sqrt(c)*x**6/6, True)) + a**4*c**3*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**
5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**
2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) +
5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sq
rt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 4*a**3*c**3*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)
/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, T
rue)) + a**2*c**3*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2
- 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*
sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2
*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a*c**3*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2,
0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) - c**3*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 -
1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(
-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))

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Giac [A]  time = 1.17002, size = 190, normalized size = 1.24 \begin{align*} \frac{45 \, c^{4} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{128 \, \sqrt{-c}{\left | a \right |}} + \frac{1}{896} \, \sqrt{-a^{2} c x^{2} + c}{\left (\frac{256 \, c^{3}}{a} -{\left (581 \, c^{3} + 2 \,{\left (384 \, a c^{3} -{\left (105 \, a^{2} c^{3} + 4 \,{\left (96 \, a^{3} c^{3} +{\left (21 \, a^{4} c^{3} - 2 \,{\left (7 \, a^{6} c^{3} x + 16 \, a^{5} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

45/128*c^4*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a)) + 1/896*sqrt(-a^2*c*x^2 + c)*(25
6*c^3/a - (581*c^3 + 2*(384*a*c^3 - (105*a^2*c^3 + 4*(96*a^3*c^3 + (21*a^4*c^3 - 2*(7*a^6*c^3*x + 16*a^5*c^3)*
x)*x)*x)*x)*x)*x)