3.8.1 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{3/2}} \, dx\) [701]

Optimal. Leaf size=123 \[ \frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \text {ArcSin}(a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]

[Out]

1/3*(a*x+1)^2/a^2/(c-c/a^2/x^2)^(3/2)/x-2/3*(-2*a*x+5)*(-a*x+1)*(a*x+1)^2/a^4/(c-c/a^2/x^2)^(3/2)/x^3+2*(-a*x+
1)^(3/2)*(a*x+1)^(3/2)*arcsin(a*x)/a^4/(c-c/a^2/x^2)^(3/2)/x^3

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Rubi [A]
time = 0.24, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6294, 6264, 100, 148, 41, 222} \begin {gather*} \frac {(a x+1)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (1-a x)^{3/2} (a x+1)^{3/2} \text {ArcSin}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (5-2 a x) (1-a x) (a x+1)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2),x]

[Out]

(1 + a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x) - (2*(5 - 2*a*x)*(1 - a*x)*(1 + a*x)^2)/(3*a^4*(c - c/(a^2*x^2))
^(3/2)*x^3) + (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 222

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

Rule 6264

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

Rule 6294

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

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x^3}{(1-a x)^{5/2} \sqrt {1+a x}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x (2+4 a x)}{(1-a x)^{3/2} \sqrt {1+a x}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \sin ^{-1}(a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 95, normalized size = 0.77 \begin {gather*} \frac {-10+4 a x+11 a^2 x^2-3 a^3 x^3-6 (-1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2),x]

[Out]

(-10 + 4*a*x + 11*a^2*x^2 - 3*a^3*x^3 - 6*(-1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*
c*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs. \(2(109)=218\).
time = 0.82, size = 326, normalized size = 2.65

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a^{2} c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}-\frac {\left (\frac {2 \ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {8 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\)	\(216\)
default	\(-\frac {\left (3 x^{3} a^{3} c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} x^{2}-15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a^{2} c x -4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a x -6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a c -2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}+12 c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\right ) \left (a x +1\right )}{3 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\)	\(326\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(3*x^3*a^3*c^(3/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+4*c^(3/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2*x^2-15*x^2*a^2*c
^(3/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+6*(c*(a^2*x^2-1)/a^2)^(1/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x
-1)*(a*x+1)*c/a^2)^(1/2)*a^2*c*x-4*c^(3/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a*x-6*(c*(a^2*x^2-1)/a^2)^(1/2)*ln(x*c^(1
/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*a*c-2*(c*(a^2*x^2-1)/a^2)^(1/2)*c^(3/2)+12*c^(3/2
)*((a*x-1)*(a*x+1)*c/a^2)^(1/2))*(a*x+1)/((a*x-1)*(a*x+1)*c/a^2)^(1/2)/x^3/(c*(a^2*x^2-1)/a^2/x^2)^(3/2)/a^4/c
^(3/2)

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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((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2/((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(3/2)), x)

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Fricas [A]
time = 0.42, size = 281, normalized size = 2.28 \begin {gather*} \left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)
- (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3
*(6*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) -
(3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(3/2),x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(3/2)*(a^2*x^2 - 1)),x)

[Out]

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(3/2)*(a^2*x^2 - 1)), x)

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