3.14.1 \(\int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{x (c-a^2 c x^2)^{5/4}} \, dx\) [1301]

Optimal. Leaf size=144 \[ \frac {\sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {2 \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \]

[Out]

-2*(-a^2*x^2+1)^(1/4)*arctanh((-a*x+1)^(1/2))/c/(-a^2*c*x^2+c)^(1/4)+1/2*(-a^2*x^2+1)^(1/4)*arctanh(1/2*(-a*x+
1)^(1/2)*2^(1/2))/c/(-a^2*c*x^2+c)^(1/4)*2^(1/2)+(-a^2*x^2+1)^(1/4)/c/(-a^2*c*x^2+c)^(1/4)/(-a*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {6288, 6285, 87, 162, 65, 214, 212} \begin {gather*} \frac {\sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {2 \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(ArcTanh[a*x]/2)/(x*(c - a^2*c*x^2)^(5/4)),x]

[Out]

(1 - a^2*x^2)^(1/4)/(c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) - (2*(1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]])/(
c*(c - a^2*c*x^2)^(1/4)) + ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*c*(c - a^2*c*x^2)^(1/
4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 87

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

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 212

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

Rule 214

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

Rule 6285

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

Rule 6288

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 79, normalized size = 0.55 \begin {gather*} -\frac {\sqrt [4]{1-a^2 x^2} \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )-2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-a x\right )\right )}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(ArcTanh[a*x]/2)/(x*(c - a^2*c*x^2)^(5/4)),x]

[Out]

-(((1 - a^2*x^2)^(1/4)*(Hypergeometric2F1[-1/2, 1, 1/2, (1 - a*x)/2] - 2*Hypergeometric2F1[-1/2, 1, 1/2, 1 - a
*x]))/(c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x)

[Out]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x)

________________________________________________________________________________________

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)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(5/4)*x), x)

________________________________________________________________________________________

Fricas [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(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x/(-a**2*c*x**2+c)**(5/4),x)

[Out]

Timed out

________________________________________________________________________________________

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*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x, algorithm="giac")

[Out]

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(5/4)*x), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x*(c - a^2*c*x^2)^(5/4)),x)

[Out]

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x*(c - a^2*c*x^2)^(5/4)), x)

________________________________________________________________________________________