3.1.78 \(\int \frac {e^{\frac {3}{2} \tanh ^{-1}(a x)}}{x^4} \, dx\) [78]

Optimal. Leaf size=139 \[ -\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {17}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]

[Out]

-1/3*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x^3-7/12*a*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x^2-23/24*a^2*(-a*x+1)^(1/4)*(a*x+1)
^(3/4)/x+17/8*a^3*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))-17/8*a^3*arctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))

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Rubi [A]
time = 0.04, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6261, 101, 156, 12, 95, 304, 209, 212} \begin {gather*} \frac {17}{8} a^3 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {23 a^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}{24 x}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (a x+1)^{3/4}}{12 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*ArcTanh[a*x])/2)/x^4,x]

[Out]

-1/3*((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x^3 - (7*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(12*x^2) - (23*a^2*(1 - a*x
)^(1/4)*(1 + a*x)^(3/4))/(24*x) + (17*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/8 - (17*a^3*ArcTanh[(1 + a*
x)^(1/4)/(1 - a*x)^(1/4)])/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 209

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

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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 6261

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rubi steps

\begin {align*} \int \frac {e^{\frac {3}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+a x)^{3/4}}{x^4 (1-a x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}+\frac {1}{3} \int \frac {\frac {7 a}{2}+2 a^2 x}{x^3 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {1}{6} \int \frac {-\frac {23 a^2}{4}-\frac {7 a^3 x}{2}}{x^2 (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{6} \int \frac {51 a^3}{8 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{16} \left (17 a^3\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {1}{4} \left (17 a^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}-\frac {1}{8} \left (17 a^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {1}{8} \left (17 a^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{3 x^3}-\frac {7 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{12 x^2}-\frac {23 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 x}+\frac {17}{8} a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {17}{8} a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 78, normalized size = 0.56 \begin {gather*} -\frac {\sqrt [4]{1-a x} \left (8+22 a x+37 a^2 x^2+23 a^3 x^3+102 a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{1+a x}\right )\right )}{24 x^3 \sqrt [4]{1+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((3*ArcTanh[a*x])/2)/x^4,x]

[Out]

-1/24*((1 - a*x)^(1/4)*(8 + 22*a*x + 37*a^2*x^2 + 23*a^3*x^3 + 102*a^3*x^3*Hypergeometric2F1[1/4, 1, 5/4, (1 -
a*x)/(1 + a*x)]))/(x^3*(1 + a*x)^(1/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x)

[Out]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x)

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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)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/sqrt(-a^2*x^2 + 1))^(3/2)/x^4, x)

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Fricas [A]
time = 0.38, size = 157, normalized size = 1.13 \begin {gather*} \frac {102 \, a^{3} x^{3} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (23 \, a^{2} x^{2} + 14 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm="fricas")

[Out]

1/48*(102*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x -
1)) + 1) + 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(23*a^2*x^2 + 14*a*x + 8)*sqrt(-a^2*x^
2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**4,x)

[Out]

Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x**4, 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)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,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 (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}}{x^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))^(3/2)/x^4,x)

[Out]

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(3/2)/x^4, x)

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