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

Optimal. Leaf size=96 \[ -\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \]

[Out]

7/3/a/(c-c/a/x)^(3/2)-x/(c-c/a/x)^(3/2)-7*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a/c^(3/2)+7/a/c/(c-c/a/x)^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 78, 51, 63, 208} \[ -\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(3*a*(c - c/(a*x))^(3/2)) + 7/(a*c*Sqrt[c - c/(a*x)]) - x/(c - c/(a*x))^(3/2) - (7*ArcTanh[Sqrt[c - c/(a*x)]
/Sqrt[c]])/(a*c^(3/2))

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 78

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

Rule 208

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

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6133

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[(u*(c + d/x)^p*(1 + a*x)^(n/
2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 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 x}\right )^{3/2}} \, dx &=\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} x} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {(7 c) \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}\\ &=\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 55, normalized size = 0.57 \[ \frac {x \left (7 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {1}{a x}\right )-3 a x\right )}{3 c (a x-1) \sqrt {c-\frac {c}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-3*a*x + 7*Hypergeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c*Sqrt[c - c/(a*x)]*(-1 + a*x))

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fricas [A]  time = 0.66, size = 238, normalized size = 2.48 \[ \left [\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(3*a^3*x^
3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3*(21*(a^2*x^2 - 2*a*
x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (3*a^3*x^3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c
)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

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giac [A]  time = 0.20, size = 136, normalized size = 1.42 \[ \frac {a {\left (\frac {2 \, {\left (2 \, c + \frac {9 \, {\left (a c x - c\right )}}{a x}\right )} x}{{\left (a c x - c\right )} a \sqrt {\frac {a c x - c}{a x}}} + \frac {21 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {3 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )}}\right )}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*(2*(2*c + 9*(a*c*x - c)/(a*x))*x/((a*c*x - c)*a*sqrt((a*c*x - c)/(a*x))) + 21*arctan(sqrt((a*c*x - c)/(a
*x))/sqrt(-c))/(a^2*sqrt(-c)) - 3*sqrt((a*c*x - c)/(a*x))/(a^2*(c - (a*c*x - c)/(a*x))))/c

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maple [B]  time = 0.05, size = 260, normalized size = 2.71 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (42 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}+21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{3} a^{3}-36 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -126 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}-63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a^{2}+28 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+126 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x +63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x a -42 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{6 \sqrt {\left (a x -1\right ) x}\, c^{2} \left (a x -1\right )^{3} \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(c*(a*x-1)/a/x)^(1/2)*x*(42*a^(7/2)*((a*x-1)*x)^(1/2)*x^3+21*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)
/a^(1/2))*x^3*a^3-36*a^(5/2)*((a*x-1)*x)^(3/2)*x-126*a^(5/2)*((a*x-1)*x)^(1/2)*x^2-63*ln(1/2*(2*((a*x-1)*x)^(1
/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^2*a^2+28*a^(3/2)*((a*x-1)*x)^(3/2)+126*a^(3/2)*((a*x-1)*x)^(1/2)*x+63*ln(1/2*(
2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x*a-42*((a*x-1)*x)^(1/2)*a^(1/2)-21*ln(1/2*(2*((a*x-1)*x)^(1/2)*
a^(1/2)+2*a*x-1)/a^(1/2)))/((a*x-1)*x)^(1/2)/c^2/(a*x-1)^3/a^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{a c x \sqrt {c - \frac {c}{a x}} - 2 c \sqrt {c - \frac {c}{a x}} + \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\, dx - \int \frac {1}{a c x \sqrt {c - \frac {c}{a x}} - 2 c \sqrt {c - \frac {c}{a x}} + \frac {c \sqrt {c - \frac {c}{a x}}}{a x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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