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

Optimal. Leaf size=145 \[ \frac {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}-\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}} \]

[Out]

-11/7/a/(c-c/a/x)^(7/2)-11/5/a/c/(c-c/a/x)^(5/2)-11/3/a/c^2/(c-c/a/x)^(3/2)+x/(c-c/a/x)^(7/2)+11*arctanh((c-c/
a/x)^(1/2)/c^(1/2))/a/c^(7/2)-11/a/c^3/(c-c/a/x)^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 78, 51, 63, 208} \[ -\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-11/(7*a*(c - c/(a*x))^(7/2)) - 11/(5*a*c*(c - c/(a*x))^(5/2)) - 11/(3*a*c^2*(c - c/(a*x))^(3/2)) - 11/(a*c^3*
Sqrt[c - c/(a*x)]) + x/(c - c/(a*x))^(7/2) + (11*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(7/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\\ &=-\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{7/2} (1-a x)} \, dx\\ &=\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{9/2} x} \, dx}{a}\\ &=\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx}{a}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {(11 c) \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^3}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^4}\\ &=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 46, normalized size = 0.32 \[ \frac {7 x-\frac {11 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};1-\frac {1}{a x}\right )}{a}}{7 \left (c-\frac {c}{a x}\right )^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(7*x - (11*Hypergeometric2F1[-7/2, 1, -5/2, 1 - 1/(a*x)])/a)/(7*(c - c/(a*x))^(7/2))

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fricas [A]  time = 0.60, size = 346, normalized size = 2.39 \[ \left [\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x -
c)/(a*x)) + c) + 2*(105*a^5*x^5 - 1936*a^4*x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x
)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4), -1/105*(1155*(a^4*x^4 - 4*a^3*x^3 + 6
*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (105*a^5*x^5 - 1936*a^4*x^4 + 4466
*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*
a^2*c^4*x + a*c^4)]

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giac [A]  time = 0.14, size = 187, normalized size = 1.29 \[ -\frac {a {\left (\frac {2 \, {\left (30 \, c^{3} + \frac {63 \, {\left (a c x - c\right )} c^{2}}{a x} + \frac {140 \, {\left (a c x - c\right )}^{2} c}{a^{2} x^{2}} + \frac {525 \, {\left (a c x - c\right )}^{3}}{a^{3} x^{3}}\right )} a x^{3}}{{\left (a c x - c\right )}^{3} c^{2} \sqrt {\frac {a c x - c}{a x}}} + \frac {1155 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} - \frac {105 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{2}}\right )}}{105 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*a*(2*(30*c^3 + 63*(a*c*x - c)*c^2/(a*x) + 140*(a*c*x - c)^2*c/(a^2*x^2) + 525*(a*c*x - c)^3/(a^3*x^3))*
a*x^3/((a*c*x - c)^3*c^2*sqrt((a*c*x - c)/(a*x))) + 1155*arctan(sqrt((a*c*x - c)/(a*x))/sqrt(-c))/(a^2*sqrt(-c
)*c^2) - 105*sqrt((a*c*x - c)/(a*x))/(a^2*(c - (a*c*x - c)/(a*x))*c^2))/c

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maple [B]  time = 0.06, size = 396, normalized size = 2.73 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2310 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} x^{5}+2100 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{3}-1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{5} a^{5}+11550 a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, x^{4}-5368 a^{\frac {7}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}+5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{4} a^{4}-23100 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}+4928 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{3} a^{3}+23100 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}-1540 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a^{2}-11550 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x -5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x a +2310 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{210 \sqrt {\left (a x -1\right ) x}\, c^{4} \sqrt {a}\, \left (a x -1\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(c*(a*x-1)/a/x)^(1/2)*x*(-2310*((a*x-1)*x)^(1/2)*a^(11/2)*x^5+2100*((a*x-1)*x)^(3/2)*a^(9/2)*x^3-1155*l
n(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^5*a^5+11550*a^(9/2)*((a*x-1)*x)^(1/2)*x^4-5368*a^(7/2)*
((a*x-1)*x)^(3/2)*x^2+5775*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^4*a^4-23100*a^(7/2)*((a*x-1
)*x)^(1/2)*x^3+4928*a^(5/2)*((a*x-1)*x)^(3/2)*x-11550*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^
3*a^3+23100*a^(5/2)*((a*x-1)*x)^(1/2)*x^2-1540*a^(3/2)*((a*x-1)*x)^(3/2)+11550*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(
1/2)+2*a*x-1)/a^(1/2))*x^2*a^2-11550*a^(3/2)*((a*x-1)*x)^(1/2)*x-5775*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*
x-1)/a^(1/2))*x*a+2310*((a*x-1)*x)^(1/2)*a^(1/2)+1155*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^4/a^(1/2)/(a*x-1)^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x - 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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