3.706 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\)

Optimal. Leaf size=301 \[ -\frac {3 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 x^7 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}} \]

[Out]

-1/6*(c-c/a^2/x^2)^(7/2)*x/(-a^2*x^2+1)^(7/2)-3/5*a*(c-c/a^2/x^2)^(7/2)*x^2/(-a^2*x^2+1)^(7/2)-1/4*a^2*(c-c/a^
2/x^2)^(7/2)*x^3/(-a^2*x^2+1)^(7/2)+5/3*a^3*(c-c/a^2/x^2)^(7/2)*x^4/(-a^2*x^2+1)^(7/2)+5/2*a^4*(c-c/a^2/x^2)^(
7/2)*x^5/(-a^2*x^2+1)^(7/2)-a^5*(c-c/a^2/x^2)^(7/2)*x^6/(-a^2*x^2+1)^(7/2)+a^7*(c-c/a^2/x^2)^(7/2)*x^8/(-a^2*x
^2+1)^(7/2)+3*a^6*(c-c/a^2/x^2)^(7/2)*x^7*ln(x)/(-a^2*x^2+1)^(7/2)

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Rubi [A]  time = 0.20, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6160, 6150, 88} \[ \frac {a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 x^7 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2),x]

[Out]

-((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) - (3*a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2
)) - (a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) + (5*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a
^2*x^2)^(7/2)) + (5*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) - (a^5*(c - c/(a^2*x^2))^(7/2)*x^
6)/(1 - a^2*x^2)^(7/2) + (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) + (3*a^6*(c - c/(a^2*x^2))^(7/2
)*x^7*Log[x])/(1 - a^2*x^2)^(7/2)

Rule 88

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

Rule 6150

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 6160

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

Rubi steps

\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2}}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^2 (1+a x)^5}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (a^7+\frac {1}{x^7}+\frac {3 a}{x^6}+\frac {a^2}{x^5}-\frac {5 a^3}{x^4}-\frac {5 a^4}{x^3}+\frac {a^5}{x^2}+\frac {3 a^6}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 \left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}+\frac {a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 98, normalized size = 0.33 \[ -\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (60 a^7 x^7+180 a^6 x^6 \log (x)-60 a^5 x^5+150 a^4 x^4+100 a^3 x^3-15 a^2 x^2-36 a x-10\right )}{60 a^6 x^5 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2),x]

[Out]

-1/60*(c^3*Sqrt[c - c/(a^2*x^2)]*(-10 - 36*a*x - 15*a^2*x^2 + 100*a^3*x^3 + 150*a^4*x^4 - 60*a^5*x^5 + 60*a^7*
x^7 + 180*a^6*x^6*Log[x]))/(a^6*x^5*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.77, size = 542, normalized size = 1.80 \[ \left [\frac {90 \, {\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) + {\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} + 150 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} - 60 \, a^{5} + 150 \, a^{4} + 100 \, a^{3} - 15 \, a^{2} - 36 \, a - 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} - 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x - 10 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left (a^{8} x^{7} - a^{6} x^{5}\right )}}, \frac {180 \, {\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) + {\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} + 150 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} - 60 \, a^{5} + 150 \, a^{4} + 100 \, a^{3} - 15 \, a^{2} - 36 \, a - 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} - 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x - 10 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left (a^{8} x^{7} - a^{6} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

[1/60*(90*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^
2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 + 150
*a^4*c^3*x^4 - (60*a^7 - 60*a^5 + 150*a^4 + 100*a^3 - 15*a^2 - 36*a - 10)*c^3*x^6 + 100*a^3*c^3*x^3 - 15*a^2*c
^3*x^2 - 36*a*c^3*x - 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), 1/60*(1
80*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a
^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 + 150*a^4*c^3*x^4 - (60*a^7 - 6
0*a^5 + 150*a^4 + 100*a^3 - 15*a^2 - 36*a - 10)*c^3*x^6 + 100*a^3*c^3*x^3 - 15*a^2*c^3*x^2 - 36*a*c^3*x - 10*c
^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(7/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 102, normalized size = 0.34 \[ \frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (60 a^{7} x^{7}+180 a^{6} \ln \relax (x ) x^{6}-60 x^{5} a^{5}+150 x^{4} a^{4}+100 x^{3} a^{3}-15 a^{2} x^{2}-36 a x -10\right )}{60 \left (a^{2} x^{2}-1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*x/(a^2*x^2-1)^4*(-a^2*x^2+1)^(1/2)*(60*a^7*x^7+180*a^6*ln(x)*x^6-60*x^5*a^5
+150*x^4*a^4+100*x^3*a^3-15*a^2*x^2-36*a*x-10)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(7/2),x)

[Out]

Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)

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