3.7.87 \(\int e^{\tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\) [687]

Optimal. Leaf size=299 \[ -\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{\left (1-a^2 x^2\right )^{7/2}}-\frac {3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}} \]

[Out]

-1/6*(c-c/a^2/x^2)^(7/2)*x/(-a^2*x^2+1)^(7/2)-1/5*a*(c-c/a^2/x^2)^(7/2)*x^2/(-a^2*x^2+1)^(7/2)+3/4*a^2*(c-c/a^
2/x^2)^(7/2)*x^3/(-a^2*x^2+1)^(7/2)+a^3*(c-c/a^2/x^2)^(7/2)*x^4/(-a^2*x^2+1)^(7/2)-3/2*a^4*(c-c/a^2/x^2)^(7/2)
*x^5/(-a^2*x^2+1)^(7/2)-3*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)-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 = 0.13, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6295, 6285, 90} \begin {gather*} -\frac {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^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 {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 {3 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 {3 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 {a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 90

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 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 6295

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/x^(2*p))*(1 + c*(x^2/d))^p*E^(n*ArcTanh[a*x]), 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^{\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^{\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)^3 (1+a x)^4}{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 {a}{x^6}-\frac {3 a^2}{x^5}-\frac {3 a^3}{x^4}+\frac {3 a^4}{x^3}+\frac {3 a^5}{x^2}-\frac {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 {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 {3 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 {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{\left (1-a^2 x^2\right )^{7/2}}-\frac {3 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 {3 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 {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 = 0.04, size = 98, normalized size = 0.33 \begin {gather*} \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (10+12 a x-45 a^2 x^2-60 a^3 x^3+90 a^4 x^4+180 a^5 x^5+60 a^7 x^7+60 a^6 x^6 \log (x)\right )}{60 a^6 x^5 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*Sqrt[c - c/(a^2*x^2)]*(10 + 12*a*x - 45*a^2*x^2 - 60*a^3*x^3 + 90*a^4*x^4 + 180*a^5*x^5 + 60*a^7*x^7 + 60
*a^6*x^6*Log[x]))/(60*a^6*x^5*Sqrt[1 - a^2*x^2])

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Maple [A]
time = 0.42, size = 102, normalized size = 0.34

method	result	size
default	\(-\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (60 a^{7} x^{7}+60 \ln \left (x \right ) a^{6} x^{6}+180 a^{5} x^{5}+90 a^{4} x^{4}-60 a^{3} x^{3}-45 a^{2} x^{2}+12 a x +10\right )}{60 \left (a^{2} x^{2}-1\right )^{4}}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+60*ln(x)*a^6*x^6+180*a^5*x^
5+90*a^4*x^4-60*a^3*x^3-45*a^2*x^2+12*a*x+10)

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

[Out]

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

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Fricas [A]
time = 0.41, size = 542, normalized size = 1.81 \begin {gather*} \left [\frac {30 \, {\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} + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} + 180 \, a^{5} + 90 \, a^{4} - 60 \, a^{3} - 45 \, a^{2} + 12 \, a + 10\right )} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, 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 {60 \, {\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} + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} + 180 \, a^{5} + 90 \, a^{4} - 60 \, a^{3} - 45 \, a^{2} + 12 \, a + 10\right )} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, 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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(30*(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 + 180*a^5*c^3*x^5 + 90
*a^4*c^3*x^4 - (60*a^7 + 180*a^5 + 90*a^4 - 60*a^3 - 45*a^2 + 12*a + 10)*c^3*x^6 - 60*a^3*c^3*x^3 - 45*a^2*c^3
*x^2 + 12*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*(60
*(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 + 180*a^5*c^3*x^5 + 90*a^4*c^3*x^4 - (60*a^7 + 180
*a^5 + 90*a^4 - 60*a^3 - 45*a^2 + 12*a + 10)*c^3*x^6 - 60*a^3*c^3*x^3 - 45*a^2*c^3*x^2 + 12*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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)*(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)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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