3.4.24 \(\int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^5} \, dx\) [324]

Optimal. Leaf size=110 \[ -\frac {c^4 \sqrt {1-a^2 x^2}}{4 x^4}+\frac {a c^4 \sqrt {1-a^2 x^2}}{x^3}-\frac {11 a^2 c^4 \sqrt {1-a^2 x^2}}{8 x^2}+a^4 c^4 \text {ArcSin}(a x)+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

a^4*c^4*arcsin(a*x)+13/8*a^4*c^4*arctanh((-a^2*x^2+1)^(1/2))-1/4*c^4*(-a^2*x^2+1)^(1/2)/x^4+a*c^4*(-a^2*x^2+1)
^(1/2)/x^3-11/8*a^2*c^4*(-a^2*x^2+1)^(1/2)/x^2

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Rubi [A]
time = 0.17, antiderivative size = 116, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 1821, 825, 858, 222, 272, 65, 214} \begin {gather*} a^4 c^4 \text {ArcSin}(a x)-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^5,x]

[Out]

-1/8*(a^2*c^4*(13 - 8*a*x)*Sqrt[1 - a^2*x^2])/x^2 - (c^4*(1 - a^2*x^2)^(3/2))/(4*x^4) + (a*c^4*(1 - a^2*x^2)^(
3/2))/x^3 + a^4*c^4*ArcSin[a*x] + (13*a^4*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

Rule 65

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 214

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

Rule 222

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 825

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^
(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 6263

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^5} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} c \int \frac {\sqrt {1-a^2 x^2} \left (12 a c^3-13 a^2 c^3 x+4 a^3 c^3 x^2\right )}{x^4} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+\frac {1}{12} c \int \frac {\left (39 a^2 c^3-12 a^3 c^3 x\right ) \sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac {1}{48} c \int \frac {78 a^4 c^3-48 a^5 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac {1}{8} \left (13 a^4 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\left (a^5 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)-\frac {1}{16} \left (13 a^4 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac {1}{8} \left (13 a^2 c^4\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a^2 c^4 (13-8 a x) \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac {13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 125, normalized size = 1.14 \begin {gather*} \frac {1}{16} c^4 \left (-13 a^4 \text {ArcSin}(a x)-\frac {2 \left (2-8 a x+9 a^2 x^2+8 a^3 x^3-11 a^4 x^4+29 a^4 x^4 \sqrt {1-a^2 x^2} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{x^4 \sqrt {1-a^2 x^2}}+26 a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^5,x]

[Out]

(c^4*(-13*a^4*ArcSin[a*x] - (2*(2 - 8*a*x + 9*a^2*x^2 + 8*a^3*x^3 - 11*a^4*x^4 + 29*a^4*x^4*Sqrt[1 - a^2*x^2]*
ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(x^4*Sqrt[1 - a^2*x^2]) + 26*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]))/16

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Maple [A]
time = 1.27, size = 172, normalized size = 1.56

method	result	size
risch	\(\frac {\left (11 a^{4} x^{4}-8 a^{3} x^{3}-9 a^{2} x^{2}+8 a x -2\right ) c^{4}}{8 x^{4} \sqrt {-a^{2} x^{2}+1}}+\left (\frac {a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {13 a^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}\right ) c^{4}\)	\(104\)
default	\(c^{4} \left (\frac {a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {11 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}-\frac {2 a^{3} \sqrt {-a^{2} x^{2}+1}}{x}+3 a^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(172\)
meijerg	\(a^{4} c^{4} \arcsin \left (a x \right )-\frac {3 a^{4} c^{4} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}-\frac {2 a^{3} c^{4} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a^{4} c^{4} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{\sqrt {\pi }}+\frac {a \,c^{4} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{x^{3}}+\frac {a^{4} c^{4} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^5,x,method=_RETURNVERBOSE)

[Out]

c^4*(a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-3*a*(-1/3*(-a^2*x^2+1)^(1/2)/x^3-2/3*a^2*(-a^2*x
^2+1)^(1/2)/x)-1/4*(-a^2*x^2+1)^(1/2)/x^4+11/4*a^2*(-1/2*(-a^2*x^2+1)^(1/2)/x^2-1/2*a^2*arctanh(1/(-a^2*x^2+1)
^(1/2)))-2*a^3*(-a^2*x^2+1)^(1/2)/x+3*a^4*arctanh(1/(-a^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.46, size = 109, normalized size = 0.99 \begin {gather*} a^{4} c^{4} \arcsin \left (a x\right ) + \frac {13}{8} \, a^{4} c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{4}}{x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^5,x, algorithm="maxima")

[Out]

a^4*c^4*arcsin(a*x) + 13/8*a^4*c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 11/8*sqrt(-a^2*x^2 + 1)*a^2*c
^4/x^2 + sqrt(-a^2*x^2 + 1)*a*c^4/x^3 - 1/4*sqrt(-a^2*x^2 + 1)*c^4/x^4

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Fricas [A]
time = 0.40, size = 106, normalized size = 0.96 \begin {gather*} -\frac {16 \, a^{4} c^{4} x^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 13 \, a^{4} c^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (11 \, a^{2} c^{4} x^{2} - 8 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^5,x, algorithm="fricas")

[Out]

-1/8*(16*a^4*c^4*x^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 13*a^4*c^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) +
(11*a^2*c^4*x^2 - 8*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/x^4

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Sympy [C] Result contains complex when optimal does not.
time = 6.52, size = 503, normalized size = 4.57 \begin {gather*} a^{5} c^{4} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\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)*(-a*c*x+c)**4/x**5,x)

[Out]

a**5*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 <
0)) - 3*a**4*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**3*c**4*Pi
ecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 2*a**2*c**4*Piecewi
se((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a
**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) - 3*a*c**4*Piecewise((-2*I
*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**
2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*
sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs
(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/
(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (96) = 192\).
time = 0.45, size = 316, normalized size = 2.87 \begin {gather*} \frac {a^{5} c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {13 \, a^{5} c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {{\left (a^{5} c^{4} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{4}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c^{4}}{x^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a x^{3}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{4} {\left | a \right |}}{x} - \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c^{4} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{4} {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^5,x, algorithm="giac")

[Out]

a^5*c^4*arcsin(a*x)*sgn(a)/abs(a) + 13/8*a^5*c^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))
/abs(a) + 1/64*(a^5*c^4 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3*c^4/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a
*c^4/x^2 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a))
+ 1/64*(8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*c^4*abs(a)/x - 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*c^4*abs
(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c^4*abs(a)/x^3 - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4*abs(a)/
(a*x^4))/a^4

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Mupad [B]
time = 0.05, size = 113, normalized size = 1.03 \begin {gather*} \frac {a\,c^4\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {11\,a^2\,c^4\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {a^4\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^4*(a*x + 1))/(x^5*(1 - a^2*x^2)^(1/2)),x)

[Out]

(a*c^4*(1 - a^2*x^2)^(1/2))/x^3 - (c^4*(1 - a^2*x^2)^(1/2))/(4*x^4) - (a^4*c^4*atan((1 - a^2*x^2)^(1/2)*1i)*13
i)/8 + (a^5*c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (11*a^2*c^4*(1 - a^2*x^2)^(1/2))/(8*x^2)

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