3.12.40 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [1140]

Optimal. Leaf size=91 \[ -\frac {5 c \sqrt {1-a^2 x^2}}{2 a}-\frac {5 c (1+a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {c (1+a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c \text {ArcSin}(a x)}{2 a} \]

[Out]

5/2*c*arcsin(a*x)/a-5/2*c*(-a^2*x^2+1)^(1/2)/a-5/6*c*(a*x+1)*(-a^2*x^2+1)^(1/2)/a-1/3*c*(a*x+1)^2*(-a^2*x^2+1)
^(1/2)/a

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Rubi [A]
time = 0.04, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6273, 685, 655, 222} \begin {gather*} -\frac {c \sqrt {1-a^2 x^2} (a x+1)^2}{3 a}-\frac {5 c \sqrt {1-a^2 x^2} (a x+1)}{6 a}-\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c \text {ArcSin}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*c*Sqrt[1 - a^2*x^2])/(2*a) - (5*c*(1 + a*x)*Sqrt[1 - a^2*x^2])/(6*a) - (c*(1 + a*x)^2*Sqrt[1 - a^2*x^2])/(
3*a) + (5*c*ArcSin[a*x])/(2*a)

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

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

Rule 6273

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.63 \begin {gather*} -\frac {c \left (\sqrt {1-a^2 x^2} \left (22+9 a x+2 a^2 x^2\right )+30 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(c*(Sqrt[1 - a^2*x^2]*(22 + 9*a*x + 2*a^2*x^2) + 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(77)=154\).
time = 0.07, size = 280, normalized size = 3.08

method	result	size
risch	\(\frac {\left (2 a^{2} x^{2}+9 a x +22\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\)	\(71\)
meijerg	\(-\frac {2 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {2 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(269\)
default	\(-c \left (a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-2 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {3}{a \sqrt {-a^{2} x^{2}+1}}-\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 106, normalized size = 1.16 \begin {gather*} \frac {a^{3} c x^{4}}{3 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {10 \, a c x^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, c x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, c \arcsin \left (a x\right )}{2 \, a} - \frac {11 \, c}{3 \, \sqrt {-a^{2} x^{2} + 1} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*c*x^4/sqrt(-a^2*x^2 + 1) + 3/2*a^2*c*x^3/sqrt(-a^2*x^2 + 1) + 10/3*a*c*x^2/sqrt(-a^2*x^2 + 1) - 3/2*c*
x/sqrt(-a^2*x^2 + 1) + 5/2*c*arcsin(a*x)/a - 11/3*c/(sqrt(-a^2*x^2 + 1)*a)

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Fricas [A]
time = 0.33, size = 62, normalized size = 0.68 \begin {gather*} -\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c x^{2} + 9 \, a c x + 22 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(30*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c*x^2 + 9*a*c*x + 22*c)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]
time = 8.58, size = 218, normalized size = 2.40 \begin {gather*} a^{3} c \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + c \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, T
rue)) + 3*a**2*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x
**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 3*a*c*Piecewise(
(x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a*
*2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))

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Giac [A]
time = 0.44, size = 46, normalized size = 0.51 \begin {gather*} \frac {5 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c x + 9 \, c\right )} x + \frac {22 \, c}{a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*c*arcsin(a*x)*sgn(a)/abs(a) - 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c*x + 9*c)*x + 22*c/a)

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Mupad [B]
time = 0.04, size = 74, normalized size = 0.81 \begin {gather*} \frac {5\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {3\,c\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {11\,c\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {a\,c\,x^2\,\sqrt {1-a^2\,x^2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c*asinh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2)) - (3*c*x*(1 - a^2*x^2)^(1/2))/2 - (11*c*(1 - a^2*x^2)^(1/2))/(3*a
) - (a*c*x^2*(1 - a^2*x^2)^(1/2))/3

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