3.1148 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=143 \[ -\frac {c^3 (a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}-\frac {3 c^3 (a x+1) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {9 c^3 \sin ^{-1}(a x)}{16 a} \]

[Out]

3/8*c^3*x*(-a^2*x^2+1)^(3/2)-3/10*c^3*(-a^2*x^2+1)^(5/2)/a-3/14*c^3*(a*x+1)*(-a^2*x^2+1)^(5/2)/a-1/7*c^3*(a*x+
1)^2*(-a^2*x^2+1)^(5/2)/a+9/16*c^3*arcsin(a*x)/a+9/16*c^3*x*(-a^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac {c^3 (a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}-\frac {3 c^3 (a x+1) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {9 c^3 \sin ^{-1}(a x)}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*c^3*x*Sqrt[1 - a^2*x^2])/16 + (3*c^3*x*(1 - a^2*x^2)^(3/2))/8 - (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) - (3*c^3
*(1 + a*x)*(1 - a^2*x^2)^(5/2))/(14*a) - (c^3*(1 + a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16
*a)

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

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 641

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 671

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 6138

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 )^3 \, dx &=c^3 \int (1+a x)^3 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{7} \left (9 c^3\right ) \int (1+a x)^2 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{2} \left (3 c^3\right ) \int (1+a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{2} \left (3 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{8} \left (9 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{16} \left (9 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {9 c^3 \sin ^{-1}(a x)}{16 a}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 91, normalized size = 0.64 \[ -\frac {c^3 \left (\sqrt {1-a^2 x^2} \left (80 a^6 x^6+280 a^5 x^5+208 a^4 x^4-350 a^3 x^3-656 a^2 x^2-245 a x+368\right )+630 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{560 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/560*(c^3*(Sqrt[1 - a^2*x^2]*(368 - 245*a*x - 656*a^2*x^2 - 350*a^3*x^3 + 208*a^4*x^4 + 280*a^5*x^5 + 80*a^6
*x^6) + 630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a

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fricas [A]  time = 0.64, size = 114, normalized size = 0.80 \[ -\frac {630 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (80 \, a^{6} c^{3} x^{6} + 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} - 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} - 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a} \]

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)^3,x, algorithm="fricas")

[Out]

-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (80*a^6*c^3*x^6 + 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 -
350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 - 245*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a

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giac [A]  time = 0.22, size = 102, normalized size = 0.71 \[ \frac {9 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, {\left | a \right |}} - \frac {1}{560} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {368 \, c^{3}}{a} - {\left (245 \, c^{3} + 2 \, {\left (328 \, a c^{3} + {\left (175 \, a^{2} c^{3} - 4 \, {\left (26 \, a^{3} c^{3} + 5 \, {\left (2 \, a^{5} c^{3} x + 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

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)^3,x, algorithm="giac")

[Out]

9/16*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/560*sqrt(-a^2*x^2 + 1)*(368*c^3/a - (245*c^3 + 2*(328*a*c^3 + (175*a^2*
c^3 - 4*(26*a^3*c^3 + 5*(2*a^5*c^3*x + 7*a^4*c^3)*x)*x)*x)*x)*x)

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maple [A]  time = 0.11, size = 229, normalized size = 1.60 \[ \frac {c^{3} a^{7} x^{8}}{7 \sqrt {-a^{2} x^{2}+1}}+\frac {8 c^{3} a^{5} x^{6}}{35 \sqrt {-a^{2} x^{2}+1}}-\frac {54 c^{3} a^{3} x^{4}}{35 \sqrt {-a^{2} x^{2}+1}}+\frac {64 c^{3} a \,x^{2}}{35 \sqrt {-a^{2} x^{2}+1}}-\frac {23 c^{3}}{35 a \sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} a^{6} x^{7}}{2 \sqrt {-a^{2} x^{2}+1}}-\frac {9 c^{3} a^{4} x^{5}}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {3 c^{3} a^{2} x^{3}}{16 \sqrt {-a^{2} x^{2}+1}}+\frac {7 c^{3} x}{16 \sqrt {-a^{2} x^{2}+1}}+\frac {9 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 \sqrt {a^{2}}} \]

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)^3,x)

[Out]

1/7*c^3*a^7*x^8/(-a^2*x^2+1)^(1/2)+8/35*c^3*a^5*x^6/(-a^2*x^2+1)^(1/2)-54/35*c^3*a^3*x^4/(-a^2*x^2+1)^(1/2)+64
/35*c^3*a*x^2/(-a^2*x^2+1)^(1/2)-23/35*c^3/a/(-a^2*x^2+1)^(1/2)+1/2*c^3*a^6*x^7/(-a^2*x^2+1)^(1/2)-9/8*c^3*a^4
*x^5/(-a^2*x^2+1)^(1/2)+3/16*c^3*a^2*x^3/(-a^2*x^2+1)^(1/2)+7/16*c^3*x/(-a^2*x^2+1)^(1/2)+9/16*c^3/(a^2)^(1/2)
*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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maxima [A]  time = 0.44, size = 210, normalized size = 1.47 \[ \frac {a^{7} c^{3} x^{8}}{7 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {a^{6} c^{3} x^{7}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, a^{5} c^{3} x^{6}}{35 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {9 \, a^{4} c^{3} x^{5}}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {54 \, a^{3} c^{3} x^{4}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c^{3} x^{3}}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {64 \, a c^{3} x^{2}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c^{3} x}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {23 \, c^{3}}{35 \, \sqrt {-a^{2} x^{2} + 1} a} \]

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)^3,x, algorithm="maxima")

[Out]

1/7*a^7*c^3*x^8/sqrt(-a^2*x^2 + 1) + 1/2*a^6*c^3*x^7/sqrt(-a^2*x^2 + 1) + 8/35*a^5*c^3*x^6/sqrt(-a^2*x^2 + 1)
- 9/8*a^4*c^3*x^5/sqrt(-a^2*x^2 + 1) - 54/35*a^3*c^3*x^4/sqrt(-a^2*x^2 + 1) + 3/16*a^2*c^3*x^3/sqrt(-a^2*x^2 +
1) + 64/35*a*c^3*x^2/sqrt(-a^2*x^2 + 1) + 7/16*c^3*x/sqrt(-a^2*x^2 + 1) + 9/16*c^3*arcsin(a*x)/a - 23/35*c^3/
(sqrt(-a^2*x^2 + 1)*a)

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mupad [B]  time = 0.05, size = 174, normalized size = 1.22 \[ \frac {7\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {9\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}-\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{35\,a}+\frac {41\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{35}+\frac {5\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{8}-\frac {13\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{2}-\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (9*c^3*asinh(x*(-a^2)^(1/2)))/(16*(-a^2)^(1/2)) - (23*c^3*(1 - a^2*x^2)^(1/
2))/(35*a) + (41*a*c^3*x^2*(1 - a^2*x^2)^(1/2))/35 + (5*a^2*c^3*x^3*(1 - a^2*x^2)^(1/2))/8 - (13*a^3*c^3*x^4*(
1 - a^2*x^2)^(1/2))/35 - (a^4*c^3*x^5*(1 - a^2*x^2)^(1/2))/2 - (a^5*c^3*x^6*(1 - a^2*x^2)^(1/2))/7

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sympy [C]  time = 25.47, size = 632, normalized size = 4.42 \[ - a^{5} c^{3} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{3} \left (\begin {cases} \frac {i a^{2} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{3} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{3} \left (\begin {cases} \frac {i a^{2} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) + 3 a c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {i a^{2} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]

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)**3,x)

[Out]

-a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x*
*2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 3*a**4*c**3*Piecewise((I*
a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) +
I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x*
*2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2
+ 1)) + asin(a*x)/(16*a**5), True)) - 2*a**3*c**3*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x*
*2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**2*c**3*Piecewise((I*a*
*2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acos
h(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) -
x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) + 3*a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-
a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**
2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))

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