\(\int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx\) [337]

Optimal result

Integrand size = 19, antiderivative size = 189 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=\frac {29 c^4 x \sqrt {1-a^2 x^2}}{128 a^3}+\frac {29 c^4 x^3 \sqrt {1-a^2 x^2}}{192 a}-\frac {29}{48} a c^4 x^5 \sqrt {1-a^2 x^2}-\frac {4 c^4 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}+\frac {1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a^4}-\frac {3 c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^4}-\frac {29 c^4 \arcsin (a x)}{128 a^4} \] Output:

29/128*c^4*x*(-a^2*x^2+1)^(1/2)/a^3+29/192*c^4*x^3*(-a^2*x^2+1)^(1/2)/a-29 
/48*a*c^4*x^5*(-a^2*x^2+1)^(1/2)-4/3*c^4*(-a^2*x^2+1)^(3/2)/a^4+1/8*a*c^4* 
x^5*(-a^2*x^2+1)^(3/2)+7/5*c^4*(-a^2*x^2+1)^(5/2)/a^4-3/7*c^4*(-a^2*x^2+1) 
^(7/2)/a^4-29/128*c^4*arcsin(a*x)/a^4
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=-\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (4864-3045 a x+2432 a^2 x^2-2030 a^3 x^3-1536 a^4 x^4+6440 a^5 x^5-5760 a^6 x^6+1680 a^7 x^7\right )-6090 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{13440 a^4} \] Input:

Integrate[E^ArcTanh[a*x]*x^3*(c - a*c*x)^4,x]
 

Output:

-1/13440*(c^4*(Sqrt[1 - a^2*x^2]*(4864 - 3045*a*x + 2432*a^2*x^2 - 2030*a^ 
3*x^3 - 1536*a^4*x^4 + 6440*a^5*x^5 - 5760*a^6*x^6 + 1680*a^7*x^7) - 6090* 
ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a^4
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.947, Rules used = {6678, 27, 541, 25, 2340, 25, 27, 533, 27, 533, 25, 27, 533, 25, 27, 455, 211, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 e^{\text {arctanh}(a x)} (c-a c x)^4 \, dx\)

\(\Big \downarrow \) 6678

\(\displaystyle c \int c^3 x^3 (1-a x)^3 \sqrt {1-a^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \int x^3 (1-a x)^3 \sqrt {1-a^2 x^2}dx\)

\(\Big \downarrow \) 541

\(\displaystyle c^4 \left (\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}-\frac {\int -x^3 \sqrt {1-a^2 x^2} \left (24 x^2 a^4-29 x a^3+8 a^2\right )dx}{8 a^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\int x^3 \sqrt {1-a^2 x^2} \left (24 x^2 a^4-29 x a^3+8 a^2\right )dx}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 2340

\(\displaystyle c^4 \left (\frac {-\frac {\int -a^4 x^3 (152-203 a x) \sqrt {1-a^2 x^2}dx}{7 a^2}-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\frac {\int a^4 x^3 (152-203 a x) \sqrt {1-a^2 x^2}dx}{7 a^2}-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \int x^3 (152-203 a x) \sqrt {1-a^2 x^2}dx-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 533

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {\int -3 a x^2 (203-304 a x) \sqrt {1-a^2 x^2}dx}{6 a^2}+\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\int x^2 (203-304 a x) \sqrt {1-a^2 x^2}dx}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 533

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {\int -a x (608-1015 a x) \sqrt {1-a^2 x^2}dx}{5 a^2}+\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\int a x (608-1015 a x) \sqrt {1-a^2 x^2}dx}{5 a^2}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\int x (608-1015 a x) \sqrt {1-a^2 x^2}dx}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 533

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {\int -a (1015-2432 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}+\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int a (1015-2432 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int (1015-2432 a x) \sqrt {1-a^2 x^2}dx}{4 a}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {1015 \int \sqrt {1-a^2 x^2}dx+\frac {2432 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {1015 \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {2432 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle c^4 \left (\frac {\frac {1}{7} a^2 \left (\frac {203 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {\frac {304 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {1015 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {1015 \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {2432 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}}{2 a}\right )-\frac {24}{7} a^2 x^4 \left (1-a^2 x^2\right )^{3/2}}{8 a^2}+\frac {1}{8} a x^5 \left (1-a^2 x^2\right )^{3/2}\right )\)

Input:

Int[E^ArcTanh[a*x]*x^3*(c - a*c*x)^4,x]
 

Output:

c^4*((a*x^5*(1 - a^2*x^2)^(3/2))/8 + ((-24*a^2*x^4*(1 - a^2*x^2)^(3/2))/7 
+ (a^2*((203*x^3*(1 - a^2*x^2)^(3/2))/(6*a) - ((304*x^2*(1 - a^2*x^2)^(3/2 
))/(5*a) - ((1015*x*(1 - a^2*x^2)^(3/2))/(4*a) - ((2432*(1 - a^2*x^2)^(3/2 
))/(3*a) + 1015*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/(4*a))/(5*a 
))/(2*a)))/7)/(8*a^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

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 455

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

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6678

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {\left (1680 a^{7} x^{7}-5760 x^{6} a^{6}+6440 a^{5} x^{5}-1536 a^{4} x^{4}-2030 a^{3} x^{3}+2432 a^{2} x^{2}-3045 a x +4864\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{13440 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {29 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{128 a^{3} \sqrt {a^{2}}}\)	\(118\)
meijerg	\(\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (144 x^{6} a^{6}+168 a^{4} x^{4}+210 a^{2} x^{2}+315\right ) \sqrt {-a^{2} x^{2}+1}}{576 a^{8}}+\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{64 a^{9}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{4} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{6} a^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{2 a^{4} \sqrt {\pi }}-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{a^{4} \sqrt {\pi }}-\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{4} \sqrt {\pi }}\)	\(410\)
default	\(c^{4} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}+a^{5} \left (-\frac {x^{7} \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {-\frac {7 x^{5} \sqrt {-a^{2} x^{2}+1}}{48 a^{2}}+\frac {7 \left (-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}\right )}{8 a^{2}}}{a^{2}}\right )-3 a \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{2} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )\right )\)	\(522\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/13440*(1680*a^7*x^7-5760*a^6*x^6+6440*a^5*x^5-1536*a^4*x^4-2030*a^3*x^3+ 
2432*a^2*x^2-3045*a*x+4864)*(a^2*x^2-1)/a^4/(-a^2*x^2+1)^(1/2)*c^4-29/128/ 
a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c^4
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=\frac {6090 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (1680 \, a^{7} c^{4} x^{7} - 5760 \, a^{6} c^{4} x^{6} + 6440 \, a^{5} c^{4} x^{5} - 1536 \, a^{4} c^{4} x^{4} - 2030 \, a^{3} c^{4} x^{3} + 2432 \, a^{2} c^{4} x^{2} - 3045 \, a c^{4} x + 4864 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{13440 \, a^{4}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x, algorithm="fricas 
")
 

Output:

1/13440*(6090*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (1680*a^7*c^4*x 
^7 - 5760*a^6*c^4*x^6 + 6440*a^5*c^4*x^5 - 1536*a^4*c^4*x^4 - 2030*a^3*c^4 
*x^3 + 2432*a^2*c^4*x^2 - 3045*a*c^4*x + 4864*c^4)*sqrt(-a^2*x^2 + 1))/a^4
 

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{3} c^{4} x^{7}}{8} + \frac {3 a^{2} c^{4} x^{6}}{7} - \frac {23 a c^{4} x^{5}}{48} + \frac {4 c^{4} x^{4}}{35} + \frac {29 c^{4} x^{3}}{192 a} - \frac {19 c^{4} x^{2}}{105 a^{2}} + \frac {29 c^{4} x}{128 a^{3}} - \frac {38 c^{4}}{105 a^{4}}\right ) - \frac {29 c^{4} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{128 a^{3} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{5} c^{4} x^{9}}{9} - \frac {3 a^{4} c^{4} x^{8}}{8} + \frac {2 a^{3} c^{4} x^{7}}{7} + \frac {a^{2} c^{4} x^{6}}{3} - \frac {3 a c^{4} x^{5}}{5} + \frac {c^{4} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a*c*x+c)**4,x)
 

Output:

Piecewise((sqrt(-a**2*x**2 + 1)*(-a**3*c**4*x**7/8 + 3*a**2*c**4*x**6/7 - 
23*a*c**4*x**5/48 + 4*c**4*x**4/35 + 29*c**4*x**3/(192*a) - 19*c**4*x**2/( 
105*a**2) + 29*c**4*x/(128*a**3) - 38*c**4/(105*a**4)) - 29*c**4*log(-2*a* 
*2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(128*a**3*sqrt(-a**2)), Ne(a**2 
, 0)), (a**5*c**4*x**9/9 - 3*a**4*c**4*x**8/8 + 2*a**3*c**4*x**7/7 + a**2* 
c**4*x**6/3 - 3*a*c**4*x**5/5 + c**4*x**4/4, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.99 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=-\frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{7} + \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{6} - \frac {23}{48} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{5} + \frac {4}{35} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{4} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{3}}{192 \, a} - \frac {19 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2}}{105 \, a^{2}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{128 \, a^{3}} - \frac {29 \, c^{4} \arcsin \left (a x\right )}{128 \, a^{4}} - \frac {38 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{105 \, a^{4}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x, algorithm="maxima 
")
 

Output:

-1/8*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^7 + 3/7*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^6 - 
 23/48*sqrt(-a^2*x^2 + 1)*a*c^4*x^5 + 4/35*sqrt(-a^2*x^2 + 1)*c^4*x^4 + 29 
/192*sqrt(-a^2*x^2 + 1)*c^4*x^3/a - 19/105*sqrt(-a^2*x^2 + 1)*c^4*x^2/a^2 
+ 29/128*sqrt(-a^2*x^2 + 1)*c^4*x/a^3 - 29/128*c^4*arcsin(a*x)/a^4 - 38/10 
5*sqrt(-a^2*x^2 + 1)*c^4/a^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.62 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=-\frac {29 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{128 \, a^{3} {\left | a \right |}} - \frac {1}{13440} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (\frac {1216 \, c^{4}}{a^{2}} - {\left (\frac {1015 \, c^{4}}{a} + 4 \, {\left (192 \, c^{4} - 5 \, {\left (161 \, a c^{4} + 6 \, {\left (7 \, a^{3} c^{4} x - 24 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac {3045 \, c^{4}}{a^{3}}\right )} x + \frac {4864 \, c^{4}}{a^{4}}\right )} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x, algorithm="giac")
 

Output:

-29/128*c^4*arcsin(a*x)*sgn(a)/(a^3*abs(a)) - 1/13440*sqrt(-a^2*x^2 + 1)*( 
(2*(1216*c^4/a^2 - (1015*c^4/a + 4*(192*c^4 - 5*(161*a*c^4 + 6*(7*a^3*c^4* 
x - 24*a^2*c^4)*x)*x)*x)*x)*x - 3045*c^4/a^3)*x + 4864*c^4/a^4)
 

Mupad [B] (verification not implemented)

Time = 13.98 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.06 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=\frac {4\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {38\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4}+\frac {29\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128\,a^3}-\frac {23\,a\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}-\frac {29\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,a^3\,\sqrt {-a^2}}+\frac {29\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192\,a}-\frac {19\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{105\,a^2}+\frac {3\,a^2\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{7}-\frac {a^3\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8} \] Input:

int((x^3*(c - a*c*x)^4*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
 

Output:

(4*c^4*x^4*(1 - a^2*x^2)^(1/2))/35 - (38*c^4*(1 - a^2*x^2)^(1/2))/(105*a^4 
) + (29*c^4*x*(1 - a^2*x^2)^(1/2))/(128*a^3) - (23*a*c^4*x^5*(1 - a^2*x^2) 
^(1/2))/48 - (29*c^4*asinh(x*(-a^2)^(1/2)))/(128*a^3*(-a^2)^(1/2)) + (29*c 
^4*x^3*(1 - a^2*x^2)^(1/2))/(192*a) - (19*c^4*x^2*(1 - a^2*x^2)^(1/2))/(10 
5*a^2) + (3*a^2*c^4*x^6*(1 - a^2*x^2)^(1/2))/7 - (a^3*c^4*x^7*(1 - a^2*x^2 
)^(1/2))/8
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84 \[ \int e^{\text {arctanh}(a x)} x^3 (c-a c x)^4 \, dx=\frac {c^{4} \left (-3045 \mathit {asin} \left (a x \right )-1680 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+5760 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-6440 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+1536 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+2030 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-2432 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+3045 \sqrt {-a^{2} x^{2}+1}\, a x -4864 \sqrt {-a^{2} x^{2}+1}+4864\right )}{13440 a^{4}} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x)
 

Output:

(c**4*( - 3045*asin(a*x) - 1680*sqrt( - a**2*x**2 + 1)*a**7*x**7 + 5760*sq 
rt( - a**2*x**2 + 1)*a**6*x**6 - 6440*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 1 
536*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 2030*sqrt( - a**2*x**2 + 1)*a**3*x* 
*3 - 2432*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 3045*sqrt( - a**2*x**2 + 1)*a 
*x - 4864*sqrt( - a**2*x**2 + 1) + 4864))/(13440*a**4)