3.31 \(\int x^3 (d+c d x)^4 (a+b \tanh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=224 \[ \frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {769 b d^4 \log (1-c x)}{560 c^4}-\frac {b d^4 \log (c x+1)}{560 c^4}+\frac {1}{56} b c^3 d^4 x^7+\frac {11 b d^4 x}{8 c^3}+\frac {2}{21} b c^2 d^4 x^6+\frac {24 b d^4 x^2}{35 c^2}+\frac {9}{40} b c d^4 x^5+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4 \]

[Out]

11/8*b*d^4*x/c^3+24/35*b*d^4*x^2/c^2+11/24*b*d^4*x^3/c+12/35*b*d^4*x^4+9/40*b*c*d^4*x^5+2/21*b*c^2*d^4*x^6+1/5
6*b*c^3*d^4*x^7+1/4*d^4*x^4*(a+b*arctanh(c*x))+4/5*c*d^4*x^5*(a+b*arctanh(c*x))+c^2*d^4*x^6*(a+b*arctanh(c*x))
+4/7*c^3*d^4*x^7*(a+b*arctanh(c*x))+1/8*c^4*d^4*x^8*(a+b*arctanh(c*x))+769/560*b*d^4*ln(-c*x+1)/c^4-1/560*b*d^
4*ln(c*x+1)/c^4

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Rubi [A]  time = 0.21, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {43, 5936, 12, 1802, 633, 31} \[ \frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{56} b c^3 d^4 x^7+\frac {2}{21} b c^2 d^4 x^6+\frac {24 b d^4 x^2}{35 c^2}+\frac {11 b d^4 x}{8 c^3}+\frac {769 b d^4 \log (1-c x)}{560 c^4}-\frac {b d^4 \log (c x+1)}{560 c^4}+\frac {9}{40} b c d^4 x^5+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]

[Out]

(11*b*d^4*x)/(8*c^3) + (24*b*d^4*x^2)/(35*c^2) + (11*b*d^4*x^3)/(24*c) + (12*b*d^4*x^4)/35 + (9*b*c*d^4*x^5)/4
0 + (2*b*c^2*d^4*x^6)/21 + (b*c^3*d^4*x^7)/56 + (d^4*x^4*(a + b*ArcTanh[c*x]))/4 + (4*c*d^4*x^5*(a + b*ArcTanh
[c*x]))/5 + c^2*d^4*x^6*(a + b*ArcTanh[c*x]) + (4*c^3*d^4*x^7*(a + b*ArcTanh[c*x]))/7 + (c^4*d^4*x^8*(a + b*Ar
cTanh[c*x]))/8 + (769*b*d^4*Log[1 - c*x])/(560*c^4) - (b*d^4*Log[1 + c*x])/(560*c^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 5936

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 - c^2*
x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q,
0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rubi steps

\begin {align*} \int x^3 (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac {d^4 x^4 \left (70+224 c x+280 c^2 x^2+160 c^3 x^3+35 c^4 x^4\right )}{280 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{280} \left (b c d^4\right ) \int \frac {x^4 \left (70+224 c x+280 c^2 x^2+160 c^3 x^3+35 c^4 x^4\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{280} \left (b c d^4\right ) \int \left (-\frac {385}{c^4}-\frac {384 x}{c^3}-\frac {385 x^2}{c^2}-\frac {384 x^3}{c}-315 x^4-160 c x^5-35 c^2 x^6+\frac {385+384 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 b d^4 x^2}{35 c^2}+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} b c^2 d^4 x^6+\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^4\right ) \int \frac {385+384 c x}{1-c^2 x^2} \, dx}{280 c^3}\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 b d^4 x^2}{35 c^2}+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} b c^2 d^4 x^6+\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac {\left (b d^4\right ) \int \frac {1}{-c-c^2 x} \, dx}{560 c^2}-\frac {\left (769 b d^4\right ) \int \frac {1}{c-c^2 x} \, dx}{560 c^2}\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 b d^4 x^2}{35 c^2}+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} b c^2 d^4 x^6+\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac {769 b d^4 \log (1-c x)}{560 c^4}-\frac {b d^4 \log (1+c x)}{560 c^4}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 177, normalized size = 0.79 \[ \frac {d^4 \left (210 a c^8 x^8+960 a c^7 x^7+1680 a c^6 x^6+1344 a c^5 x^5+420 a c^4 x^4+30 b c^7 x^7+160 b c^6 x^6+378 b c^5 x^5+576 b c^4 x^4+770 b c^3 x^3+1152 b c^2 x^2+6 b c^4 x^4 \left (35 c^4 x^4+160 c^3 x^3+280 c^2 x^2+224 c x+70\right ) \tanh ^{-1}(c x)+2310 b c x+2307 b \log (1-c x)-3 b \log (c x+1)\right )}{1680 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]

[Out]

(d^4*(2310*b*c*x + 1152*b*c^2*x^2 + 770*b*c^3*x^3 + 420*a*c^4*x^4 + 576*b*c^4*x^4 + 1344*a*c^5*x^5 + 378*b*c^5
*x^5 + 1680*a*c^6*x^6 + 160*b*c^6*x^6 + 960*a*c^7*x^7 + 30*b*c^7*x^7 + 210*a*c^8*x^8 + 6*b*c^4*x^4*(70 + 224*c
*x + 280*c^2*x^2 + 160*c^3*x^3 + 35*c^4*x^4)*ArcTanh[c*x] + 2307*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(1680*c^4
)

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fricas [A]  time = 0.55, size = 222, normalized size = 0.99 \[ \frac {210 \, a c^{8} d^{4} x^{8} + 30 \, {\left (32 \, a + b\right )} c^{7} d^{4} x^{7} + 80 \, {\left (21 \, a + 2 \, b\right )} c^{6} d^{4} x^{6} + 42 \, {\left (32 \, a + 9 \, b\right )} c^{5} d^{4} x^{5} + 12 \, {\left (35 \, a + 48 \, b\right )} c^{4} d^{4} x^{4} + 770 \, b c^{3} d^{4} x^{3} + 1152 \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 3 \, b d^{4} \log \left (c x + 1\right ) + 2307 \, b d^{4} \log \left (c x - 1\right ) + 3 \, {\left (35 \, b c^{8} d^{4} x^{8} + 160 \, b c^{7} d^{4} x^{7} + 280 \, b c^{6} d^{4} x^{6} + 224 \, b c^{5} d^{4} x^{5} + 70 \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{1680 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="fricas")

[Out]

1/1680*(210*a*c^8*d^4*x^8 + 30*(32*a + b)*c^7*d^4*x^7 + 80*(21*a + 2*b)*c^6*d^4*x^6 + 42*(32*a + 9*b)*c^5*d^4*
x^5 + 12*(35*a + 48*b)*c^4*d^4*x^4 + 770*b*c^3*d^4*x^3 + 1152*b*c^2*d^4*x^2 + 2310*b*c*d^4*x - 3*b*d^4*log(c*x
+ 1) + 2307*b*d^4*log(c*x - 1) + 3*(35*b*c^8*d^4*x^8 + 160*b*c^7*d^4*x^7 + 280*b*c^6*d^4*x^6 + 224*b*c^5*d^4*
x^5 + 70*b*c^4*d^4*x^4)*log(-(c*x + 1)/(c*x - 1)))/c^4

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giac [B]  time = 0.34, size = 817, normalized size = 3.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="giac")

[Out]

-4/105*c*(36*b*d^4*log(-(c*x + 1)/(c*x - 1) + 1)/c^5 - 12*(35*(c*x + 1)^7*b*d^4/(c*x - 1)^7 - 70*(c*x + 1)^6*b
*d^4/(c*x - 1)^6 + 175*(c*x + 1)^5*b*d^4/(c*x - 1)^5 - 210*(c*x + 1)^4*b*d^4/(c*x - 1)^4 + 168*(c*x + 1)^3*b*d
^4/(c*x - 1)^3 - 84*(c*x + 1)^2*b*d^4/(c*x - 1)^2 + 24*(c*x + 1)*b*d^4/(c*x - 1) - 3*b*d^4)*log(-(c*x + 1)/(c*
x - 1))/((c*x + 1)^8*c^5/(c*x - 1)^8 - 8*(c*x + 1)^7*c^5/(c*x - 1)^7 + 28*(c*x + 1)^6*c^5/(c*x - 1)^6 - 56*(c*
x + 1)^5*c^5/(c*x - 1)^5 + 70*(c*x + 1)^4*c^5/(c*x - 1)^4 - 56*(c*x + 1)^3*c^5/(c*x - 1)^3 + 28*(c*x + 1)^2*c^
5/(c*x - 1)^2 - 8*(c*x + 1)*c^5/(c*x - 1) + c^5) - 36*b*d^4*log(-(c*x + 1)/(c*x - 1))/c^5 - (840*(c*x + 1)^7*a
*d^4/(c*x - 1)^7 - 1680*(c*x + 1)^6*a*d^4/(c*x - 1)^6 + 4200*(c*x + 1)^5*a*d^4/(c*x - 1)^5 - 5040*(c*x + 1)^4*
a*d^4/(c*x - 1)^4 + 4032*(c*x + 1)^3*a*d^4/(c*x - 1)^3 - 2016*(c*x + 1)^2*a*d^4/(c*x - 1)^2 + 576*(c*x + 1)*a*
d^4/(c*x - 1) - 72*a*d^4 + 384*(c*x + 1)^7*b*d^4/(c*x - 1)^7 - 1830*(c*x + 1)^6*b*d^4/(c*x - 1)^6 + 4304*(c*x
+ 1)^5*b*d^4/(c*x - 1)^5 - 6031*(c*x + 1)^4*b*d^4/(c*x - 1)^4 + 5228*(c*x + 1)^3*b*d^4/(c*x - 1)^3 - 2782*(c*x
+ 1)^2*b*d^4/(c*x - 1)^2 + 836*(c*x + 1)*b*d^4/(c*x - 1) - 109*b*d^4)/((c*x + 1)^8*c^5/(c*x - 1)^8 - 8*(c*x +
1)^7*c^5/(c*x - 1)^7 + 28*(c*x + 1)^6*c^5/(c*x - 1)^6 - 56*(c*x + 1)^5*c^5/(c*x - 1)^5 + 70*(c*x + 1)^4*c^5/(
c*x - 1)^4 - 56*(c*x + 1)^3*c^5/(c*x - 1)^3 + 28*(c*x + 1)^2*c^5/(c*x - 1)^2 - 8*(c*x + 1)*c^5/(c*x - 1) + c^5
))

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maple [A]  time = 0.03, size = 237, normalized size = 1.06 \[ \frac {c^{4} d^{4} a \,x^{8}}{8}+\frac {4 c^{3} d^{4} a \,x^{7}}{7}+c^{2} d^{4} a \,x^{6}+\frac {4 c \,d^{4} a \,x^{5}}{5}+\frac {d^{4} a \,x^{4}}{4}+\frac {c^{4} d^{4} b \arctanh \left (c x \right ) x^{8}}{8}+\frac {4 c^{3} d^{4} b \arctanh \left (c x \right ) x^{7}}{7}+c^{2} d^{4} b \arctanh \left (c x \right ) x^{6}+\frac {4 c \,d^{4} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {d^{4} b \arctanh \left (c x \right ) x^{4}}{4}+\frac {b \,c^{3} d^{4} x^{7}}{56}+\frac {2 b \,c^{2} d^{4} x^{6}}{21}+\frac {9 b c \,d^{4} x^{5}}{40}+\frac {12 b \,d^{4} x^{4}}{35}+\frac {11 b \,d^{4} x^{3}}{24 c}+\frac {24 b \,d^{4} x^{2}}{35 c^{2}}+\frac {11 b \,d^{4} x}{8 c^{3}}+\frac {769 d^{4} b \ln \left (c x -1\right )}{560 c^{4}}-\frac {b \,d^{4} \ln \left (c x +1\right )}{560 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x)

[Out]

1/8*c^4*d^4*a*x^8+4/7*c^3*d^4*a*x^7+c^2*d^4*a*x^6+4/5*c*d^4*a*x^5+1/4*d^4*a*x^4+1/8*c^4*d^4*b*arctanh(c*x)*x^8
+4/7*c^3*d^4*b*arctanh(c*x)*x^7+c^2*d^4*b*arctanh(c*x)*x^6+4/5*c*d^4*b*arctanh(c*x)*x^5+1/4*d^4*b*arctanh(c*x)
*x^4+1/56*b*c^3*d^4*x^7+2/21*b*c^2*d^4*x^6+9/40*b*c*d^4*x^5+12/35*b*d^4*x^4+11/24*b*d^4*x^3/c+24/35*b*d^4*x^2/
c^2+11/8*b*d^4*x/c^3+769/560/c^4*d^4*b*ln(c*x-1)-1/560*b*d^4*ln(c*x+1)/c^4

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maxima [A]  time = 0.33, size = 373, normalized size = 1.67 \[ \frac {1}{8} \, a c^{4} d^{4} x^{8} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + a c^{2} d^{4} x^{6} + \frac {4}{5} \, a c d^{4} x^{5} + \frac {1}{1680} \, {\left (210 \, x^{8} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (15 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} + 105 \, x\right )}}{c^{8}} - \frac {105 \, \log \left (c x + 1\right )}{c^{9}} + \frac {105 \, \log \left (c x - 1\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} + \frac {1}{21} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac {1}{4} \, a d^{4} x^{4} + \frac {1}{30} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac {1}{5} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="maxima")

[Out]

1/8*a*c^4*d^4*x^8 + 4/7*a*c^3*d^4*x^7 + a*c^2*d^4*x^6 + 4/5*a*c*d^4*x^5 + 1/1680*(210*x^8*arctanh(c*x) + c*(2*
(15*c^6*x^7 + 21*c^4*x^5 + 35*c^2*x^3 + 105*x)/c^8 - 105*log(c*x + 1)/c^9 + 105*log(c*x - 1)/c^9))*b*c^4*d^4 +
1/21*(12*x^7*arctanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^8))*b*c^3*d^4 + 1/4
*a*d^4*x^4 + 1/30*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*lo
g(c*x - 1)/c^7))*b*c^2*d^4 + 1/5*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*b*c
*d^4 + 1/24*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*b*d^4

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mupad [B]  time = 1.84, size = 337, normalized size = 1.50 \[ \frac {a\,d^4\,x^4}{4}+\frac {12\,b\,d^4\,x^4}{35}+a\,c^2\,d^4\,x^6+\frac {4\,a\,c^3\,d^4\,x^7}{7}+\frac {a\,c^4\,d^4\,x^8}{8}+\frac {11\,b\,d^4\,x^3}{24\,c}+\frac {24\,b\,d^4\,x^2}{35\,c^2}+\frac {2\,b\,c^2\,d^4\,x^6}{21}+\frac {b\,c^3\,d^4\,x^7}{56}+\frac {769\,b\,d^4\,\ln \left (c\,x-1\right )}{560\,c^4}-\frac {b\,d^4\,\ln \left (c\,x+1\right )}{560\,c^4}+\frac {b\,d^4\,x^4\,\ln \left (c\,x+1\right )}{8}-\frac {b\,d^4\,x^4\,\ln \left (1-c\,x\right )}{8}+\frac {4\,a\,c\,d^4\,x^5}{5}+\frac {11\,b\,d^4\,x}{8\,c^3}+\frac {9\,b\,c\,d^4\,x^5}{40}+\frac {b\,c^2\,d^4\,x^6\,\ln \left (c\,x+1\right )}{2}-\frac {b\,c^2\,d^4\,x^6\,\ln \left (1-c\,x\right )}{2}+\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (c\,x+1\right )}{7}-\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (1-c\,x\right )}{7}+\frac {b\,c^4\,d^4\,x^8\,\ln \left (c\,x+1\right )}{16}-\frac {b\,c^4\,d^4\,x^8\,\ln \left (1-c\,x\right )}{16}+\frac {2\,b\,c\,d^4\,x^5\,\ln \left (c\,x+1\right )}{5}-\frac {2\,b\,c\,d^4\,x^5\,\ln \left (1-c\,x\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c*x))*(d + c*d*x)^4,x)

[Out]

(a*d^4*x^4)/4 + (12*b*d^4*x^4)/35 + a*c^2*d^4*x^6 + (4*a*c^3*d^4*x^7)/7 + (a*c^4*d^4*x^8)/8 + (11*b*d^4*x^3)/(
24*c) + (24*b*d^4*x^2)/(35*c^2) + (2*b*c^2*d^4*x^6)/21 + (b*c^3*d^4*x^7)/56 + (769*b*d^4*log(c*x - 1))/(560*c^
4) - (b*d^4*log(c*x + 1))/(560*c^4) + (b*d^4*x^4*log(c*x + 1))/8 - (b*d^4*x^4*log(1 - c*x))/8 + (4*a*c*d^4*x^5
)/5 + (11*b*d^4*x)/(8*c^3) + (9*b*c*d^4*x^5)/40 + (b*c^2*d^4*x^6*log(c*x + 1))/2 - (b*c^2*d^4*x^6*log(1 - c*x)
)/2 + (2*b*c^3*d^4*x^7*log(c*x + 1))/7 - (2*b*c^3*d^4*x^7*log(1 - c*x))/7 + (b*c^4*d^4*x^8*log(c*x + 1))/16 -
(b*c^4*d^4*x^8*log(1 - c*x))/16 + (2*b*c*d^4*x^5*log(c*x + 1))/5 - (2*b*c*d^4*x^5*log(1 - c*x))/5

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sympy [A]  time = 3.76, size = 294, normalized size = 1.31 \[ \begin {cases} \frac {a c^{4} d^{4} x^{8}}{8} + \frac {4 a c^{3} d^{4} x^{7}}{7} + a c^{2} d^{4} x^{6} + \frac {4 a c d^{4} x^{5}}{5} + \frac {a d^{4} x^{4}}{4} + \frac {b c^{4} d^{4} x^{8} \operatorname {atanh}{\left (c x \right )}}{8} + \frac {4 b c^{3} d^{4} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {b c^{3} d^{4} x^{7}}{56} + b c^{2} d^{4} x^{6} \operatorname {atanh}{\left (c x \right )} + \frac {2 b c^{2} d^{4} x^{6}}{21} + \frac {4 b c d^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {9 b c d^{4} x^{5}}{40} + \frac {b d^{4} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {12 b d^{4} x^{4}}{35} + \frac {11 b d^{4} x^{3}}{24 c} + \frac {24 b d^{4} x^{2}}{35 c^{2}} + \frac {11 b d^{4} x}{8 c^{3}} + \frac {48 b d^{4} \log {\left (x - \frac {1}{c} \right )}}{35 c^{4}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{280 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{4} x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*d*x+d)**4*(a+b*atanh(c*x)),x)

[Out]

Piecewise((a*c**4*d**4*x**8/8 + 4*a*c**3*d**4*x**7/7 + a*c**2*d**4*x**6 + 4*a*c*d**4*x**5/5 + a*d**4*x**4/4 +
b*c**4*d**4*x**8*atanh(c*x)/8 + 4*b*c**3*d**4*x**7*atanh(c*x)/7 + b*c**3*d**4*x**7/56 + b*c**2*d**4*x**6*atanh
(c*x) + 2*b*c**2*d**4*x**6/21 + 4*b*c*d**4*x**5*atanh(c*x)/5 + 9*b*c*d**4*x**5/40 + b*d**4*x**4*atanh(c*x)/4 +
12*b*d**4*x**4/35 + 11*b*d**4*x**3/(24*c) + 24*b*d**4*x**2/(35*c**2) + 11*b*d**4*x/(8*c**3) + 48*b*d**4*log(x
- 1/c)/(35*c**4) - b*d**4*atanh(c*x)/(280*c**4), Ne(c, 0)), (a*d**4*x**4/4, True))

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