3.1.23 \(\int (d+e x)^3 (a+b \tanh ^{-1}(c x^2)) \, dx\) [23]

Optimal. Leaf size=182 \[ \frac {2 b d e^2 x}{c}+\frac {b e^3 x^2}{4 c}+\frac {b d \left (c d^2-e^2\right ) \text {ArcTan}\left (\sqrt {c} x\right )}{c^{3/2}}-\frac {b d \left (c d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {c} x\right )}{c^{3/2}}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{4 e}+\frac {b \left (c^2 d^4+6 c d^2 e^2+e^4\right ) \log \left (1-c x^2\right )}{8 c^2 e}-\frac {b \left (c^2 d^4-6 c d^2 e^2+e^4\right ) \log \left (1+c x^2\right )}{8 c^2 e} \]

[Out]

2*b*d*e^2*x/c+1/4*b*e^3*x^2/c+b*d*(c*d^2-e^2)*arctan(x*c^(1/2))/c^(3/2)+1/4*(e*x+d)^4*(a+b*arctanh(c*x^2))/e-b
*d*(c*d^2+e^2)*arctanh(x*c^(1/2))/c^(3/2)+1/8*b*(c^2*d^4+6*c*d^2*e^2+e^4)*ln(-c*x^2+1)/c^2/e-1/8*b*(c^2*d^4-6*
c*d^2*e^2+e^4)*ln(c*x^2+1)/c^2/e

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Rubi [A]
time = 0.20, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6071, 1847, 1294, 1181, 211, 214, 1833, 1824, 647, 31} \begin {gather*} \frac {(d+e x)^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{4 e}+\frac {b d \text {ArcTan}\left (\sqrt {c} x\right ) \left (c d^2-e^2\right )}{c^{3/2}}-\frac {b d \left (c d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {c} x\right )}{c^{3/2}}+\frac {b \left (c^2 d^4+6 c d^2 e^2+e^4\right ) \log \left (1-c x^2\right )}{8 c^2 e}-\frac {b \left (c^2 d^4-6 c d^2 e^2+e^4\right ) \log \left (c x^2+1\right )}{8 c^2 e}+\frac {2 b d e^2 x}{c}+\frac {b e^3 x^2}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*ArcTanh[c*x^2]),x]

[Out]

(2*b*d*e^2*x)/c + (b*e^3*x^2)/(4*c) + (b*d*(c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/c^(3/2) - (b*d*(c*d^2 + e^2)*ArcTa
nh[Sqrt[c]*x])/c^(3/2) + ((d + e*x)^4*(a + b*ArcTanh[c*x^2]))/(4*e) + (b*(c^2*d^4 + 6*c*d^2*e^2 + e^4)*Log[1 -
c*x^2])/(8*c^2*e) - (b*(c^2*d^4 - 6*c*d^2*e^2 + e^4)*Log[1 + c*x^2])/(8*c^2*e)

Rule 31

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

Rule 211

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

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 647

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 1181

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]

Rule 1294

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*(
(a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1824

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

Rule 1833

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]

Rule 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 6071

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(
(a + b*ArcTanh[c*x^n])/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 - c^2*
x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (d+e x)^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\int \left (a (d+e x)^3+b (d+e x)^3 \tanh ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac {a (d+e x)^4}{4 e}+b \int (d+e x)^3 \tanh ^{-1}\left (c x^2\right ) \, dx\\ &=\frac {a (d+e x)^4}{4 e}+b \int \left (d^3 \tanh ^{-1}\left (c x^2\right )+3 d^2 e x \tanh ^{-1}\left (c x^2\right )+3 d e^2 x^2 \tanh ^{-1}\left (c x^2\right )+e^3 x^3 \tanh ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac {a (d+e x)^4}{4 e}+\left (b d^3\right ) \int \tanh ^{-1}\left (c x^2\right ) \, dx+\left (3 b d^2 e\right ) \int x \tanh ^{-1}\left (c x^2\right ) \, dx+\left (3 b d e^2\right ) \int x^2 \tanh ^{-1}\left (c x^2\right ) \, dx+\left (b e^3\right ) \int x^3 \tanh ^{-1}\left (c x^2\right ) \, dx\\ &=\frac {a (d+e x)^4}{4 e}+b d^3 x \tanh ^{-1}\left (c x^2\right )+\frac {3}{2} b d^2 e x^2 \tanh ^{-1}\left (c x^2\right )+b d e^2 x^3 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b e^3 x^4 \tanh ^{-1}\left (c x^2\right )-\left (2 b c d^3\right ) \int \frac {x^2}{1-c^2 x^4} \, dx-\left (3 b c d^2 e\right ) \int \frac {x^3}{1-c^2 x^4} \, dx-\left (2 b c d e^2\right ) \int \frac {x^4}{1-c^2 x^4} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^5}{1-c^2 x^4} \, dx\\ &=\frac {2 b d e^2 x}{c}+\frac {a (d+e x)^4}{4 e}+b d^3 x \tanh ^{-1}\left (c x^2\right )+\frac {3}{2} b d^2 e x^2 \tanh ^{-1}\left (c x^2\right )+b d e^2 x^3 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b e^3 x^4 \tanh ^{-1}\left (c x^2\right )+\frac {3 b d^2 e \log \left (1-c^2 x^4\right )}{4 c}-\left (b d^3\right ) \int \frac {1}{1-c x^2} \, dx+\left (b d^3\right ) \int \frac {1}{1+c x^2} \, dx-\frac {\left (2 b d e^2\right ) \int \frac {1}{1-c^2 x^4} \, dx}{c}-\frac {1}{4} \left (b c e^3\right ) \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=\frac {2 b d e^2 x}{c}+\frac {b e^3 x^2}{4 c}+\frac {a (d+e x)^4}{4 e}+\frac {b d^3 \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b d^3 \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+b d^3 x \tanh ^{-1}\left (c x^2\right )+\frac {3}{2} b d^2 e x^2 \tanh ^{-1}\left (c x^2\right )+b d e^2 x^3 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b e^3 x^4 \tanh ^{-1}\left (c x^2\right )+\frac {3 b d^2 e \log \left (1-c^2 x^4\right )}{4 c}-\frac {\left (b d e^2\right ) \int \frac {1}{1-c x^2} \, dx}{c}-\frac {\left (b d e^2\right ) \int \frac {1}{1+c x^2} \, dx}{c}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac {2 b d e^2 x}{c}+\frac {b e^3 x^2}{4 c}+\frac {a (d+e x)^4}{4 e}+\frac {b d^3 \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b d e^2 \tan ^{-1}\left (\sqrt {c} x\right )}{c^{3/2}}-\frac {b d^3 \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b d e^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{c^{3/2}}-\frac {b e^3 \tanh ^{-1}\left (c x^2\right )}{4 c^2}+b d^3 x \tanh ^{-1}\left (c x^2\right )+\frac {3}{2} b d^2 e x^2 \tanh ^{-1}\left (c x^2\right )+b d e^2 x^3 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b e^3 x^4 \tanh ^{-1}\left (c x^2\right )+\frac {3 b d^2 e \log \left (1-c^2 x^4\right )}{4 c}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 254, normalized size = 1.40 \begin {gather*} \frac {1}{8} \left (\frac {8 d \left (a c d^2+2 b e^2\right ) x}{c}+\frac {2 e \left (6 a c d^2+b e^2\right ) x^2}{c}+8 a d e^2 x^3+2 a e^3 x^4+\frac {8 b d \left (c d^2-e^2\right ) \text {ArcTan}\left (\sqrt {c} x\right )}{c^{3/2}}+2 b x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \tanh ^{-1}\left (c x^2\right )+\frac {b \left (4 c^{3/2} d^3+4 \sqrt {c} d e^2+e^3\right ) \log \left (1-\sqrt {c} x\right )}{c^2}+\frac {b \left (-4 c^2 d^3-4 c d e^2+\sqrt {c} e^3\right ) \log \left (1+\sqrt {c} x\right )}{c^{5/2}}-\frac {b e^3 \log \left (1+c x^2\right )}{c^2}+\frac {6 b d^2 e \log \left (1-c^2 x^4\right )}{c}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*ArcTanh[c*x^2]),x]

[Out]

((8*d*(a*c*d^2 + 2*b*e^2)*x)/c + (2*e*(6*a*c*d^2 + b*e^2)*x^2)/c + 8*a*d*e^2*x^3 + 2*a*e^3*x^4 + (8*b*d*(c*d^2
- e^2)*ArcTan[Sqrt[c]*x])/c^(3/2) + 2*b*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x^2] + (b*(4*
c^(3/2)*d^3 + 4*Sqrt[c]*d*e^2 + e^3)*Log[1 - Sqrt[c]*x])/c^2 + (b*(-4*c^2*d^3 - 4*c*d*e^2 + Sqrt[c]*e^3)*Log[1
+ Sqrt[c]*x])/c^(5/2) - (b*e^3*Log[1 + c*x^2])/c^2 + (6*b*d^2*e*Log[1 - c^2*x^4])/c)/8

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Maple [A]
time = 0.35, size = 276, normalized size = 1.52

method	result	size
default	\(\frac {\left (e x +d \right )^{4} a}{4 e}+\frac {b \,e^{3} \arctanh \left (c \,x^{2}\right ) x^{4}}{4}+b \,e^{2} \arctanh \left (c \,x^{2}\right ) x^{3} d +\frac {3 b e \arctanh \left (c \,x^{2}\right ) x^{2} d^{2}}{2}+b \arctanh \left (c \,x^{2}\right ) x \,d^{3}+\frac {b \arctanh \left (c \,x^{2}\right ) d^{4}}{4 e}+\frac {b \,e^{3} x^{2}}{4 c}+\frac {2 b d \,e^{2} x}{c}+\frac {b \ln \left (c \,x^{2}-1\right ) d^{4}}{8 e}+\frac {3 b e \ln \left (c \,x^{2}-1\right ) d^{2}}{4 c}+\frac {b \,e^{3} \ln \left (c \,x^{2}-1\right )}{8 c^{2}}-\frac {b \arctanh \left (x \sqrt {c}\right ) d^{3}}{\sqrt {c}}-\frac {b \,e^{2} \arctanh \left (x \sqrt {c}\right ) d}{c^{\frac {3}{2}}}-\frac {b \ln \left (c \,x^{2}+1\right ) d^{4}}{8 e}+\frac {3 b e \ln \left (c \,x^{2}+1\right ) d^{2}}{4 c}-\frac {b \,e^{3} \ln \left (c \,x^{2}+1\right )}{8 c^{2}}+\frac {b \arctan \left (x \sqrt {c}\right ) d^{3}}{\sqrt {c}}-\frac {b \,e^{2} \arctan \left (x \sqrt {c}\right ) d}{c^{\frac {3}{2}}}\)	\(276\)
risch	\(\text {Expression too large to display}\)	\(4746\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*arctanh(c*x^2)),x,method=_RETURNVERBOSE)

[Out]

1/4*(e*x+d)^4*a/e+1/4*b*e^3*arctanh(c*x^2)*x^4+b*e^2*arctanh(c*x^2)*x^3*d+3/2*b*e*arctanh(c*x^2)*x^2*d^2+b*arc
tanh(c*x^2)*x*d^3+1/4*b/e*arctanh(c*x^2)*d^4+1/4*b*e^3*x^2/c+2*b*d*e^2*x/c+1/8*b/e*ln(c*x^2-1)*d^4+3/4*b*e/c*l
n(c*x^2-1)*d^2+1/8*b*e^3/c^2*ln(c*x^2-1)-b/c^(1/2)*arctanh(x*c^(1/2))*d^3-b*e^2/c^(3/2)*arctanh(x*c^(1/2))*d-1
/8*b/e*ln(c*x^2+1)*d^4+3/4*b*e/c*ln(c*x^2+1)*d^2-1/8*b*e^3/c^2*ln(c*x^2+1)+b/c^(1/2)*arctan(x*c^(1/2))*d^3-b*e
^2/c^(3/2)*arctan(x*c^(1/2))*d

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Maxima [A]
time = 0.47, size = 238, normalized size = 1.31 \begin {gather*} \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + \frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {3}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} + 2 \, x \operatorname {artanh}\left (c x^{2}\right )\right )} b d^{3} + a d^{3} x + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{2}\right ) + c {\left (\frac {4 \, x}{c^{2}} - \frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )}\right )} b d e^{2} + \frac {3 \, {\left (2 \, c x^{2} \operatorname {artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} b d^{2} e}{4 \, c} + \frac {1}{8} \, {\left (2 \, x^{4} \operatorname {artanh}\left (c x^{2}\right ) + c {\left (\frac {2 \, x^{2}}{c^{2}} - \frac {\log \left (c x^{2} + 1\right )}{c^{3}} + \frac {\log \left (c x^{2} - 1\right )}{c^{3}}\right )}\right )} b e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arctanh(c*x^2)),x, algorithm="maxima")

[Out]

1/4*a*x^4*e^3 + a*d*x^3*e^2 + 3/2*a*d^2*x^2*e + 1/2*(c*(2*arctan(sqrt(c)*x)/c^(3/2) + log((c*x - sqrt(c))/(c*x
+ sqrt(c)))/c^(3/2)) + 2*x*arctanh(c*x^2))*b*d^3 + a*d^3*x + 1/2*(2*x^3*arctanh(c*x^2) + c*(4*x/c^2 - 2*arcta
n(sqrt(c)*x)/c^(5/2) + log((c*x - sqrt(c))/(c*x + sqrt(c)))/c^(5/2)))*b*d*e^2 + 3/4*(2*c*x^2*arctanh(c*x^2) +
log(-c^2*x^4 + 1))*b*d^2*e/c + 1/8*(2*x^4*arctanh(c*x^2) + c*(2*x^2/c^2 - log(c*x^2 + 1)/c^3 + log(c*x^2 - 1)/
c^3))*b*e^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (156) = 312\).
time = 0.40, size = 1137, normalized size = 6.25 \begin {gather*} \left [\frac {12 \, a c^{2} d^{2} x^{2} \cosh \left (1\right ) + 8 \, a c^{2} d^{3} x + 2 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )^{3} + 2 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \sinh \left (1\right )^{3} + 8 \, {\left (a c^{2} d x^{3} + 2 \, b c d x\right )} \cosh \left (1\right )^{2} + 2 \, {\left (4 \, a c^{2} d x^{3} + 8 \, b c d x + 3 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 8 \, {\left (b c d^{3} - b d \cosh \left (1\right )^{2} - 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) - b d \sinh \left (1\right )^{2}\right )} \sqrt {c} \arctan \left (\sqrt {c} x\right ) + 4 \, {\left (b c d^{3} + b d \cosh \left (1\right )^{2} + 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) + b d \sinh \left (1\right )^{2}\right )} \sqrt {c} \log \left (\frac {c x^{2} - 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right ) + {\left (6 \, b c d^{2} \cosh \left (1\right ) - b \cosh \left (1\right )^{3} - 3 \, b \cosh \left (1\right ) \sinh \left (1\right )^{2} - b \sinh \left (1\right )^{3} + 3 \, {\left (2 \, b c d^{2} - b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x^{2} + 1\right ) + {\left (6 \, b c d^{2} \cosh \left (1\right ) + b \cosh \left (1\right )^{3} + 3 \, b \cosh \left (1\right ) \sinh \left (1\right )^{2} + b \sinh \left (1\right )^{3} + 3 \, {\left (2 \, b c d^{2} + b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x^{2} - 1\right ) + {\left (b c^{2} x^{4} \cosh \left (1\right )^{3} + b c^{2} x^{4} \sinh \left (1\right )^{3} + 4 \, b c^{2} d x^{3} \cosh \left (1\right )^{2} + 6 \, b c^{2} d^{2} x^{2} \cosh \left (1\right ) + 4 \, b c^{2} d^{3} x + {\left (3 \, b c^{2} x^{4} \cosh \left (1\right ) + 4 \, b c^{2} d x^{3}\right )} \sinh \left (1\right )^{2} + {\left (3 \, b c^{2} x^{4} \cosh \left (1\right )^{2} + 8 \, b c^{2} d x^{3} \cosh \left (1\right ) + 6 \, b c^{2} d^{2} x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, {\left (6 \, a c^{2} d^{2} x^{2} + 3 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )^{2} + 8 \, {\left (a c^{2} d x^{3} + 2 \, b c d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{8 \, c^{2}}, \frac {12 \, a c^{2} d^{2} x^{2} \cosh \left (1\right ) + 8 \, a c^{2} d^{3} x + 2 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )^{3} + 2 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \sinh \left (1\right )^{3} + 8 \, {\left (a c^{2} d x^{3} + 2 \, b c d x\right )} \cosh \left (1\right )^{2} + 2 \, {\left (4 \, a c^{2} d x^{3} + 8 \, b c d x + 3 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 8 \, {\left (b c d^{3} + b d \cosh \left (1\right )^{2} + 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) + b d \sinh \left (1\right )^{2}\right )} \sqrt {-c} \arctan \left (\sqrt {-c} x\right ) + 4 \, {\left (b c d^{3} - b d \cosh \left (1\right )^{2} - 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) - b d \sinh \left (1\right )^{2}\right )} \sqrt {-c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right ) + {\left (6 \, b c d^{2} \cosh \left (1\right ) - b \cosh \left (1\right )^{3} - 3 \, b \cosh \left (1\right ) \sinh \left (1\right )^{2} - b \sinh \left (1\right )^{3} + 3 \, {\left (2 \, b c d^{2} - b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x^{2} + 1\right ) + {\left (6 \, b c d^{2} \cosh \left (1\right ) + b \cosh \left (1\right )^{3} + 3 \, b \cosh \left (1\right ) \sinh \left (1\right )^{2} + b \sinh \left (1\right )^{3} + 3 \, {\left (2 \, b c d^{2} + b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x^{2} - 1\right ) + {\left (b c^{2} x^{4} \cosh \left (1\right )^{3} + b c^{2} x^{4} \sinh \left (1\right )^{3} + 4 \, b c^{2} d x^{3} \cosh \left (1\right )^{2} + 6 \, b c^{2} d^{2} x^{2} \cosh \left (1\right ) + 4 \, b c^{2} d^{3} x + {\left (3 \, b c^{2} x^{4} \cosh \left (1\right ) + 4 \, b c^{2} d x^{3}\right )} \sinh \left (1\right )^{2} + {\left (3 \, b c^{2} x^{4} \cosh \left (1\right )^{2} + 8 \, b c^{2} d x^{3} \cosh \left (1\right ) + 6 \, b c^{2} d^{2} x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, {\left (6 \, a c^{2} d^{2} x^{2} + 3 \, {\left (a c^{2} x^{4} + b c x^{2}\right )} \cosh \left (1\right )^{2} + 8 \, {\left (a c^{2} d x^{3} + 2 \, b c d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{8 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arctanh(c*x^2)),x, algorithm="fricas")

[Out]

[1/8*(12*a*c^2*d^2*x^2*cosh(1) + 8*a*c^2*d^3*x + 2*(a*c^2*x^4 + b*c*x^2)*cosh(1)^3 + 2*(a*c^2*x^4 + b*c*x^2)*s
inh(1)^3 + 8*(a*c^2*d*x^3 + 2*b*c*d*x)*cosh(1)^2 + 2*(4*a*c^2*d*x^3 + 8*b*c*d*x + 3*(a*c^2*x^4 + b*c*x^2)*cosh
(1))*sinh(1)^2 + 8*(b*c*d^3 - b*d*cosh(1)^2 - 2*b*d*cosh(1)*sinh(1) - b*d*sinh(1)^2)*sqrt(c)*arctan(sqrt(c)*x)
+ 4*(b*c*d^3 + b*d*cosh(1)^2 + 2*b*d*cosh(1)*sinh(1) + b*d*sinh(1)^2)*sqrt(c)*log((c*x^2 - 2*sqrt(c)*x + 1)/(
c*x^2 - 1)) + (6*b*c*d^2*cosh(1) - b*cosh(1)^3 - 3*b*cosh(1)*sinh(1)^2 - b*sinh(1)^3 + 3*(2*b*c*d^2 - b*cosh(1
)^2)*sinh(1))*log(c*x^2 + 1) + (6*b*c*d^2*cosh(1) + b*cosh(1)^3 + 3*b*cosh(1)*sinh(1)^2 + b*sinh(1)^3 + 3*(2*b
*c*d^2 + b*cosh(1)^2)*sinh(1))*log(c*x^2 - 1) + (b*c^2*x^4*cosh(1)^3 + b*c^2*x^4*sinh(1)^3 + 4*b*c^2*d*x^3*cos
h(1)^2 + 6*b*c^2*d^2*x^2*cosh(1) + 4*b*c^2*d^3*x + (3*b*c^2*x^4*cosh(1) + 4*b*c^2*d*x^3)*sinh(1)^2 + (3*b*c^2*
x^4*cosh(1)^2 + 8*b*c^2*d*x^3*cosh(1) + 6*b*c^2*d^2*x^2)*sinh(1))*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*(6*a*c^2*d
^2*x^2 + 3*(a*c^2*x^4 + b*c*x^2)*cosh(1)^2 + 8*(a*c^2*d*x^3 + 2*b*c*d*x)*cosh(1))*sinh(1))/c^2, 1/8*(12*a*c^2*
d^2*x^2*cosh(1) + 8*a*c^2*d^3*x + 2*(a*c^2*x^4 + b*c*x^2)*cosh(1)^3 + 2*(a*c^2*x^4 + b*c*x^2)*sinh(1)^3 + 8*(a
*c^2*d*x^3 + 2*b*c*d*x)*cosh(1)^2 + 2*(4*a*c^2*d*x^3 + 8*b*c*d*x + 3*(a*c^2*x^4 + b*c*x^2)*cosh(1))*sinh(1)^2
+ 8*(b*c*d^3 + b*d*cosh(1)^2 + 2*b*d*cosh(1)*sinh(1) + b*d*sinh(1)^2)*sqrt(-c)*arctan(sqrt(-c)*x) + 4*(b*c*d^3
- b*d*cosh(1)^2 - 2*b*d*cosh(1)*sinh(1) - b*d*sinh(1)^2)*sqrt(-c)*log((c*x^2 + 2*sqrt(-c)*x - 1)/(c*x^2 + 1))
+ (6*b*c*d^2*cosh(1) - b*cosh(1)^3 - 3*b*cosh(1)*sinh(1)^2 - b*sinh(1)^3 + 3*(2*b*c*d^2 - b*cosh(1)^2)*sinh(1
))*log(c*x^2 + 1) + (6*b*c*d^2*cosh(1) + b*cosh(1)^3 + 3*b*cosh(1)*sinh(1)^2 + b*sinh(1)^3 + 3*(2*b*c*d^2 + b*
cosh(1)^2)*sinh(1))*log(c*x^2 - 1) + (b*c^2*x^4*cosh(1)^3 + b*c^2*x^4*sinh(1)^3 + 4*b*c^2*d*x^3*cosh(1)^2 + 6*
b*c^2*d^2*x^2*cosh(1) + 4*b*c^2*d^3*x + (3*b*c^2*x^4*cosh(1) + 4*b*c^2*d*x^3)*sinh(1)^2 + (3*b*c^2*x^4*cosh(1)
^2 + 8*b*c^2*d*x^3*cosh(1) + 6*b*c^2*d^2*x^2)*sinh(1))*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*(6*a*c^2*d^2*x^2 + 3*
(a*c^2*x^4 + b*c*x^2)*cosh(1)^2 + 8*(a*c^2*d*x^3 + 2*b*c*d*x)*cosh(1))*sinh(1))/c^2]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2786 vs. \(2 (168) = 336\).
time = 8.67, size = 2786, normalized size = 15.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*atanh(c*x**2)),x)

[Out]

Piecewise((8*a*c**2*d**3*x*sqrt(-1/c)*sqrt(1/c)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/
2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 12*a*c**2*d**2*e*x**2*sqrt(-1/c)*sqrt(1/c)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c
) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 8*a*c**2*d*e**2*x**3*sqrt(-1/c)*sqrt(1/c)
/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 2*a*c**2*e
**3*x**4*sqrt(-1/c)*sqrt(1/c)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(
-1/c)*sqrt(1/c)) + 8*b*c**2*d**3*x*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3
*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 2*b*c**2*d**3*(-1/c)**(3/2)*log(x + sqrt(-1/c))/(2*
c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 2*b*c**2*d**3*
(1/c)**(3/2)*log(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sq
rt(-1/c)*sqrt(1/c)) + 12*b*c**2*d**2*e*x**2*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c)
- 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 3*b*c**2*d**2*e*(-1/c)**(3/2)*sqrt(1/c)*lo
g(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1
/c)) - 3*b*c**2*d**2*e*sqrt(-1/c)*(1/c)**(3/2)*log(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sq
rt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 8*b*c**2*d*e**2*x**3*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2
)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 2*b*c**2*
e**3*x**4*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2)
+ 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 4*b*c*d**3*sqrt(-1/c)*log(x - sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) -
2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 6*b*c*d**3*sqrt(-1/c)*log(x + sqrt(-1/c))/(2*
c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 8*b*c*d**3*sqr
t(-1/c)*log(x - sqrt(1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/
c)*sqrt(1/c)) + 8*b*c*d**3*sqrt(-1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)*
*(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 4*b*c*d**3*sqrt(1/c)*log(x - sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1
/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 6*b*c*d**3*sqrt(1/c)*log(x + sqrt(-1/c)
)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 12*b*c*d*
*2*e*sqrt(-1/c)*sqrt(1/c)*log(x - sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2)
+ 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 18*b*c*d**2*e*sqrt(-1/c)*sqrt(1/c)*log(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2
)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 12*b*c*d**2*e*sqrt(-1/c)*sqrt(1
/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1
/c)) + 16*b*c*d*e**2*x*sqrt(-1/c)*sqrt(1/c)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) +
12*c**2*sqrt(-1/c)*sqrt(1/c)) + 2*b*c*e**3*x**2*sqrt(-1/c)*sqrt(1/c)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3
*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 4*b*d*e**2*sqrt(-1/c)*log(x - sqrt(-1/c))/(2*c**3*(
-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 4*b*d*e**2*sqrt(-1/c
)*log(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sq
rt(1/c)) + 8*b*d*e**2*sqrt(-1/c)*log(x - sqrt(1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)*
*(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 8*b*d*e**2*sqrt(-1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) -
2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) - 4*b*d*e**2*sqrt(1/c)*log(x - sqrt(-1/c))/(2*
c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)) + 4*b*d*e**2*sqr
t(1/c)*log(x + sqrt(-1/c))/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c)*(1/c)**(3/2) + 12*c**2*sqrt(-1/
c)*sqrt(1/c)) - 2*b*e**3*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2)/(2*c**3*(-1/c)**(3/2)*sqrt(1/c) - 2*c**3*sqrt(-1/c
)*(1/c)**(3/2) + 12*c**2*sqrt(-1/c)*sqrt(1/c)), Ne(c, 0)), (a*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x
**4/4), True))

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Giac [A]
time = 7.83, size = 227, normalized size = 1.25 \begin {gather*} \frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {{\left (6 \, a c d^{2} e + b e^{3}\right )} x^{2}}{4 \, c} + \frac {1}{8} \, {\left (b e^{3} x^{4} + 4 \, b d e^{2} x^{3} + 6 \, b d^{2} e x^{2} + 4 \, b d^{3} x\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + \frac {{\left (a c d^{3} + 2 \, b d e^{2}\right )} x}{c} + \frac {{\left (b c d^{3} - b d e^{2}\right )} \arctan \left (\sqrt {c} x\right )}{c^{\frac {3}{2}}} + \frac {{\left (b c d^{3} + b d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {{\left (6 \, b c d^{2} e - b e^{3}\right )} \log \left (c x^{2} + 1\right )}{8 \, c^{2}} + \frac {{\left (6 \, b c d^{2} e + b e^{3}\right )} \log \left (c x^{2} - 1\right )}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arctanh(c*x^2)),x, algorithm="giac")

[Out]

1/4*a*e^3*x^4 + a*d*e^2*x^3 + 1/4*(6*a*c*d^2*e + b*e^3)*x^2/c + 1/8*(b*e^3*x^4 + 4*b*d*e^2*x^3 + 6*b*d^2*e*x^2
+ 4*b*d^3*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)) + (a*c*d^3 + 2*b*d*e^2)*x/c + (b*c*d^3 - b*d*e^2)*arctan(sqrt(c)*x
)/c^(3/2) + (b*c*d^3 + b*d*e^2)*arctan(c*x/sqrt(-c))/(sqrt(-c)*c) + 1/8*(6*b*c*d^2*e - b*e^3)*log(c*x^2 + 1)/c
^2 + 1/8*(6*b*c*d^2*e + b*e^3)*log(c*x^2 - 1)/c^2

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Mupad [B]
time = 2.04, size = 823, normalized size = 4.52 \begin {gather*} \ln \left (c\,x^2+1\right )\,\left (\frac {b\,d^3\,x}{2}+\frac {3\,b\,d^2\,e\,x^2}{4}+\frac {b\,d\,e^2\,x^3}{2}+\frac {b\,e^3\,x^4}{8}\right )-\ln \left (1-c\,x^2\right )\,\left (\frac {b\,d^3\,x}{2}+\frac {3\,b\,d^2\,e\,x^2}{4}+\frac {b\,d\,e^2\,x^3}{2}+\frac {b\,e^3\,x^4}{8}\right )+\frac {a\,e^3\,x^4}{4}-\frac {\ln \left (8\,c^5\,d^6-c^2\,e^6-4\,d\,e^5\,\sqrt {-c^5}+e^6\,x\,\sqrt {-c^5}+8\,c^3\,d^2\,e^4+4\,c^4\,d^3\,e^3\,x-4\,c^3\,d\,e^5\,x+4\,c\,d^3\,e^3\,\sqrt {-c^5}-8\,c^3\,d^6\,x\,\sqrt {-c^5}-8\,c\,d^2\,e^4\,x\,\sqrt {-c^5}\right )\,\left (b\,c^2\,e^3-4\,b\,c\,d^3\,\sqrt {-c^5}+4\,b\,d\,e^2\,\sqrt {-c^5}-6\,b\,c^3\,d^2\,e\right )}{8\,c^4}-\frac {\ln \left (8\,c^5\,d^6-c^2\,e^6+4\,d\,e^5\,\sqrt {-c^5}-e^6\,x\,\sqrt {-c^5}+8\,c^3\,d^2\,e^4+4\,c^4\,d^3\,e^3\,x-4\,c^3\,d\,e^5\,x-4\,c\,d^3\,e^3\,\sqrt {-c^5}+8\,c^3\,d^6\,x\,\sqrt {-c^5}+8\,c\,d^2\,e^4\,x\,\sqrt {-c^5}\right )\,\left (b\,c^2\,e^3+4\,b\,c\,d^3\,\sqrt {-c^5}-4\,b\,d\,e^2\,\sqrt {-c^5}-6\,b\,c^3\,d^2\,e\right )}{8\,c^4}+\frac {x\,\left (2\,a\,c^2\,d^3+4\,b\,c\,d\,e^2\right )}{2\,c^2}+\frac {\ln \left (64\,c^2\,d^{12}\,{\left (c^5\right )}^{7/2}+128\,d^8\,e^4\,{\left (c^5\right )}^{7/2}-64\,c^{20}\,d^{12}\,x-c^{14}\,e^{12}\,x+c\,e^{12}\,{\left (c^5\right )}^{5/2}-32\,c^{16}\,d^4\,e^8\,x-128\,c^{18}\,d^8\,e^4\,x+32\,c^3\,d^4\,e^8\,{\left (c^5\right )}^{5/2}\right )\,\left (b\,c^2\,e^3+4\,b\,c\,d^3\,\sqrt {c^5}+4\,b\,d\,e^2\,\sqrt {c^5}+6\,b\,c^3\,d^2\,e\right )}{8\,c^4}+\frac {\ln \left (8\,c^{10}\,d^6+c^7\,e^6+8\,c^8\,d^2\,e^4-4\,d\,e^5\,{\left (c^5\right )}^{3/2}+e^6\,x\,{\left (c^5\right )}^{3/2}-4\,c^9\,d^3\,e^3\,x-4\,c\,d^3\,e^3\,{\left (c^5\right )}^{3/2}+8\,c^3\,d^6\,x\,{\left (c^5\right )}^{3/2}-4\,c^8\,d\,e^5\,x+8\,c\,d^2\,e^4\,x\,{\left (c^5\right )}^{3/2}\right )\,\left (b\,c^2\,e^3-4\,b\,c\,d^3\,\sqrt {c^5}-4\,b\,d\,e^2\,\sqrt {c^5}+6\,b\,c^3\,d^2\,e\right )}{8\,c^4}+\frac {x^2\,\left (6\,a\,c^2\,d^2\,e+b\,c\,e^3\right )}{4\,c^2}+a\,d\,e^2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^2))*(d + e*x)^3,x)

[Out]

log(c*x^2 + 1)*((b*e^3*x^4)/8 + (b*d^3*x)/2 + (3*b*d^2*e*x^2)/4 + (b*d*e^2*x^3)/2) - log(1 - c*x^2)*((b*e^3*x^
4)/8 + (b*d^3*x)/2 + (3*b*d^2*e*x^2)/4 + (b*d*e^2*x^3)/2) + (a*e^3*x^4)/4 - (log(8*c^5*d^6 - c^2*e^6 - 4*d*e^5
*(-c^5)^(1/2) + e^6*x*(-c^5)^(1/2) + 8*c^3*d^2*e^4 + 4*c^4*d^3*e^3*x - 4*c^3*d*e^5*x + 4*c*d^3*e^3*(-c^5)^(1/2
) - 8*c^3*d^6*x*(-c^5)^(1/2) - 8*c*d^2*e^4*x*(-c^5)^(1/2))*(b*c^2*e^3 - 4*b*c*d^3*(-c^5)^(1/2) + 4*b*d*e^2*(-c
^5)^(1/2) - 6*b*c^3*d^2*e))/(8*c^4) - (log(8*c^5*d^6 - c^2*e^6 + 4*d*e^5*(-c^5)^(1/2) - e^6*x*(-c^5)^(1/2) + 8
*c^3*d^2*e^4 + 4*c^4*d^3*e^3*x - 4*c^3*d*e^5*x - 4*c*d^3*e^3*(-c^5)^(1/2) + 8*c^3*d^6*x*(-c^5)^(1/2) + 8*c*d^2
*e^4*x*(-c^5)^(1/2))*(b*c^2*e^3 + 4*b*c*d^3*(-c^5)^(1/2) - 4*b*d*e^2*(-c^5)^(1/2) - 6*b*c^3*d^2*e))/(8*c^4) +
(x*(2*a*c^2*d^3 + 4*b*c*d*e^2))/(2*c^2) + (log(64*c^2*d^12*(c^5)^(7/2) + 128*d^8*e^4*(c^5)^(7/2) - 64*c^20*d^1
2*x - c^14*e^12*x + c*e^12*(c^5)^(5/2) - 32*c^16*d^4*e^8*x - 128*c^18*d^8*e^4*x + 32*c^3*d^4*e^8*(c^5)^(5/2))*
(b*c^2*e^3 + 4*b*c*d^3*(c^5)^(1/2) + 4*b*d*e^2*(c^5)^(1/2) + 6*b*c^3*d^2*e))/(8*c^4) + (log(8*c^10*d^6 + c^7*e
^6 + 8*c^8*d^2*e^4 - 4*d*e^5*(c^5)^(3/2) + e^6*x*(c^5)^(3/2) - 4*c^9*d^3*e^3*x - 4*c*d^3*e^3*(c^5)^(3/2) + 8*c
^3*d^6*x*(c^5)^(3/2) - 4*c^8*d*e^5*x + 8*c*d^2*e^4*x*(c^5)^(3/2))*(b*c^2*e^3 - 4*b*c*d^3*(c^5)^(1/2) - 4*b*d*e
^2*(c^5)^(1/2) + 6*b*c^3*d^2*e))/(8*c^4) + (x^2*(b*c*e^3 + 6*a*c^2*d^2*e))/(4*c^2) + a*d*e^2*x^3

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