3.1.31 \(\int (e+f x)^3 (a+b \tanh ^{-1}(c+d x)) \, dx\) [31]

Optimal. Leaf size=168 \[ \frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}+\frac {b (d e+f-c f)^4 \log (1-c-d x)}{8 d^4 f}-\frac {b (d e-f-c f)^4 \log (1+c+d x)}{8 d^4 f} \]

[Out]

1/4*b*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*x/d^3+1/2*b*f^2*(-c*f+d*e)*(d*x+c)^2/d^4+1/12*b*f^3*(d*x+c)^3/d^4
+1/4*(f*x+e)^4*(a+b*arctanh(d*x+c))/f+1/8*b*(-c*f+d*e+f)^4*ln(-d*x-c+1)/d^4/f-1/8*b*(-c*f+d*e-f)^4*ln(d*x+c+1)
/d^4/f

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Rubi [A]
time = 0.24, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6246, 6063, 716, 647, 31} \begin {gather*} \frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}+\frac {b f x \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac {b f^2 (c+d x)^2 (d e-c f)}{2 d^4}-\frac {b (-c f+d e-f)^4 \log (c+d x+1)}{8 d^4 f}+\frac {b (-c f+d e+f)^4 \log (-c-d x+1)}{8 d^4 f}+\frac {b f^3 (c+d x)^3}{12 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*
(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcTanh[c + d*x]))/(4*f) + (b*(d*e + f - c*f)^4*Log[1 - c - d*x])/
(8*d^4*f) - (b*(d*e - f - c*f)^4*Log[1 + c + d*x])/(8*d^4*f)

Rule 31

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

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 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 6063

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

Rule 6246

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rubi steps

\begin {align*} \int (e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^3 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^4}{1-x^2} \, dx,x,c+d x\right )}{4 f}\\ &=\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2 \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right )}{d^4}-\frac {4 f^3 (d e-c f) x}{d^4}-\frac {f^4 x^2}{d^4}+\frac {d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4+4 f (d e-c f) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) x}{d^4 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{4 f}\\ &=\frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}-\frac {b \text {Subst}\left (\int \frac {d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4+4 f (d e-c f) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) x}{1-x^2} \, dx,x,c+d x\right )}{4 d^4 f}\\ &=\frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}+\frac {\left (b (d e-f-c f)^4\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{8 d^4 f}-\frac {\left (b (d e+f-c f)^4\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{8 d^4 f}\\ &=\frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 f}+\frac {b (d e+f-c f)^4 \log (1-c-d x)}{8 d^4 f}-\frac {b (d e-f-c f)^4 \log (1+c+d x)}{8 d^4 f}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 270, normalized size = 1.61 \begin {gather*} \frac {6 d \left (4 a d^3 e^3+b f \left (6 d^2 e^2-8 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) x+6 d^2 f \left (6 a d^2 e^2+b f (2 d e-c f)\right ) x^2+2 d^3 f^2 (12 a d e+b f) x^3+6 a d^4 f^3 x^4+6 b d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \tanh ^{-1}(c+d x)-3 b (-1+c) \left (4 d^3 e^3-6 (-1+c) d^2 e^2 f+4 (-1+c)^2 d e f^2-(-1+c)^3 f^3\right ) \log (1-c-d x)-3 b (1+c) \left (-4 d^3 e^3+6 (1+c) d^2 e^2 f-4 (1+c)^2 d e f^2+(1+c)^3 f^3\right ) \log (1+c+d x)}{24 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*d*(4*a*d^3*e^3 + b*f*(6*d^2*e^2 - 8*c*d*e*f + (1 + 3*c^2)*f^2))*x + 6*d^2*f*(6*a*d^2*e^2 + b*f*(2*d*e - c*f
))*x^2 + 2*d^3*f^2*(12*a*d*e + b*f)*x^3 + 6*a*d^4*f^3*x^4 + 6*b*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x
^3)*ArcTanh[c + d*x] - 3*b*(-1 + c)*(4*d^3*e^3 - 6*(-1 + c)*d^2*e^2*f + 4*(-1 + c)^2*d*e*f^2 - (-1 + c)^3*f^3)
*Log[1 - c - d*x] - 3*b*(1 + c)*(-4*d^3*e^3 + 6*(1 + c)*d^2*e^2*f - 4*(1 + c)^2*d*e*f^2 + (1 + c)^3*f^3)*Log[1
+ c + d*x])/(24*d^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(981\) vs. \(2(156)=312\).
time = 1.40, size = 982, normalized size = 5.85

method	result	size
risch	\(\frac {b \,e^{3} \ln \left (d x +c +1\right )}{2 d}+\frac {b \,e^{3} \ln \left (-d x -c +1\right )}{2 d}-\frac {2 f^{2} b c e x}{d^{2}}+\frac {f^{2} \ln \left (d x +c +1\right ) b \,c^{3} e}{2 d^{3}}-\frac {3 f \ln \left (d x +c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}-\frac {f^{2} \ln \left (-d x -c +1\right ) b \,c^{3} e}{2 d^{3}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (d x +c +1\right ) b c \,e^{2}}{2 d^{2}}+\frac {3 f^{2} \ln \left (-d x -c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (-d x -c +1\right ) b c \,e^{2}}{2 d^{2}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b c e}{2 d^{3}}-\frac {3 f^{2} \ln \left (-d x -c +1\right ) b c e}{2 d^{3}}+\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}+\frac {3 f b \,e^{2} x}{2 d}-\frac {f^{2} b e \,x^{3} \ln \left (-d x -c +1\right )}{2}-\frac {3 f b \,e^{2} x^{2} \ln \left (-d x -c +1\right )}{4}+\frac {\ln \left (d x +c +1\right ) b c \,e^{3}}{2 d}-\frac {\ln \left (-d x -c +1\right ) b c \,e^{3}}{2 d}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{3}}{2 d^{4}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{3}}{2 d^{4}}-\frac {3 f^{3} \ln \left (d x +c +1\right ) b \,c^{2}}{4 d^{4}}-\frac {3 f \ln \left (d x +c +1\right ) b \,e^{2}}{4 d^{2}}+\frac {3 f^{3} \ln \left (-d x -c +1\right ) b \,c^{2}}{4 d^{4}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,e^{2}}{4 d^{2}}-\frac {f^{3} \ln \left (d x +c +1\right ) b c}{2 d^{4}}+\frac {f^{2} \ln \left (d x +c +1\right ) b e}{2 d^{3}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b c}{2 d^{4}}+\frac {f^{2} \ln \left (-d x -c +1\right ) b e}{2 d^{3}}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{4}}{8 d^{4}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{4}}{8 d^{4}}+f^{2} a e \,x^{3}+\frac {3 f a \,e^{2} x^{2}}{2}+a \,e^{3} x -\frac {f^{3} b c \,x^{2}}{4 d^{2}}+\frac {f^{2} b e \,x^{2}}{2 d}+\frac {f^{3} a \,x^{4}}{4}+\frac {f^{3} b \,x^{3}}{12 d}+\frac {f^{3} b x}{4 d^{3}}-\frac {b \,e^{3} x \ln \left (-d x -c +1\right )}{2}-\frac {f^{3} b \,x^{4} \ln \left (-d x -c +1\right )}{8}-\frac {\ln \left (d x +c +1\right ) b \,e^{4}}{8 f}-\frac {f^{3} \ln \left (d x +c +1\right ) b}{8 d^{4}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b}{8 d^{4}}+\frac {\left (f x +e \right )^{4} b \ln \left (d x +c +1\right )}{8 f}\)	\(780\)
derivativedivides	\(\text {Expression too large to display}\)	\(982\)
default	\(\text {Expression too large to display}\)	\(982\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/2*b*e^3*ln(d*x+c-1)+1/2*b*e^3*ln(d*x+c+1)-3*b/d^2*f^2*c*e*(d*x+c)+3/2*b/d^3*f^3*c^2*(d*x+c)+1/8*b/d^3*f
^3*ln(d*x+c-1)*c^4-1/2*b/d^3*f^3*ln(d*x+c-1)*c^3+3/4*b/d^3*f^3*ln(d*x+c-1)*c^2-1/2*b/d^3*f^3*ln(d*x+c-1)*c+1/4
*b/d^3*f^3*arctanh(d*x+c)*c^4+1/4*b/d^3*f^3*arctanh(d*x+c)*(d*x+c)^4+1/4*b*d/f*arctanh(d*x+c)*e^4-3/4*b/d*f*ln
(d*x+c+1)*e^2+1/2*b/d^2*f^2*ln(d*x+c+1)*e+3/4*b/d*f*ln(d*x+c-1)*e^2+1/2*b/d^2*f^2*ln(d*x+c-1)*e-1/8*b*d/f*ln(d
*x+c+1)*e^4+1/8*b*d/f*ln(d*x+c-1)*e^4-1/8*b/d^3*f^3*ln(d*x+c+1)*c^4-1/2*b/d^3*f^3*ln(d*x+c+1)*c^3-3/4*b/d^3*f^
3*ln(d*x+c+1)*c^2-1/2*b/d^3*f^3*ln(d*x+c+1)*c+3/2*b/d*f*e^2*(d*x+c)+1/2*b/d^2*f^2*e*(d*x+c)^2-1/2*b/d^3*f^3*c*
(d*x+c)^2-b/d^3*f^3*arctanh(d*x+c)*c^3*(d*x+c)+3/2*b/d^3*f^3*arctanh(d*x+c)*c^2*(d*x+c)^2-b/d^3*f^3*arctanh(d*
x+c)*c*(d*x+c)^3-1/2*b/d^2*f^2*ln(d*x+c-1)*c^3*e+3/4*b/d*f*ln(d*x+c-1)*c^2*e^2+3/2*b/d^2*f^2*ln(d*x+c-1)*c^2*e
-3/2*b/d*f*ln(d*x+c-1)*c*e^2-3/2*b/d^2*f^2*ln(d*x+c-1)*c*e+1/2*b/d^2*f^2*ln(d*x+c+1)*c^3*e-3/4*b/d*f*ln(d*x+c+
1)*c^2*e^2+3/2*b/d^2*f^2*ln(d*x+c+1)*c^2*e-3/2*b/d*f*ln(d*x+c+1)*c*e^2+3/2*b/d^2*f^2*ln(d*x+c+1)*c*e-b/d^2*f^2
*arctanh(d*x+c)*c^3*e+3/2*b/d*f*arctanh(d*x+c)*c^2*e^2+3/2*b/d*f*arctanh(d*x+c)*e^2*(d*x+c)^2+b/d^2*f^2*arctan
h(d*x+c)*e*(d*x+c)^3+1/4*(c*f-d*e-f*(d*x+c))^4*a/d^3/f+1/4*b/d^3*f^3*(d*x+c)+1/12*b/d^3*f^3*(d*x+c)^3-1/2*b*ln
(d*x+c-1)*c*e^3+1/2*b*ln(d*x+c+1)*c*e^3-b*arctanh(d*x+c)*c*e^3+b*arctanh(d*x+c)*e^3*(d*x+c)+1/8*b/d^3*f^3*ln(d
*x+c-1)-1/8*b/d^3*f^3*ln(d*x+c+1)+3*b/d^2*f^2*arctanh(d*x+c)*c^2*e*(d*x+c)-3*b/d*f*arctanh(d*x+c)*c*e^2*(d*x+c
)-3*b/d^2*f^2*arctanh(d*x+c)*c*e*(d*x+c)^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (161) = 322\).
time = 0.26, size = 331, normalized size = 1.97 \begin {gather*} \frac {1}{4} \, a f^{3} x^{4} + a f^{2} x^{3} e + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} b f^{3} + \frac {3}{2} \, a f x^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b f^{2} e + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b f e^{2} + a x e^{3} + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*f^3*x^4 + a*f^2*x^3*e + 1/24*(6*x^4*arctanh(d*x + c) + d*(2*(d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 + 1)*x)/d^4
- 3*(c^4 + 4*c^3 + 6*c^2 + 4*c + 1)*log(d*x + c + 1)/d^5 + 3*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*log(d*x + c - 1)/
d^5))*b*f^3 + 3/2*a*f*x^2*e^2 + 1/2*(2*x^3*arctanh(d*x + c) + d*((d*x^2 - 4*c*x)/d^3 + (c^3 + 3*c^2 + 3*c + 1)
*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d*x + c - 1)/d^4))*b*f^2*e + 3/4*(2*x^2*arctanh(d*x + c) +
d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*b*f*e^2 + a*x*e^3
+ 1/2*(2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*b*e^3/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (161) = 322\).
time = 0.41, size = 818, normalized size = 4.87 \begin {gather*} \frac {6 \, a d^{4} f^{3} x^{4} + 2 \, b d^{3} f^{3} x^{3} - 6 \, b c d^{2} f^{3} x^{2} + 24 \, a d^{4} x \cosh \left (1\right )^{3} + 24 \, a d^{4} x \sinh \left (1\right )^{3} + 6 \, {\left (3 \, b c^{2} + b\right )} d f^{3} x + 36 \, {\left (a d^{4} f x^{2} + b d^{3} f x\right )} \cosh \left (1\right )^{2} + 36 \, {\left (a d^{4} f x^{2} + 2 \, a d^{4} x \cosh \left (1\right ) + b d^{3} f x\right )} \sinh \left (1\right )^{2} + 12 \, {\left (2 \, a d^{4} f^{2} x^{3} + b d^{3} f^{2} x^{2} - 4 \, b c d^{2} f^{2} x\right )} \cosh \left (1\right ) + 3 \, {\left (4 \, {\left (b c + b\right )} d^{3} \cosh \left (1\right )^{3} + 4 \, {\left (b c + b\right )} d^{3} \sinh \left (1\right )^{3} - 6 \, {\left (b c^{2} + 2 \, b c + b\right )} d^{2} f \cosh \left (1\right )^{2} + 4 \, {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} d f^{2} \cosh \left (1\right ) - {\left (b c^{4} + 4 \, b c^{3} + 6 \, b c^{2} + 4 \, b c + b\right )} f^{3} + 6 \, {\left (2 \, {\left (b c + b\right )} d^{3} \cosh \left (1\right ) - {\left (b c^{2} + 2 \, b c + b\right )} d^{2} f\right )} \sinh \left (1\right )^{2} + 4 \, {\left (3 \, {\left (b c + b\right )} d^{3} \cosh \left (1\right )^{2} - 3 \, {\left (b c^{2} + 2 \, b c + b\right )} d^{2} f \cosh \left (1\right ) + {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} d f^{2}\right )} \sinh \left (1\right )\right )} \log \left (d x + c + 1\right ) - 3 \, {\left (4 \, {\left (b c - b\right )} d^{3} \cosh \left (1\right )^{3} + 4 \, {\left (b c - b\right )} d^{3} \sinh \left (1\right )^{3} - 6 \, {\left (b c^{2} - 2 \, b c + b\right )} d^{2} f \cosh \left (1\right )^{2} + 4 \, {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} d f^{2} \cosh \left (1\right ) - {\left (b c^{4} - 4 \, b c^{3} + 6 \, b c^{2} - 4 \, b c + b\right )} f^{3} + 6 \, {\left (2 \, {\left (b c - b\right )} d^{3} \cosh \left (1\right ) - {\left (b c^{2} - 2 \, b c + b\right )} d^{2} f\right )} \sinh \left (1\right )^{2} + 4 \, {\left (3 \, {\left (b c - b\right )} d^{3} \cosh \left (1\right )^{2} - 3 \, {\left (b c^{2} - 2 \, b c + b\right )} d^{2} f \cosh \left (1\right ) + {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} d f^{2}\right )} \sinh \left (1\right )\right )} \log \left (d x + c - 1\right ) + 3 \, {\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} f^{2} x^{3} \cosh \left (1\right ) + 6 \, b d^{4} f x^{2} \cosh \left (1\right )^{2} + 4 \, b d^{4} x \cosh \left (1\right )^{3} + 4 \, b d^{4} x \sinh \left (1\right )^{3} + 6 \, {\left (b d^{4} f x^{2} + 2 \, b d^{4} x \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 4 \, {\left (b d^{4} f^{2} x^{3} + 3 \, b d^{4} f x^{2} \cosh \left (1\right ) + 3 \, b d^{4} x \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 12 \, {\left (2 \, a d^{4} f^{2} x^{3} + b d^{3} f^{2} x^{2} + 6 \, a d^{4} x \cosh \left (1\right )^{2} - 4 \, b c d^{2} f^{2} x + 6 \, {\left (a d^{4} f x^{2} + b d^{3} f x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{24 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*a*d^4*f^3*x^4 + 2*b*d^3*f^3*x^3 - 6*b*c*d^2*f^3*x^2 + 24*a*d^4*x*cosh(1)^3 + 24*a*d^4*x*sinh(1)^3 + 6*
(3*b*c^2 + b)*d*f^3*x + 36*(a*d^4*f*x^2 + b*d^3*f*x)*cosh(1)^2 + 36*(a*d^4*f*x^2 + 2*a*d^4*x*cosh(1) + b*d^3*f
*x)*sinh(1)^2 + 12*(2*a*d^4*f^2*x^3 + b*d^3*f^2*x^2 - 4*b*c*d^2*f^2*x)*cosh(1) + 3*(4*(b*c + b)*d^3*cosh(1)^3
+ 4*(b*c + b)*d^3*sinh(1)^3 - 6*(b*c^2 + 2*b*c + b)*d^2*f*cosh(1)^2 + 4*(b*c^3 + 3*b*c^2 + 3*b*c + b)*d*f^2*co
sh(1) - (b*c^4 + 4*b*c^3 + 6*b*c^2 + 4*b*c + b)*f^3 + 6*(2*(b*c + b)*d^3*cosh(1) - (b*c^2 + 2*b*c + b)*d^2*f)*
sinh(1)^2 + 4*(3*(b*c + b)*d^3*cosh(1)^2 - 3*(b*c^2 + 2*b*c + b)*d^2*f*cosh(1) + (b*c^3 + 3*b*c^2 + 3*b*c + b)
*d*f^2)*sinh(1))*log(d*x + c + 1) - 3*(4*(b*c - b)*d^3*cosh(1)^3 + 4*(b*c - b)*d^3*sinh(1)^3 - 6*(b*c^2 - 2*b*
c + b)*d^2*f*cosh(1)^2 + 4*(b*c^3 - 3*b*c^2 + 3*b*c - b)*d*f^2*cosh(1) - (b*c^4 - 4*b*c^3 + 6*b*c^2 - 4*b*c +
b)*f^3 + 6*(2*(b*c - b)*d^3*cosh(1) - (b*c^2 - 2*b*c + b)*d^2*f)*sinh(1)^2 + 4*(3*(b*c - b)*d^3*cosh(1)^2 - 3*
(b*c^2 - 2*b*c + b)*d^2*f*cosh(1) + (b*c^3 - 3*b*c^2 + 3*b*c - b)*d*f^2)*sinh(1))*log(d*x + c - 1) + 3*(b*d^4*
f^3*x^4 + 4*b*d^4*f^2*x^3*cosh(1) + 6*b*d^4*f*x^2*cosh(1)^2 + 4*b*d^4*x*cosh(1)^3 + 4*b*d^4*x*sinh(1)^3 + 6*(b
*d^4*f*x^2 + 2*b*d^4*x*cosh(1))*sinh(1)^2 + 4*(b*d^4*f^2*x^3 + 3*b*d^4*f*x^2*cosh(1) + 3*b*d^4*x*cosh(1)^2)*si
nh(1))*log(-(d*x + c + 1)/(d*x + c - 1)) + 12*(2*a*d^4*f^2*x^3 + b*d^3*f^2*x^2 + 6*a*d^4*x*cosh(1)^2 - 4*b*c*d
^2*f^2*x + 6*(a*d^4*f*x^2 + b*d^3*f*x)*cosh(1))*sinh(1))/d^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (151) = 302\).
time = 2.07, size = 644, normalized size = 3.83 \begin {gather*} \begin {cases} a e^{3} x + \frac {3 a e^{2} f x^{2}}{2} + a e f^{2} x^{3} + \frac {a f^{3} x^{4}}{4} - \frac {b c^{4} f^{3} \operatorname {atanh}{\left (c + d x \right )}}{4 d^{4}} + \frac {b c^{3} e f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} - \frac {b c^{3} f^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{4}} + \frac {b c^{3} f^{3} \operatorname {atanh}{\left (c + d x \right )}}{d^{4}} - \frac {3 b c^{2} e^{2} f \operatorname {atanh}{\left (c + d x \right )}}{2 d^{2}} + \frac {3 b c^{2} e f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {3 b c^{2} e f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + \frac {3 b c^{2} f^{3} x}{4 d^{3}} - \frac {3 b c^{2} f^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d^{4}} + \frac {b c e^{3} \operatorname {atanh}{\left (c + d x \right )}}{d} - \frac {3 b c e^{2} f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {3 b c e^{2} f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} - \frac {2 b c e f^{2} x}{d^{2}} - \frac {b c f^{3} x^{2}}{4 d^{2}} + \frac {3 b c e f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} - \frac {b c f^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{4}} + \frac {b c f^{3} \operatorname {atanh}{\left (c + d x \right )}}{d^{4}} + b e^{3} x \operatorname {atanh}{\left (c + d x \right )} + \frac {3 b e^{2} f x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + b e f^{2} x^{3} \operatorname {atanh}{\left (c + d x \right )} + \frac {b f^{3} x^{4} \operatorname {atanh}{\left (c + d x \right )}}{4} + \frac {b e^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e^{3} \operatorname {atanh}{\left (c + d x \right )}}{d} + \frac {3 b e^{2} f x}{2 d} + \frac {b e f^{2} x^{2}}{2 d} + \frac {b f^{3} x^{3}}{12 d} - \frac {3 b e^{2} f \operatorname {atanh}{\left (c + d x \right )}}{2 d^{2}} + \frac {b e f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {b e f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + \frac {b f^{3} x}{4 d^{3}} - \frac {b f^{3} \operatorname {atanh}{\left (c + d x \right )}}{4 d^{4}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {atanh}{\left (c \right )}\right ) \left (e^{3} x + \frac {3 e^{2} f x^{2}}{2} + e f^{2} x^{3} + \frac {f^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*e**3*x + 3*a*e**2*f*x**2/2 + a*e*f**2*x**3 + a*f**3*x**4/4 - b*c**4*f**3*atanh(c + d*x)/(4*d**4)
+ b*c**3*e*f**2*atanh(c + d*x)/d**3 - b*c**3*f**3*log(c/d + x + 1/d)/d**4 + b*c**3*f**3*atanh(c + d*x)/d**4 -
3*b*c**2*e**2*f*atanh(c + d*x)/(2*d**2) + 3*b*c**2*e*f**2*log(c/d + x + 1/d)/d**3 - 3*b*c**2*e*f**2*atanh(c +
d*x)/d**3 + 3*b*c**2*f**3*x/(4*d**3) - 3*b*c**2*f**3*atanh(c + d*x)/(2*d**4) + b*c*e**3*atanh(c + d*x)/d - 3*b
*c*e**2*f*log(c/d + x + 1/d)/d**2 + 3*b*c*e**2*f*atanh(c + d*x)/d**2 - 2*b*c*e*f**2*x/d**2 - b*c*f**3*x**2/(4*
d**2) + 3*b*c*e*f**2*atanh(c + d*x)/d**3 - b*c*f**3*log(c/d + x + 1/d)/d**4 + b*c*f**3*atanh(c + d*x)/d**4 + b
*e**3*x*atanh(c + d*x) + 3*b*e**2*f*x**2*atanh(c + d*x)/2 + b*e*f**2*x**3*atanh(c + d*x) + b*f**3*x**4*atanh(c
+ d*x)/4 + b*e**3*log(c/d + x + 1/d)/d - b*e**3*atanh(c + d*x)/d + 3*b*e**2*f*x/(2*d) + b*e*f**2*x**2/(2*d) +
b*f**3*x**3/(12*d) - 3*b*e**2*f*atanh(c + d*x)/(2*d**2) + b*e*f**2*log(c/d + x + 1/d)/d**3 - b*e*f**2*atanh(c
+ d*x)/d**3 + b*f**3*x/(4*d**3) - b*f**3*atanh(c + d*x)/(4*d**4), Ne(d, 0)), ((a + b*atanh(c))*(e**3*x + 3*e*
*2*f*x**2/2 + e*f**2*x**3 + f**3*x**4/4), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2336 vs. \(2 (156) = 312\).
time = 0.46, size = 2336, normalized size = 13.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((c + 1)*d - (c - 1)*d)*(3*((d*x + c + 1)^3*b*d^3*e^3/(d*x + c - 1)^3 - 3*(d*x + c + 1)^2*b*d^3*e^3/(d*x +
c - 1)^2 + 3*(d*x + c + 1)*b*d^3*e^3/(d*x + c - 1) - b*d^3*e^3 - 3*(d*x + c + 1)^3*b*c*d^2*e^2*f/(d*x + c - 1
)^3 + 9*(d*x + c + 1)^2*b*c*d^2*e^2*f/(d*x + c - 1)^2 - 9*(d*x + c + 1)*b*c*d^2*e^2*f/(d*x + c - 1) + 3*b*c*d^
2*e^2*f + 3*(d*x + c + 1)^3*b*c^2*d*e*f^2/(d*x + c - 1)^3 - 9*(d*x + c + 1)^2*b*c^2*d*e*f^2/(d*x + c - 1)^2 +
9*(d*x + c + 1)*b*c^2*d*e*f^2/(d*x + c - 1) - 3*b*c^2*d*e*f^2 - (d*x + c + 1)^3*b*c^3*f^3/(d*x + c - 1)^3 + 3*
(d*x + c + 1)^2*b*c^3*f^3/(d*x + c - 1)^2 - 3*(d*x + c + 1)*b*c^3*f^3/(d*x + c - 1) + b*c^3*f^3 + 3*(d*x + c +
1)^3*b*d^2*e^2*f/(d*x + c - 1)^3 - 6*(d*x + c + 1)^2*b*d^2*e^2*f/(d*x + c - 1)^2 + 3*(d*x + c + 1)*b*d^2*e^2*
f/(d*x + c - 1) - 6*(d*x + c + 1)^3*b*c*d*e*f^2/(d*x + c - 1)^3 + 12*(d*x + c + 1)^2*b*c*d*e*f^2/(d*x + c - 1)
^2 - 6*(d*x + c + 1)*b*c*d*e*f^2/(d*x + c - 1) + 3*(d*x + c + 1)^3*b*c^2*f^3/(d*x + c - 1)^3 - 6*(d*x + c + 1)
^2*b*c^2*f^3/(d*x + c - 1)^2 + 3*(d*x + c + 1)*b*c^2*f^3/(d*x + c - 1) + 3*(d*x + c + 1)^3*b*d*e*f^2/(d*x + c
- 1)^3 - 3*(d*x + c + 1)^2*b*d*e*f^2/(d*x + c - 1)^2 + (d*x + c + 1)*b*d*e*f^2/(d*x + c - 1) - b*d*e*f^2 - 3*(
d*x + c + 1)^3*b*c*f^3/(d*x + c - 1)^3 + 3*(d*x + c + 1)^2*b*c*f^3/(d*x + c - 1)^2 - (d*x + c + 1)*b*c*f^3/(d*
x + c - 1) + b*c*f^3 + (d*x + c + 1)^3*b*f^3/(d*x + c - 1)^3 + (d*x + c + 1)*b*f^3/(d*x + c - 1))*log(-(d*x +
c + 1)/(d*x + c - 1))/((d*x + c + 1)^4*d^5/(d*x + c - 1)^4 - 4*(d*x + c + 1)^3*d^5/(d*x + c - 1)^3 + 6*(d*x +
c + 1)^2*d^5/(d*x + c - 1)^2 - 4*(d*x + c + 1)*d^5/(d*x + c - 1) + d^5) + (6*(d*x + c + 1)^3*a*d^3*e^3/(d*x +
c - 1)^3 - 18*(d*x + c + 1)^2*a*d^3*e^3/(d*x + c - 1)^2 + 18*(d*x + c + 1)*a*d^3*e^3/(d*x + c - 1) - 6*a*d^3*e
^3 - 18*(d*x + c + 1)^3*a*c*d^2*e^2*f/(d*x + c - 1)^3 + 54*(d*x + c + 1)^2*a*c*d^2*e^2*f/(d*x + c - 1)^2 - 54*
(d*x + c + 1)*a*c*d^2*e^2*f/(d*x + c - 1) + 18*a*c*d^2*e^2*f + 18*(d*x + c + 1)^3*a*c^2*d*e*f^2/(d*x + c - 1)^
3 - 54*(d*x + c + 1)^2*a*c^2*d*e*f^2/(d*x + c - 1)^2 + 54*(d*x + c + 1)*a*c^2*d*e*f^2/(d*x + c - 1) - 18*a*c^2
*d*e*f^2 - 6*(d*x + c + 1)^3*a*c^3*f^3/(d*x + c - 1)^3 + 18*(d*x + c + 1)^2*a*c^3*f^3/(d*x + c - 1)^2 - 18*(d*
x + c + 1)*a*c^3*f^3/(d*x + c - 1) + 6*a*c^3*f^3 + 18*(d*x + c + 1)^3*a*d^2*e^2*f/(d*x + c - 1)^3 - 36*(d*x +
c + 1)^2*a*d^2*e^2*f/(d*x + c - 1)^2 + 18*(d*x + c + 1)*a*d^2*e^2*f/(d*x + c - 1) + 9*(d*x + c + 1)^3*b*d^2*e^
2*f/(d*x + c - 1)^3 - 27*(d*x + c + 1)^2*b*d^2*e^2*f/(d*x + c - 1)^2 + 27*(d*x + c + 1)*b*d^2*e^2*f/(d*x + c -
1) - 9*b*d^2*e^2*f - 36*(d*x + c + 1)^3*a*c*d*e*f^2/(d*x + c - 1)^3 + 72*(d*x + c + 1)^2*a*c*d*e*f^2/(d*x + c
- 1)^2 - 36*(d*x + c + 1)*a*c*d*e*f^2/(d*x + c - 1) - 18*(d*x + c + 1)^3*b*c*d*e*f^2/(d*x + c - 1)^3 + 54*(d*
x + c + 1)^2*b*c*d*e*f^2/(d*x + c - 1)^2 - 54*(d*x + c + 1)*b*c*d*e*f^2/(d*x + c - 1) + 18*b*c*d*e*f^2 + 18*(d
*x + c + 1)^3*a*c^2*f^3/(d*x + c - 1)^3 - 36*(d*x + c + 1)^2*a*c^2*f^3/(d*x + c - 1)^2 + 18*(d*x + c + 1)*a*c^
2*f^3/(d*x + c - 1) + 9*(d*x + c + 1)^3*b*c^2*f^3/(d*x + c - 1)^3 - 27*(d*x + c + 1)^2*b*c^2*f^3/(d*x + c - 1)
^2 + 27*(d*x + c + 1)*b*c^2*f^3/(d*x + c - 1) - 9*b*c^2*f^3 + 18*(d*x + c + 1)^3*a*d*e*f^2/(d*x + c - 1)^3 - 1
8*(d*x + c + 1)^2*a*d*e*f^2/(d*x + c - 1)^2 + 6*(d*x + c + 1)*a*d*e*f^2/(d*x + c - 1) - 6*a*d*e*f^2 + 6*(d*x +
c + 1)^3*b*d*e*f^2/(d*x + c - 1)^3 - 12*(d*x + c + 1)^2*b*d*e*f^2/(d*x + c - 1)^2 + 6*(d*x + c + 1)*b*d*e*f^2
/(d*x + c - 1) - 18*(d*x + c + 1)^3*a*c*f^3/(d*x + c - 1)^3 + 18*(d*x + c + 1)^2*a*c*f^3/(d*x + c - 1)^2 - 6*(
d*x + c + 1)*a*c*f^3/(d*x + c - 1) + 6*a*c*f^3 - 6*(d*x + c + 1)^3*b*c*f^3/(d*x + c - 1)^3 + 12*(d*x + c + 1)^
2*b*c*f^3/(d*x + c - 1)^2 - 6*(d*x + c + 1)*b*c*f^3/(d*x + c - 1) + 6*(d*x + c + 1)^3*a*f^3/(d*x + c - 1)^3 +
6*(d*x + c + 1)*a*f^3/(d*x + c - 1) + 3*(d*x + c + 1)^3*b*f^3/(d*x + c - 1)^3 - 6*(d*x + c + 1)^2*b*f^3/(d*x +
c - 1)^2 + 5*(d*x + c + 1)*b*f^3/(d*x + c - 1) - 2*b*f^3)/((d*x + c + 1)^4*d^5/(d*x + c - 1)^4 - 4*(d*x + c +
1)^3*d^5/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^5/(d*x + c - 1)^2 - 4*(d*x + c + 1)*d^5/(d*x + c - 1) + d^5) -
3*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3 + b*d*e*f^2 - b*c*f^3)*log(-(d*x + c + 1)/(d*x +
c - 1) + 1)/d^5 + 3*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3 + b*d*e*f^2 - b*c*f^3)*log(-(d
*x + c + 1)/(d*x + c - 1))/d^5)

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Mupad [B]
time = 1.72, size = 737, normalized size = 4.39 \begin {gather*} \ln \left (c+d\,x+1\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )-\ln \left (1-d\,x-c\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )+x\,\left (\frac {e\,\left (6\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+2\,a\,d^2\,e^2+3\,b\,d\,e\,f-6\,a\,f^2\right )}{2\,d^2}-\frac {\left (4\,c^2-4\right )\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{4\,d^2}+\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{4\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{4\,d^2}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{8\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{8\,d^2}\right )+x^3\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{12\,d}-\frac {2\,a\,c\,f^3}{3\,d}\right )+\frac {a\,f^3\,x^4}{4}+\frac {\ln \left (c+d\,x-1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2-4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f+12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3-12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2-4\,b\,c\,f^3+4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f+4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4}-\frac {\ln \left (c+d\,x+1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2+4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f-12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3+12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2+4\,b\,c\,f^3-4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f-4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c + d*x + 1)*((b*f^3*x^4)/8 + (b*e^3*x)/2 + (3*b*e^2*f*x^2)/4 + (b*e*f^2*x^3)/2) - log(1 - d*x - c)*((b*f^
3*x^4)/8 + (b*e^3*x)/2 + (3*b*e^2*f*x^2)/4 + (b*e*f^2*x^3)/2) + x*((e*(6*a*c^2*f^2 - 6*a*f^2 + 2*a*d^2*e^2 + 3
*b*d*e*f + 12*a*c*d*e*f))/(2*d^2) - ((4*c^2 - 4)*((f^2*(b*f + 8*a*c*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/(4*
d^2) + (2*c*((2*c*((f^2*(b*f + 8*a*c*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/d - (4*a*c^2*f^3 - 4*a*f^3 + 4*b*d
*e*f^2 + 12*a*d^2*e^2*f + 24*a*c*d*e*f^2)/(4*d^2) + (a*f^3*(4*c^2 - 4))/(4*d^2)))/d) - x^2*((c*((f^2*(b*f + 8*
a*c*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/d - (4*a*c^2*f^3 - 4*a*f^3 + 4*b*d*e*f^2 + 12*a*d^2*e^2*f + 24*a*c*
d*e*f^2)/(8*d^2) + (a*f^3*(4*c^2 - 4))/(8*d^2)) + x^3*((f^2*(b*f + 8*a*c*f + 12*a*d*e))/(12*d) - (2*a*c*f^3)/(
3*d)) + (a*f^3*x^4)/4 + (log(c + d*x - 1)*(b*f^3 + 6*b*c^2*f^3 - 4*b*c^3*f^3 + 4*b*d^3*e^3 + b*c^4*f^3 - 4*b*c
*f^3 + 4*b*d*e*f^2 - 4*b*c*d^3*e^3 + 6*b*d^2*e^2*f - 12*b*c*d^2*e^2*f + 12*b*c^2*d*e*f^2 - 4*b*c^3*d*e*f^2 + 6
*b*c^2*d^2*e^2*f - 12*b*c*d*e*f^2))/(8*d^4) - (log(c + d*x + 1)*(b*f^3 + 6*b*c^2*f^3 + 4*b*c^3*f^3 - 4*b*d^3*e
^3 + b*c^4*f^3 + 4*b*c*f^3 - 4*b*d*e*f^2 - 4*b*c*d^3*e^3 + 6*b*d^2*e^2*f + 12*b*c*d^2*e^2*f - 12*b*c^2*d*e*f^2
- 4*b*c^3*d*e*f^2 + 6*b*c^2*d^2*e^2*f - 12*b*c*d*e*f^2))/(8*d^4)

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