3.1.24 \(\int (d+e x)^2 (a+b \tanh ^{-1}(c x^2)) \, dx\) [24]

Optimal. Leaf size=158 \[ \frac {2 b e^2 x}{3 c}+\frac {b \left (3 c d^2-e^2\right ) \text {ArcTan}\left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {b \left (3 c d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 e}+\frac {b d \left (c d^2+3 e^2\right ) \log \left (1-c x^2\right )}{6 c e}-\frac {b d \left (c d^2-3 e^2\right ) \log \left (1+c x^2\right )}{6 c e} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6071, 1845, 1262, 647, 31, 1294, 1181, 211, 214} \begin {gather*} \frac {(d+e x)^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 e}+\frac {b \text {ArcTan}\left (\sqrt {c} x\right ) \left (3 c d^2-e^2\right )}{3 c^{3/2}}-\frac {b \left (3 c d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {b d \left (c d^2+3 e^2\right ) \log \left (1-c x^2\right )}{6 c e}-\frac {b d \left (c d^2-3 e^2\right ) \log \left (c x^2+1\right )}{6 c e}+\frac {2 b e^2 x}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*e^2*x)/(3*c) + (b*(3*c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) - (b*(3*c*d^2 + e^2)*ArcTanh[Sqrt[c]*x])
/(3*c^(3/2)) + ((d + e*x)^3*(a + b*ArcTanh[c*x^2]))/(3*e) + (b*d*(c*d^2 + 3*e^2)*Log[1 - c*x^2])/(6*c*e) - (b*
d*(c*d^2 - 3*e^2)*Log[1 + c*x^2])/(6*c*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 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

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 1845

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(c*x)^(m + ii)*((Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

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

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Mathematica [A]
time = 0.10, size = 170, normalized size = 1.08 \begin {gather*} \frac {1}{6} \left (6 a d^2 x+\frac {4 b e^2 x}{c}+6 a d e x^2+2 a e^2 x^3+\frac {2 b \left (3 c d^2-e^2\right ) \text {ArcTan}\left (\sqrt {c} x\right )}{c^{3/2}}+2 b x \left (3 d^2+3 d e x+e^2 x^2\right ) \tanh ^{-1}\left (c x^2\right )+\frac {b \left (3 c d^2+e^2\right ) \log \left (1-\sqrt {c} x\right )}{c^{3/2}}-\frac {b \left (3 c d^2+e^2\right ) \log \left (1+\sqrt {c} x\right )}{c^{3/2}}+\frac {3 b d e \log \left (1-c^2 x^4\right )}{c}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.28, size = 205, normalized size = 1.30

method	result	size
default	\(\frac {\left (e x +d \right )^{3} a}{3 e}+\frac {b \,e^{2} \arctanh \left (c \,x^{2}\right ) x^{3}}{3}+b e \arctanh \left (c \,x^{2}\right ) x^{2} d +b \arctanh \left (c \,x^{2}\right ) x \,d^{2}+\frac {b \arctanh \left (c \,x^{2}\right ) d^{3}}{3 e}+\frac {2 b \,e^{2} x}{3 c}+\frac {b \ln \left (c \,x^{2}-1\right ) d^{3}}{6 e}+\frac {b e \ln \left (c \,x^{2}-1\right ) d}{2 c}-\frac {b \arctanh \left (x \sqrt {c}\right ) d^{2}}{\sqrt {c}}-\frac {b \,e^{2} \arctanh \left (x \sqrt {c}\right )}{3 c^{\frac {3}{2}}}-\frac {b \ln \left (c \,x^{2}+1\right ) d^{3}}{6 e}+\frac {b e \ln \left (c \,x^{2}+1\right ) d}{2 c}+\frac {b \arctan \left (x \sqrt {c}\right ) d^{2}}{\sqrt {c}}-\frac {b \,e^{2} \arctan \left (x \sqrt {c}\right )}{3 c^{\frac {3}{2}}}\)	\(205\)
risch	\(\text {Expression too large to display}\)	\(3662\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 171, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d 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^{2} + a d^{2} x + \frac {1}{6} \, {\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 e^{2} + \frac {{\left (2 \, c x^{2} \operatorname {artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} b d e}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (131) = 262\).
time = 0.46, size = 677, normalized size = 4.28 \begin {gather*} \left [\frac {6 \, a c^{2} d x^{2} \cosh \left (1\right ) + 6 \, a c^{2} d^{2} x + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \sinh \left (1\right )^{2} + 2 \, {\left (3 \, b c d^{2} - b \cosh \left (1\right )^{2} - 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) - b \sinh \left (1\right )^{2}\right )} \sqrt {c} \arctan \left (\sqrt {c} x\right ) + {\left (3 \, b c d^{2} + b \cosh \left (1\right )^{2} + 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) + b \sinh \left (1\right )^{2}\right )} \sqrt {c} \log \left (\frac {c x^{2} - 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right ) + 3 \, {\left (b c d \cosh \left (1\right ) + b c d \sinh \left (1\right )\right )} \log \left (c x^{2} + 1\right ) + 3 \, {\left (b c d \cosh \left (1\right ) + b c d \sinh \left (1\right )\right )} \log \left (c x^{2} - 1\right ) + {\left (b c^{2} x^{3} \cosh \left (1\right )^{2} + b c^{2} x^{3} \sinh \left (1\right )^{2} + 3 \, b c^{2} d x^{2} \cosh \left (1\right ) + 3 \, b c^{2} d^{2} x + {\left (2 \, b c^{2} x^{3} \cosh \left (1\right ) + 3 \, b c^{2} d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, {\left (3 \, a c^{2} d x^{2} + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{6 \, c^{2}}, \frac {6 \, a c^{2} d x^{2} \cosh \left (1\right ) + 6 \, a c^{2} d^{2} x + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \sinh \left (1\right )^{2} + 2 \, {\left (3 \, b c d^{2} + b \cosh \left (1\right )^{2} + 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) + b \sinh \left (1\right )^{2}\right )} \sqrt {-c} \arctan \left (\sqrt {-c} x\right ) + {\left (3 \, b c d^{2} - b \cosh \left (1\right )^{2} - 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) - b \sinh \left (1\right )^{2}\right )} \sqrt {-c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right ) + 3 \, {\left (b c d \cosh \left (1\right ) + b c d \sinh \left (1\right )\right )} \log \left (c x^{2} + 1\right ) + 3 \, {\left (b c d \cosh \left (1\right ) + b c d \sinh \left (1\right )\right )} \log \left (c x^{2} - 1\right ) + {\left (b c^{2} x^{3} \cosh \left (1\right )^{2} + b c^{2} x^{3} \sinh \left (1\right )^{2} + 3 \, b c^{2} d x^{2} \cosh \left (1\right ) + 3 \, b c^{2} d^{2} x + {\left (2 \, b c^{2} x^{3} \cosh \left (1\right ) + 3 \, b c^{2} d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, {\left (3 \, a c^{2} d x^{2} + 2 \, {\left (a c^{2} x^{3} + 2 \, b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{6 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(6*a*c^2*d*x^2*cosh(1) + 6*a*c^2*d^2*x + 2*(a*c^2*x^3 + 2*b*c*x)*cosh(1)^2 + 2*(a*c^2*x^3 + 2*b*c*x)*sinh
(1)^2 + 2*(3*b*c*d^2 - b*cosh(1)^2 - 2*b*cosh(1)*sinh(1) - b*sinh(1)^2)*sqrt(c)*arctan(sqrt(c)*x) + (3*b*c*d^2
+ b*cosh(1)^2 + 2*b*cosh(1)*sinh(1) + b*sinh(1)^2)*sqrt(c)*log((c*x^2 - 2*sqrt(c)*x + 1)/(c*x^2 - 1)) + 3*(b*
c*d*cosh(1) + b*c*d*sinh(1))*log(c*x^2 + 1) + 3*(b*c*d*cosh(1) + b*c*d*sinh(1))*log(c*x^2 - 1) + (b*c^2*x^3*co
sh(1)^2 + b*c^2*x^3*sinh(1)^2 + 3*b*c^2*d*x^2*cosh(1) + 3*b*c^2*d^2*x + (2*b*c^2*x^3*cosh(1) + 3*b*c^2*d*x^2)*
sinh(1))*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*(3*a*c^2*d*x^2 + 2*(a*c^2*x^3 + 2*b*c*x)*cosh(1))*sinh(1))/c^2, 1/6
*(6*a*c^2*d*x^2*cosh(1) + 6*a*c^2*d^2*x + 2*(a*c^2*x^3 + 2*b*c*x)*cosh(1)^2 + 2*(a*c^2*x^3 + 2*b*c*x)*sinh(1)^
2 + 2*(3*b*c*d^2 + b*cosh(1)^2 + 2*b*cosh(1)*sinh(1) + b*sinh(1)^2)*sqrt(-c)*arctan(sqrt(-c)*x) + (3*b*c*d^2 -
b*cosh(1)^2 - 2*b*cosh(1)*sinh(1) - b*sinh(1)^2)*sqrt(-c)*log((c*x^2 + 2*sqrt(-c)*x - 1)/(c*x^2 + 1)) + 3*(b*
c*d*cosh(1) + b*c*d*sinh(1))*log(c*x^2 + 1) + 3*(b*c*d*cosh(1) + b*c*d*sinh(1))*log(c*x^2 - 1) + (b*c^2*x^3*co
sh(1)^2 + b*c^2*x^3*sinh(1)^2 + 3*b*c^2*d*x^2*cosh(1) + 3*b*c^2*d^2*x + (2*b*c^2*x^3*cosh(1) + 3*b*c^2*d*x^2)*
sinh(1))*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*(3*a*c^2*d*x^2 + 2*(a*c^2*x^3 + 2*b*c*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. 3465 vs. \(2 (139) = 278\).
time = 7.49, size = 3465, normalized size = 21.93 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((12*a*c**2*d**2*x*sqrt(-1/c)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24
*c**2*sqrt(1/c)) + 12*a*c**2*d**2*x*sqrt(1/c)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/
c) + 24*c**2*sqrt(1/c)) + 12*a*c**2*d*e*x**2*sqrt(-1/c)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**
2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*a*c**2*d*e*x**2*sqrt(1/c)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2)
+ 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 4*a*c**2*e**2*x**3*sqrt(-1/c)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1
/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 4*a*c**2*e**2*x**3*sqrt(1/c)/(12*c**3*(-1/c)**(3/2) - 1
2*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 3*b*c**3*d**2*(-1/c)**(3/2)*sqrt(1/c)*log(x -
sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 12*b*c**
3*d**2*(-1/c)**(3/2)*sqrt(1/c)*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqr
t(-1/c) + 24*c**2*sqrt(1/c)) + 3*b*c**3*d**2*(-1/c)**(3/2)*sqrt(1/c)*log(x - sqrt(1/c))/(12*c**3*(-1/c)**(3/2)
- 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 3*b*c**3*d**2*sqrt(-1/c)*(1/c)**(3/2)*log(
x - sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 12*b
*c**3*d**2*sqrt(-1/c)*(1/c)**(3/2)*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2
*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 3*b*c**3*d**2*sqrt(-1/c)*(1/c)**(3/2)*log(x - sqrt(1/c))/(12*c**3*(-1/c)**(
3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c**2*d**2*x*sqrt(-1/c)*atanh(c*x*
*2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c**2*d**2*x
*sqrt(1/c)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c
)) - 18*b*c**2*d**2*sqrt(-1/c)*sqrt(1/c)*log(x - sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 2
4*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 18*b*c**2*d**2*sqrt(-1/c)*sqrt(1/c)*log(x - sqrt(1/c))/(12*c**3*(-1/c
)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c**2*d**2*sqrt(-1/c)*sqrt(1/c
)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b
*c**2*d*e*x**2*sqrt(-1/c)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 2
4*c**2*sqrt(1/c)) + 12*b*c**2*d*e*x**2*sqrt(1/c)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) +
24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 3*b*c**2*d*e*(-1/c)**(3/2)*log(x - sqrt(-1/c))/(12*c**3*(-1/c)**(3/
2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c**2*d*e*(-1/c)**(3/2)*log(x + sqrt
(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 3*b*c**2*d*e
*(1/c)**(3/2)*log(x - sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2
*sqrt(1/c)) - 12*b*c**2*d*e*(1/c)**(3/2)*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 2
4*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 4*b*c**2*e**2*x**3*sqrt(-1/c)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) -
12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 4*b*c**2*e**2*x**3*sqrt(1/c)*atanh(c*x**2)/(1
2*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + b*c**2*e**2*(-1/c)**(3
/2)*sqrt(1/c)*log(x - sqrt(1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*
sqrt(1/c)) - b*c**2*e**2*sqrt(-1/c)*(1/c)**(3/2)*log(x - sqrt(1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3
/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 12*b*c*d**2*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c*
*3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c*d**2*log(x - sqrt(1/c))/(12*c**3*(-1/c)**(3
/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 12*b*c*d**2*atanh(c*x**2)/(12*c**3*(-1/
c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 15*b*c*d*e*sqrt(-1/c)*log(x - sqr
t(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 24*b*c*d*e*
sqrt(-1/c)*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sq
rt(1/c)) - 12*b*c*d*e*sqrt(-1/c)*atanh(c*x**2)/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1
/c) + 24*c**2*sqrt(1/c)) + 15*b*c*d*e*sqrt(1/c)*log(x - sqrt(-1/c))/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3
/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 24*b*c*d*e*sqrt(1/c)*log(x + sqrt(-1/c))/(12*c**3*(-1/c)**(3/2
) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) - 12*b*c*d*e*sqrt(1/c)*atanh(c*x**2)/(12*c*
*3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 8*b*c*e**2*x*sqrt(-1/c)/(1
2*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) + 24*c**2*sqrt(-1/c) + 24*c**2*sqrt(1/c)) + 8*b*c*e**2*x*sqrt(1/c)
/(12*c**3*(-1/c)**(3/2) - 12*c**3*(1/c)**(3/2) ...

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Giac [A]
time = 1.31, size = 171, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {b d e \log \left (c x^{2} + 1\right )}{2 \, c} + \frac {b d e \log \left (c x^{2} - 1\right )}{2 \, c} + \frac {1}{6} \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + \frac {{\left (3 \, a c d^{2} + 2 \, b e^{2}\right )} x}{3 \, c} + \frac {{\left (3 \, b c d^{2} - b e^{2}\right )} \arctan \left (\sqrt {c} x\right )}{3 \, c^{\frac {3}{2}}} + \frac {{\left (3 \, b c d^{2} + b e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {-c}}\right )}{3 \, \sqrt {-c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.39, size = 309, normalized size = 1.96 \begin {gather*} \ln \left (c\,x^2+1\right )\,\left (\frac {b\,d^2\,x}{2}+\frac {b\,d\,e\,x^2}{2}+\frac {b\,e^2\,x^3}{6}\right )-\ln \left (1-c\,x^2\right )\,\left (\frac {b\,d^2\,x}{2}+\frac {b\,d\,e\,x^2}{2}+\frac {b\,e^2\,x^3}{6}\right )+\frac {x\,\left (3\,a\,c^2\,d^2+2\,b\,c\,e^2\right )}{3\,c^2}+\frac {a\,e^2\,x^3}{3}-\frac {\ln \left (c+x\,\sqrt {c^3}\right )\,\left (b\,e^2\,\sqrt {c^3}+3\,b\,c\,d^2\,\sqrt {c^3}-3\,b\,c^2\,d\,e\right )}{6\,c^3}+\frac {\ln \left (c-x\,\sqrt {c^3}\right )\,\left (b\,e^2\,\sqrt {c^3}+3\,b\,c\,d^2\,\sqrt {c^3}+3\,b\,c^2\,d\,e\right )}{6\,c^3}+\frac {\ln \left (c+x\,\sqrt {-c^3}\right )\,\left (b\,e^2\,\sqrt {-c^3}+3\,b\,c^2\,d\,e-3\,b\,c\,d^2\,\sqrt {-c^3}\right )}{6\,c^3}+\frac {\ln \left (c-x\,\sqrt {-c^3}\right )\,\left (3\,b\,c^2\,d\,e-b\,e^2\,\sqrt {-c^3}+3\,b\,c\,d^2\,\sqrt {-c^3}\right )}{6\,c^3}+a\,d\,e\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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