3.1.1 \(\int (d+e x)^4 (a+b \tanh ^{-1}(c x)) \, dx\) [1]

Optimal. Leaf size=149 \[ \frac {b d e \left (2 c^2 d^2+e^2\right ) x}{c^3}+\frac {b e^2 \left (10 c^2 d^2+e^2\right ) x^2}{10 c^3}+\frac {b d e^3 x^3}{3 c}+\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}+\frac {b (c d+e)^5 \log (1-c x)}{10 c^5 e}-\frac {b (c d-e)^5 \log (1+c x)}{10 c^5 e} \]

[Out]

b*d*e*(2*c^2*d^2+e^2)*x/c^3+1/10*b*e^2*(10*c^2*d^2+e^2)*x^2/c^3+1/3*b*d*e^3*x^3/c+1/20*b*e^4*x^4/c+1/5*(e*x+d)
^5*(a+b*arctanh(c*x))/e+1/10*b*(c*d+e)^5*ln(-c*x+1)/c^5/e-1/10*b*(c*d-e)^5*ln(c*x+1)/c^5/e

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Rubi [A]
time = 0.11, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6063, 716, 647, 31} \begin {gather*} \frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}-\frac {b (c d-e)^5 \log (c x+1)}{10 c^5 e}+\frac {b (c d+e)^5 \log (1-c x)}{10 c^5 e}+\frac {b e^2 x^2 \left (10 c^2 d^2+e^2\right )}{10 c^3}+\frac {b d e x \left (2 c^2 d^2+e^2\right )}{c^3}+\frac {b d e^3 x^3}{3 c}+\frac {b e^4 x^4}{20 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*e*(2*c^2*d^2 + e^2)*x)/c^3 + (b*e^2*(10*c^2*d^2 + e^2)*x^2)/(10*c^3) + (b*d*e^3*x^3)/(3*c) + (b*e^4*x^4)/
(20*c) + ((d + e*x)^5*(a + b*ArcTanh[c*x]))/(5*e) + (b*(c*d + e)^5*Log[1 - c*x])/(10*c^5*e) - (b*(c*d - e)^5*L
og[1 + c*x])/(10*c^5*e)

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]

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}-\frac {(b c) \int \frac {(d+e x)^5}{1-c^2 x^2} \, dx}{5 e}\\ &=\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}-\frac {(b c) \int \left (-\frac {5 d e^2 \left (2 c^2 d^2+e^2\right )}{c^4}-\frac {e^3 \left (10 c^2 d^2+e^2\right ) x}{c^4}-\frac {5 d e^4 x^2}{c^2}-\frac {e^5 x^3}{c^2}+\frac {c^4 d^5+10 c^2 d^3 e^2+5 d e^4+e \left (5 c^4 d^4+10 c^2 d^2 e^2+e^4\right ) x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{5 e}\\ &=\frac {b d e \left (2 c^2 d^2+e^2\right ) x}{c^3}+\frac {b e^2 \left (10 c^2 d^2+e^2\right ) x^2}{10 c^3}+\frac {b d e^3 x^3}{3 c}+\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}-\frac {b \int \frac {c^4 d^5+10 c^2 d^3 e^2+5 d e^4+e \left (5 c^4 d^4+10 c^2 d^2 e^2+e^4\right ) x}{1-c^2 x^2} \, dx}{5 c^3 e}\\ &=\frac {b d e \left (2 c^2 d^2+e^2\right ) x}{c^3}+\frac {b e^2 \left (10 c^2 d^2+e^2\right ) x^2}{10 c^3}+\frac {b d e^3 x^3}{3 c}+\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}+\frac {\left (b (c d-e)^5\right ) \int \frac {1}{-c-c^2 x} \, dx}{10 c^3 e}-\frac {\left (b (c d+e)^5\right ) \int \frac {1}{c-c^2 x} \, dx}{10 c^3 e}\\ &=\frac {b d e \left (2 c^2 d^2+e^2\right ) x}{c^3}+\frac {b e^2 \left (10 c^2 d^2+e^2\right ) x^2}{10 c^3}+\frac {b d e^3 x^3}{3 c}+\frac {b e^4 x^4}{20 c}+\frac {(d+e x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 e}+\frac {b (c d+e)^5 \log (1-c x)}{10 c^5 e}-\frac {b (c d-e)^5 \log (1+c x)}{10 c^5 e}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 274, normalized size = 1.84 \begin {gather*} \frac {60 c^2 d \left (a c^3 d^3+b e \left (2 c^2 d^2+e^2\right )\right ) x+6 c^2 e \left (20 a c^3 d^3+b e \left (10 c^2 d^2+e^2\right )\right ) x^2+20 c^4 d e^2 (6 a c d+b e) x^3+3 c^4 e^3 (20 a c d+b e) x^4+12 a c^5 e^4 x^5+12 b c^5 x \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right ) \tanh ^{-1}(c x)+6 b \left (5 c^4 d^4+10 c^3 d^3 e+10 c^2 d^2 e^2+5 c d e^3+e^4\right ) \log (1-c x)+6 b \left (5 c^4 d^4-10 c^3 d^3 e+10 c^2 d^2 e^2-5 c d e^3+e^4\right ) \log (1+c x)}{60 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(60*c^2*d*(a*c^3*d^3 + b*e*(2*c^2*d^2 + e^2))*x + 6*c^2*e*(20*a*c^3*d^3 + b*e*(10*c^2*d^2 + e^2))*x^2 + 20*c^4
*d*e^2*(6*a*c*d + b*e)*x^3 + 3*c^4*e^3*(20*a*c*d + b*e)*x^4 + 12*a*c^5*e^4*x^5 + 12*b*c^5*x*(5*d^4 + 10*d^3*e*
x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4)*ArcTanh[c*x] + 6*b*(5*c^4*d^4 + 10*c^3*d^3*e + 10*c^2*d^2*e^2 + 5*
c*d*e^3 + e^4)*Log[1 - c*x] + 6*b*(5*c^4*d^4 - 10*c^3*d^3*e + 10*c^2*d^2*e^2 - 5*c*d*e^3 + e^4)*Log[1 + c*x])/
(60*c^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(137)=274\).
time = 0.15, size = 353, normalized size = 2.37

method	result	size
derivativedivides	\(\frac {\frac {b \ln \left (c x +1\right ) d^{4}}{2}+\frac {b \ln \left (c x -1\right ) d^{4}}{2}+b c \,e^{3} \arctanh \left (c x \right ) d \,x^{4}+2 b e \,d^{3} x +\frac {b \,e^{3} d x}{c^{2}}+b \,e^{2} d^{2} x^{2}+\frac {b \,e^{3} d \,x^{3}}{3}+\frac {b c \,e^{4} \arctanh \left (c x \right ) x^{5}}{5}+\frac {b \,e^{3} \ln \left (c x -1\right ) d}{2 c^{3}}+\frac {b \,e^{4} x^{2}}{10 c^{2}}+\frac {b \,e^{4} x^{4}}{20}-\frac {b c \ln \left (c x +1\right ) d^{5}}{10 e}-\frac {b e \ln \left (c x +1\right ) d^{3}}{c}+\frac {b \,e^{2} \ln \left (c x +1\right ) d^{2}}{c^{2}}-\frac {b \,e^{3} \ln \left (c x +1\right ) d}{2 c^{3}}+\frac {b c \ln \left (c x -1\right ) d^{5}}{10 e}+\frac {b e \ln \left (c x -1\right ) d^{3}}{c}+\frac {b \,e^{2} \ln \left (c x -1\right ) d^{2}}{c^{2}}+\frac {b \,e^{4} \ln \left (c x +1\right )}{10 c^{4}}+\frac {b \,e^{4} \ln \left (c x -1\right )}{10 c^{4}}+\frac {\left (c e x +d c \right )^{5} a}{5 c^{4} e}+b \arctanh \left (c x \right ) d^{4} c x +\frac {b c \arctanh \left (c x \right ) d^{5}}{5 e}+2 b c e \arctanh \left (c x \right ) d^{3} x^{2}+2 b c \,e^{2} \arctanh \left (c x \right ) d^{2} x^{3}}{c}\)	\(353\)
default	\(\frac {\frac {b \ln \left (c x +1\right ) d^{4}}{2}+\frac {b \ln \left (c x -1\right ) d^{4}}{2}+b c \,e^{3} \arctanh \left (c x \right ) d \,x^{4}+2 b e \,d^{3} x +\frac {b \,e^{3} d x}{c^{2}}+b \,e^{2} d^{2} x^{2}+\frac {b \,e^{3} d \,x^{3}}{3}+\frac {b c \,e^{4} \arctanh \left (c x \right ) x^{5}}{5}+\frac {b \,e^{3} \ln \left (c x -1\right ) d}{2 c^{3}}+\frac {b \,e^{4} x^{2}}{10 c^{2}}+\frac {b \,e^{4} x^{4}}{20}-\frac {b c \ln \left (c x +1\right ) d^{5}}{10 e}-\frac {b e \ln \left (c x +1\right ) d^{3}}{c}+\frac {b \,e^{2} \ln \left (c x +1\right ) d^{2}}{c^{2}}-\frac {b \,e^{3} \ln \left (c x +1\right ) d}{2 c^{3}}+\frac {b c \ln \left (c x -1\right ) d^{5}}{10 e}+\frac {b e \ln \left (c x -1\right ) d^{3}}{c}+\frac {b \,e^{2} \ln \left (c x -1\right ) d^{2}}{c^{2}}+\frac {b \,e^{4} \ln \left (c x +1\right )}{10 c^{4}}+\frac {b \,e^{4} \ln \left (c x -1\right )}{10 c^{4}}+\frac {\left (c e x +d c \right )^{5} a}{5 c^{4} e}+b \arctanh \left (c x \right ) d^{4} c x +\frac {b c \arctanh \left (c x \right ) d^{5}}{5 e}+2 b c e \arctanh \left (c x \right ) d^{3} x^{2}+2 b c \,e^{2} \arctanh \left (c x \right ) d^{2} x^{3}}{c}\)	\(353\)
risch	\(\frac {b \,e^{4} x^{4}}{20 c}+e^{3} a d \,x^{4}+2 e^{2} a \,d^{2} x^{3}+2 e a \,d^{3} x^{2}+a \,d^{4} x +\frac {e^{2} b \,d^{2} x^{2}}{c}+\frac {2 e b \,d^{3} x}{c}+\frac {e^{3} b d x}{c^{3}}-\frac {e \ln \left (c x +1\right ) b \,d^{3}}{c^{2}}+\frac {e \ln \left (-c x +1\right ) b \,d^{3}}{c^{2}}+\frac {e^{2} \ln \left (c x +1\right ) b \,d^{2}}{c^{3}}+\frac {e^{2} \ln \left (-c x +1\right ) b \,d^{2}}{c^{3}}-\frac {e^{3} \ln \left (c x +1\right ) b d}{2 c^{4}}+\frac {e^{3} \ln \left (-c x +1\right ) b d}{2 c^{4}}-\frac {e^{3} b d \,x^{4} \ln \left (-c x +1\right )}{2}-e^{2} b \,d^{2} x^{3} \ln \left (-c x +1\right )-e b \,d^{3} x^{2} \ln \left (-c x +1\right )+\frac {e^{4} a \,x^{5}}{5}+\frac {e^{4} b \,x^{2}}{10 c^{3}}-\frac {e^{4} b \,x^{5} \ln \left (-c x +1\right )}{10}-\frac {\ln \left (c x +1\right ) b \,d^{5}}{10 e}+\frac {\ln \left (c x +1\right ) b \,d^{4}}{2 c}+\frac {\ln \left (-c x +1\right ) b \,d^{4}}{2 c}+\frac {e^{4} \ln \left (c x +1\right ) b}{10 c^{5}}+\frac {e^{4} \ln \left (-c x +1\right ) b}{10 c^{5}}-\frac {b \,d^{4} x \ln \left (-c x +1\right )}{2}+\frac {\left (e x +d \right )^{5} b \ln \left (c x +1\right )}{10 e}+\frac {b d \,e^{3} x^{3}}{3 c}\)	\(399\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(a+b*arctanh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/2*b*ln(c*x+1)*d^4+1/2*b*ln(c*x-1)*d^4+2*b*e*d^3*x+b/c^2*e^3*d*x+b*e^2*d^2*x^2+1/3*b*e^3*d*x^3+1/5*b*c*e
^4*arctanh(c*x)*x^5+2*b*c*e*arctanh(c*x)*d^3*x^2+2*b*c*e^2*arctanh(c*x)*d^2*x^3+b*c*e^3*arctanh(c*x)*d*x^4+b*a
rctanh(c*x)*d^4*c*x+1/2*b/c^3*e^3*ln(c*x-1)*d+1/10*b/c^2*e^4*x^2+1/20*b*e^4*x^4-1/10*b*c/e*ln(c*x+1)*d^5-b/c*e
*ln(c*x+1)*d^3+b/c^2*e^2*ln(c*x+1)*d^2-1/2*b/c^3*e^3*ln(c*x+1)*d+1/10*b*c/e*ln(c*x-1)*d^5+b/c*e*ln(c*x-1)*d^3+
b/c^2*e^2*ln(c*x-1)*d^2+1/5*b*c/e*arctanh(c*x)*d^5+1/10*b/c^4*e^4*ln(c*x+1)+1/10*b/c^4*e^4*ln(c*x-1)+1/5*(c*e*
x+c*d)^5*a/c^4/e)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (133) = 266\).
time = 0.26, size = 269, normalized size = 1.81 \begin {gather*} \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x + {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d^{3} e + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{4}}{2 \, c} + {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{2} e^{2} + \frac {1}{6} \, {\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 e^{3} + \frac {1}{20} \, {\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 e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e^4 + a*d*x^4*e^3 + 2*a*d^2*x^3*e^2 + 2*a*d^3*x^2*e + a*d^4*x + (2*x^2*arctanh(c*x) + c*(2*x/c^2 - l
og(c*x + 1)/c^3 + log(c*x - 1)/c^3))*b*d^3*e + 1/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*d^4/c + (2*x^3*a
rctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*b*d^2*e^2 + 1/6*(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*e^3 + 1/20*(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*e^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (133) = 266\).
time = 0.39, size = 975, normalized size = 6.54 \begin {gather*} \frac {60 \, a c^{5} d^{4} x + 3 \, {\left (4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{4} + 3 \, {\left (4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} \sinh \left (1\right )^{4} + 20 \, {\left (3 \, a c^{5} d x^{4} + b c^{4} d x^{3} + 3 \, b c^{2} d x\right )} \cosh \left (1\right )^{3} + 4 \, {\left (15 \, a c^{5} d x^{4} + 5 \, b c^{4} d x^{3} + 15 \, b c^{2} d x + 3 \, {\left (4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{3} + 60 \, {\left (2 \, a c^{5} d^{2} x^{3} + b c^{4} d^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (20 \, a c^{5} d^{2} x^{3} + 10 \, b c^{4} d^{2} x^{2} + 3 \, {\left (4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 10 \, {\left (3 \, a c^{5} d x^{4} + b c^{4} d x^{3} + 3 \, b c^{2} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 120 \, {\left (a c^{5} d^{3} x^{2} + b c^{4} d^{3} x\right )} \cosh \left (1\right ) + 6 \, {\left (5 \, b c^{4} d^{4} - 10 \, b c^{3} d^{3} \cosh \left (1\right ) + 10 \, b c^{2} d^{2} \cosh \left (1\right )^{2} - 5 \, b c d \cosh \left (1\right )^{3} + b \cosh \left (1\right )^{4} + b \sinh \left (1\right )^{4} - {\left (5 \, b c d - 4 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )^{3} + {\left (10 \, b c^{2} d^{2} - 15 \, b c d \cosh \left (1\right ) + 6 \, b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )^{2} - {\left (10 \, b c^{3} d^{3} - 20 \, b c^{2} d^{2} \cosh \left (1\right ) + 15 \, b c d \cosh \left (1\right )^{2} - 4 \, b \cosh \left (1\right )^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + 1\right ) + 6 \, {\left (5 \, b c^{4} d^{4} + 10 \, b c^{3} d^{3} \cosh \left (1\right ) + 10 \, b c^{2} d^{2} \cosh \left (1\right )^{2} + 5 \, b c d \cosh \left (1\right )^{3} + b \cosh \left (1\right )^{4} + b \sinh \left (1\right )^{4} + {\left (5 \, b c d + 4 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )^{3} + {\left (10 \, b c^{2} d^{2} + 15 \, b c d \cosh \left (1\right ) + 6 \, b \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )^{2} + {\left (10 \, b c^{3} d^{3} + 20 \, b c^{2} d^{2} \cosh \left (1\right ) + 15 \, b c d \cosh \left (1\right )^{2} + 4 \, b \cosh \left (1\right )^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x - 1\right ) + 6 \, {\left (b c^{5} x^{5} \cosh \left (1\right )^{4} + b c^{5} x^{5} \sinh \left (1\right )^{4} + 5 \, b c^{5} d x^{4} \cosh \left (1\right )^{3} + 10 \, b c^{5} d^{2} x^{3} \cosh \left (1\right )^{2} + 10 \, b c^{5} d^{3} x^{2} \cosh \left (1\right ) + 5 \, b c^{5} d^{4} x + {\left (4 \, b c^{5} x^{5} \cosh \left (1\right ) + 5 \, b c^{5} d x^{4}\right )} \sinh \left (1\right )^{3} + {\left (6 \, b c^{5} x^{5} \cosh \left (1\right )^{2} + 15 \, b c^{5} d x^{4} \cosh \left (1\right ) + 10 \, b c^{5} d^{2} x^{3}\right )} \sinh \left (1\right )^{2} + {\left (4 \, b c^{5} x^{5} \cosh \left (1\right )^{3} + 15 \, b c^{5} d x^{4} \cosh \left (1\right )^{2} + 20 \, b c^{5} d^{2} x^{3} \cosh \left (1\right ) + 10 \, b c^{5} d^{3} x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 12 \, {\left (10 \, a c^{5} d^{3} x^{2} + 10 \, b c^{4} d^{3} x + {\left (4 \, a c^{5} x^{5} + b c^{4} x^{4} + 2 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{3} + 5 \, {\left (3 \, a c^{5} d x^{4} + b c^{4} d x^{3} + 3 \, b c^{2} d x\right )} \cosh \left (1\right )^{2} + 10 \, {\left (2 \, a c^{5} d^{2} x^{3} + b c^{4} d^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{60 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*a*c^5*d^4*x + 3*(4*a*c^5*x^5 + b*c^4*x^4 + 2*b*c^2*x^2)*cosh(1)^4 + 3*(4*a*c^5*x^5 + b*c^4*x^4 + 2*b*
c^2*x^2)*sinh(1)^4 + 20*(3*a*c^5*d*x^4 + b*c^4*d*x^3 + 3*b*c^2*d*x)*cosh(1)^3 + 4*(15*a*c^5*d*x^4 + 5*b*c^4*d*
x^3 + 15*b*c^2*d*x + 3*(4*a*c^5*x^5 + b*c^4*x^4 + 2*b*c^2*x^2)*cosh(1))*sinh(1)^3 + 60*(2*a*c^5*d^2*x^3 + b*c^
4*d^2*x^2)*cosh(1)^2 + 6*(20*a*c^5*d^2*x^3 + 10*b*c^4*d^2*x^2 + 3*(4*a*c^5*x^5 + b*c^4*x^4 + 2*b*c^2*x^2)*cosh
(1)^2 + 10*(3*a*c^5*d*x^4 + b*c^4*d*x^3 + 3*b*c^2*d*x)*cosh(1))*sinh(1)^2 + 120*(a*c^5*d^3*x^2 + b*c^4*d^3*x)*
cosh(1) + 6*(5*b*c^4*d^4 - 10*b*c^3*d^3*cosh(1) + 10*b*c^2*d^2*cosh(1)^2 - 5*b*c*d*cosh(1)^3 + b*cosh(1)^4 + b
*sinh(1)^4 - (5*b*c*d - 4*b*cosh(1))*sinh(1)^3 + (10*b*c^2*d^2 - 15*b*c*d*cosh(1) + 6*b*cosh(1)^2)*sinh(1)^2 -
(10*b*c^3*d^3 - 20*b*c^2*d^2*cosh(1) + 15*b*c*d*cosh(1)^2 - 4*b*cosh(1)^3)*sinh(1))*log(c*x + 1) + 6*(5*b*c^4
*d^4 + 10*b*c^3*d^3*cosh(1) + 10*b*c^2*d^2*cosh(1)^2 + 5*b*c*d*cosh(1)^3 + b*cosh(1)^4 + b*sinh(1)^4 + (5*b*c*
d + 4*b*cosh(1))*sinh(1)^3 + (10*b*c^2*d^2 + 15*b*c*d*cosh(1) + 6*b*cosh(1)^2)*sinh(1)^2 + (10*b*c^3*d^3 + 20*
b*c^2*d^2*cosh(1) + 15*b*c*d*cosh(1)^2 + 4*b*cosh(1)^3)*sinh(1))*log(c*x - 1) + 6*(b*c^5*x^5*cosh(1)^4 + b*c^5
*x^5*sinh(1)^4 + 5*b*c^5*d*x^4*cosh(1)^3 + 10*b*c^5*d^2*x^3*cosh(1)^2 + 10*b*c^5*d^3*x^2*cosh(1) + 5*b*c^5*d^4
*x + (4*b*c^5*x^5*cosh(1) + 5*b*c^5*d*x^4)*sinh(1)^3 + (6*b*c^5*x^5*cosh(1)^2 + 15*b*c^5*d*x^4*cosh(1) + 10*b*
c^5*d^2*x^3)*sinh(1)^2 + (4*b*c^5*x^5*cosh(1)^3 + 15*b*c^5*d*x^4*cosh(1)^2 + 20*b*c^5*d^2*x^3*cosh(1) + 10*b*c
^5*d^3*x^2)*sinh(1))*log(-(c*x + 1)/(c*x - 1)) + 12*(10*a*c^5*d^3*x^2 + 10*b*c^4*d^3*x + (4*a*c^5*x^5 + b*c^4*
x^4 + 2*b*c^2*x^2)*cosh(1)^3 + 5*(3*a*c^5*d*x^4 + b*c^4*d*x^3 + 3*b*c^2*d*x)*cosh(1)^2 + 10*(2*a*c^5*d^2*x^3 +
b*c^4*d^2*x^2)*cosh(1))*sinh(1))/c^5

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (134) = 268\).
time = 0.53, size = 381, normalized size = 2.56 \begin {gather*} \begin {cases} a d^{4} x + 2 a d^{3} e x^{2} + 2 a d^{2} e^{2} x^{3} + a d e^{3} x^{4} + \frac {a e^{4} x^{5}}{5} + b d^{4} x \operatorname {atanh}{\left (c x \right )} + 2 b d^{3} e x^{2} \operatorname {atanh}{\left (c x \right )} + 2 b d^{2} e^{2} x^{3} \operatorname {atanh}{\left (c x \right )} + b d e^{3} x^{4} \operatorname {atanh}{\left (c x \right )} + \frac {b e^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b d^{4} \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{c} + \frac {2 b d^{3} e x}{c} + \frac {b d^{2} e^{2} x^{2}}{c} + \frac {b d e^{3} x^{3}}{3 c} + \frac {b e^{4} x^{4}}{20 c} - \frac {2 b d^{3} e \operatorname {atanh}{\left (c x \right )}}{c^{2}} + \frac {2 b d^{2} e^{2} \log {\left (x - \frac {1}{c} \right )}}{c^{3}} + \frac {2 b d^{2} e^{2} \operatorname {atanh}{\left (c x \right )}}{c^{3}} + \frac {b d e^{3} x}{c^{3}} + \frac {b e^{4} x^{2}}{10 c^{3}} - \frac {b d e^{3} \operatorname {atanh}{\left (c x \right )}}{c^{4}} + \frac {b e^{4} \log {\left (x - \frac {1}{c} \right )}}{5 c^{5}} + \frac {b e^{4} \operatorname {atanh}{\left (c x \right )}}{5 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**4*x + 2*a*d**3*e*x**2 + 2*a*d**2*e**2*x**3 + a*d*e**3*x**4 + a*e**4*x**5/5 + b*d**4*x*atanh(c*
x) + 2*b*d**3*e*x**2*atanh(c*x) + 2*b*d**2*e**2*x**3*atanh(c*x) + b*d*e**3*x**4*atanh(c*x) + b*e**4*x**5*atanh
(c*x)/5 + b*d**4*log(x - 1/c)/c + b*d**4*atanh(c*x)/c + 2*b*d**3*e*x/c + b*d**2*e**2*x**2/c + b*d*e**3*x**3/(3
*c) + b*e**4*x**4/(20*c) - 2*b*d**3*e*atanh(c*x)/c**2 + 2*b*d**2*e**2*log(x - 1/c)/c**3 + 2*b*d**2*e**2*atanh(
c*x)/c**3 + b*d*e**3*x/c**3 + b*e**4*x**2/(10*c**3) - b*d*e**3*atanh(c*x)/c**4 + b*e**4*log(x - 1/c)/(5*c**5)
+ b*e**4*atanh(c*x)/(5*c**5), Ne(c, 0)), (a*(d**4*x + 2*d**3*e*x**2 + 2*d**2*e**2*x**3 + d*e**3*x**4 + e**4*x*
*5/5), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1576 vs. \(2 (137) = 274\).
time = 0.43, size = 1576, normalized size = 10.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c*(3*(5*(c*x + 1)^4*b*c^4*d^4/(c*x - 1)^4 - 20*(c*x + 1)^3*b*c^4*d^4/(c*x - 1)^3 + 30*(c*x + 1)^2*b*c^4*d
^4/(c*x - 1)^2 - 20*(c*x + 1)*b*c^4*d^4/(c*x - 1) + 5*b*c^4*d^4 + 20*(c*x + 1)^4*b*c^3*d^3*e/(c*x - 1)^4 - 60*
(c*x + 1)^3*b*c^3*d^3*e/(c*x - 1)^3 + 60*(c*x + 1)^2*b*c^3*d^3*e/(c*x - 1)^2 - 20*(c*x + 1)*b*c^3*d^3*e/(c*x -
1) + 30*(c*x + 1)^4*b*c^2*d^2*e^2/(c*x - 1)^4 - 60*(c*x + 1)^3*b*c^2*d^2*e^2/(c*x - 1)^3 + 40*(c*x + 1)^2*b*c
^2*d^2*e^2/(c*x - 1)^2 - 20*(c*x + 1)*b*c^2*d^2*e^2/(c*x - 1) + 10*b*c^2*d^2*e^2 + 20*(c*x + 1)^4*b*c*d*e^3/(c
*x - 1)^4 - 20*(c*x + 1)^3*b*c*d*e^3/(c*x - 1)^3 + 20*(c*x + 1)^2*b*c*d*e^3/(c*x - 1)^2 - 20*(c*x + 1)*b*c*d*e
^3/(c*x - 1) + 5*(c*x + 1)^4*b*e^4/(c*x - 1)^4 + 10*(c*x + 1)^2*b*e^4/(c*x - 1)^2 + b*e^4)*log(-(c*x + 1)/(c*x
- 1))/((c*x + 1)^5*c^6/(c*x - 1)^5 - 5*(c*x + 1)^4*c^6/(c*x - 1)^4 + 10*(c*x + 1)^3*c^6/(c*x - 1)^3 - 10*(c*x
+ 1)^2*c^6/(c*x - 1)^2 + 5*(c*x + 1)*c^6/(c*x - 1) - c^6) + 2*(15*(c*x + 1)^4*a*c^4*d^4/(c*x - 1)^4 - 60*(c*x
+ 1)^3*a*c^4*d^4/(c*x - 1)^3 + 90*(c*x + 1)^2*a*c^4*d^4/(c*x - 1)^2 - 60*(c*x + 1)*a*c^4*d^4/(c*x - 1) + 15*a
*c^4*d^4 + 60*(c*x + 1)^4*a*c^3*d^3*e/(c*x - 1)^4 - 180*(c*x + 1)^3*a*c^3*d^3*e/(c*x - 1)^3 + 180*(c*x + 1)^2*
a*c^3*d^3*e/(c*x - 1)^2 - 60*(c*x + 1)*a*c^3*d^3*e/(c*x - 1) + 30*(c*x + 1)^4*b*c^3*d^3*e/(c*x - 1)^4 - 120*(c
*x + 1)^3*b*c^3*d^3*e/(c*x - 1)^3 + 180*(c*x + 1)^2*b*c^3*d^3*e/(c*x - 1)^2 - 120*(c*x + 1)*b*c^3*d^3*e/(c*x -
1) + 30*b*c^3*d^3*e + 90*(c*x + 1)^4*a*c^2*d^2*e^2/(c*x - 1)^4 - 180*(c*x + 1)^3*a*c^2*d^2*e^2/(c*x - 1)^3 +
120*(c*x + 1)^2*a*c^2*d^2*e^2/(c*x - 1)^2 - 60*(c*x + 1)*a*c^2*d^2*e^2/(c*x - 1) + 30*a*c^2*d^2*e^2 + 30*(c*x
+ 1)^4*b*c^2*d^2*e^2/(c*x - 1)^4 - 90*(c*x + 1)^3*b*c^2*d^2*e^2/(c*x - 1)^3 + 90*(c*x + 1)^2*b*c^2*d^2*e^2/(c*
x - 1)^2 - 30*(c*x + 1)*b*c^2*d^2*e^2/(c*x - 1) + 60*(c*x + 1)^4*a*c*d*e^3/(c*x - 1)^4 - 60*(c*x + 1)^3*a*c*d*
e^3/(c*x - 1)^3 + 60*(c*x + 1)^2*a*c*d*e^3/(c*x - 1)^2 - 60*(c*x + 1)*a*c*d*e^3/(c*x - 1) + 30*(c*x + 1)^4*b*c
*d*e^3/(c*x - 1)^4 - 90*(c*x + 1)^3*b*c*d*e^3/(c*x - 1)^3 + 110*(c*x + 1)^2*b*c*d*e^3/(c*x - 1)^2 - 70*(c*x +
1)*b*c*d*e^3/(c*x - 1) + 20*b*c*d*e^3 + 15*(c*x + 1)^4*a*e^4/(c*x - 1)^4 + 30*(c*x + 1)^2*a*e^4/(c*x - 1)^2 +
3*a*e^4 + 6*(c*x + 1)^4*b*e^4/(c*x - 1)^4 - 12*(c*x + 1)^3*b*e^4/(c*x - 1)^3 + 12*(c*x + 1)^2*b*e^4/(c*x - 1)^
2 - 6*(c*x + 1)*b*e^4/(c*x - 1))/((c*x + 1)^5*c^6/(c*x - 1)^5 - 5*(c*x + 1)^4*c^6/(c*x - 1)^4 + 10*(c*x + 1)^3
*c^6/(c*x - 1)^3 - 10*(c*x + 1)^2*c^6/(c*x - 1)^2 + 5*(c*x + 1)*c^6/(c*x - 1) - c^6) - 3*(5*b*c^4*d^4 + 10*b*c
^2*d^2*e^2 + b*e^4)*log(-(c*x + 1)/(c*x - 1) + 1)/c^6 + 3*(5*b*c^4*d^4 + 10*b*c^2*d^2*e^2 + b*e^4)*log(-(c*x +
1)/(c*x - 1))/c^6)

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Mupad [B]
time = 1.70, size = 272, normalized size = 1.83 \begin {gather*} \frac {a\,e^4\,x^5}{5}+a\,d^4\,x+\frac {b\,d^4\,\ln \left (c^2\,x^2-1\right )}{2\,c}+\frac {b\,e^4\,\ln \left (c^2\,x^2-1\right )}{10\,c^5}+2\,a\,d^2\,e^2\,x^3+\frac {b\,e^4\,x^4}{20\,c}+\frac {b\,e^4\,x^2}{10\,c^3}+b\,d^4\,x\,\mathrm {atanh}\left (c\,x\right )+2\,a\,d^3\,e\,x^2+a\,d\,e^3\,x^4+\frac {b\,e^4\,x^5\,\mathrm {atanh}\left (c\,x\right )}{5}+\frac {2\,b\,d^3\,e\,x}{c}+\frac {b\,d\,e^3\,x}{c^3}-\frac {2\,b\,d^3\,e\,\mathrm {atanh}\left (c\,x\right )}{c^2}-\frac {b\,d\,e^3\,\mathrm {atanh}\left (c\,x\right )}{c^4}+2\,b\,d^3\,e\,x^2\,\mathrm {atanh}\left (c\,x\right )+b\,d\,e^3\,x^4\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,d\,e^3\,x^3}{3\,c}+2\,b\,d^2\,e^2\,x^3\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,d^2\,e^2\,\ln \left (c^2\,x^2-1\right )}{c^3}+\frac {b\,d^2\,e^2\,x^2}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))*(d + e*x)^4,x)

[Out]

(a*e^4*x^5)/5 + a*d^4*x + (b*d^4*log(c^2*x^2 - 1))/(2*c) + (b*e^4*log(c^2*x^2 - 1))/(10*c^5) + 2*a*d^2*e^2*x^3
+ (b*e^4*x^4)/(20*c) + (b*e^4*x^2)/(10*c^3) + b*d^4*x*atanh(c*x) + 2*a*d^3*e*x^2 + a*d*e^3*x^4 + (b*e^4*x^5*a
tanh(c*x))/5 + (2*b*d^3*e*x)/c + (b*d*e^3*x)/c^3 - (2*b*d^3*e*atanh(c*x))/c^2 - (b*d*e^3*atanh(c*x))/c^4 + 2*b
*d^3*e*x^2*atanh(c*x) + b*d*e^3*x^4*atanh(c*x) + (b*d*e^3*x^3)/(3*c) + 2*b*d^2*e^2*x^3*atanh(c*x) + (b*d^2*e^2
*log(c^2*x^2 - 1))/c^3 + (b*d^2*e^2*x^2)/c

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