3.1.8 \(\int \frac {a+b \tanh ^{-1}(c x)}{(d+e x)^4} \, dx\) [8]

Optimal. Leaf size=175 \[ \frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (1+c x)}{6 (c d-e)^3 e}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6063, 724, 815} \begin {gather*} -\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (c x+1)}{6 e (c d-e)^3}+\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 724

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

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 173, normalized size = 0.99 \begin {gather*} \frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}+\frac {b c}{\left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {4 b c^3 d}{\left (-c^2 d^2+e^2\right )^2 (d+e x)}-\frac {2 b \tanh ^{-1}(c x)}{e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{e (c d+e)^3}+\frac {b c^3 \log (1+c x)}{(c d-e)^3 e}-\frac {2 b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{\left (c^2 d^2-e^2\right )^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.22, size = 227, normalized size = 1.30

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +d c \right )^{3} e}-\frac {b \,c^{4} \arctanh \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}+\frac {b \,c^{4} \ln \left (c x +1\right )}{6 e \left (d c -e \right )^{3}}-\frac {b \,c^{4} \ln \left (c x -1\right )}{6 e \left (d c +e \right )^{3}}+\frac {b \,c^{4}}{6 \left (d c +e \right ) \left (d c -e \right ) \left (c e x +d c \right )^{2}}-\frac {b \,c^{6} \ln \left (c e x +d c \right ) d^{2}}{\left (d c +e \right )^{3} \left (d c -e \right )^{3}}-\frac {b \,c^{4} e^{2} \ln \left (c e x +d c \right )}{3 \left (d c +e \right )^{3} \left (d c -e \right )^{3}}+\frac {2 b \,c^{5} d}{3 \left (d c +e \right )^{2} \left (d c -e \right )^{2} \left (c e x +d c \right )}}{c}\)	\(227\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +d c \right )^{3} e}-\frac {b \,c^{4} \arctanh \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}+\frac {b \,c^{4} \ln \left (c x +1\right )}{6 e \left (d c -e \right )^{3}}-\frac {b \,c^{4} \ln \left (c x -1\right )}{6 e \left (d c +e \right )^{3}}+\frac {b \,c^{4}}{6 \left (d c +e \right ) \left (d c -e \right ) \left (c e x +d c \right )^{2}}-\frac {b \,c^{6} \ln \left (c e x +d c \right ) d^{2}}{\left (d c +e \right )^{3} \left (d c -e \right )^{3}}-\frac {b \,c^{4} e^{2} \ln \left (c e x +d c \right )}{3 \left (d c +e \right )^{3} \left (d c -e \right )^{3}}+\frac {2 b \,c^{5} d}{3 \left (d c +e \right )^{2} \left (d c -e \right )^{2} \left (c e x +d c \right )}}{c}\)	\(227\)
risch	\(-\frac {b \ln \left (c x +1\right )}{6 e \left (e x +d \right )^{3}}+\frac {\ln \left (-c x -1\right ) b \,c^{6} d^{3} e^{3} x^{3}-\ln \left (-c x +1\right ) b \,c^{6} d^{3} e^{3} x^{3}+3 b \,c^{2} d^{2} e^{4} \ln \left (-c x +1\right )+\ln \left (-c x -1\right ) b \,c^{3} e^{6} x^{3}-2 \ln \left (e x +d \right ) b \,c^{3} e^{6} x^{3}+\ln \left (-c x +1\right ) b \,c^{3} e^{6} x^{3}+3 \ln \left (-c x -1\right ) b \,c^{5} d^{5} e -6 \ln \left (e x +d \right ) b \,c^{5} d^{5} e +3 \ln \left (-c x +1\right ) b \,c^{5} d^{5} e +3 \ln \left (-c x -1\right ) b \,c^{4} d^{4} e^{2}-6 \ln \left (-c x +1\right ) b \,c^{4} d^{4} e^{2}+\ln \left (-c x -1\right ) b \,c^{3} d^{3} e^{3}-2 \ln \left (e x +d \right ) b \,c^{3} d^{3} e^{3}+\ln \left (-c x +1\right ) b \,c^{3} d^{3} e^{3}+4 b \,c^{5} d^{3} e^{3} x^{2}+9 b \,c^{5} d^{4} e^{2} x -4 b \,c^{3} d \,e^{5} x^{2}-10 b \,c^{3} d^{2} e^{4} x +2 a \,e^{6}+\ln \left (-c x -1\right ) b \,c^{6} d^{6}+5 b \,c^{5} d^{5} e +6 a \,c^{4} d^{4} e^{2}-6 b \,c^{3} d^{3} e^{3}-6 a \,c^{2} d^{2} e^{4}+b c \,e^{6} x +b c d \,e^{5}-2 a \,c^{6} d^{6}+3 \ln \left (-c x -1\right ) b \,c^{6} d^{4} e^{2} x^{2}+3 \ln \left (-c x -1\right ) b \,c^{5} d^{2} e^{4} x^{3}-6 \ln \left (e x +d \right ) b \,c^{5} d^{2} e^{4} x^{3}-3 \ln \left (-c x +1\right ) b \,c^{6} d^{4} e^{2} x^{2}+3 \ln \left (-c x +1\right ) b \,c^{5} d^{2} e^{4} x^{3}+3 \ln \left (-c x -1\right ) b \,c^{6} d^{5} e x +9 \ln \left (-c x -1\right ) b \,c^{5} d^{3} e^{3} x^{2}+3 \ln \left (-c x -1\right ) b \,c^{4} d \,e^{5} x^{3}-18 \ln \left (e x +d \right ) b \,c^{5} d^{3} e^{3} x^{2}-3 \ln \left (-c x +1\right ) b \,c^{6} d^{5} e x +9 \ln \left (-c x +1\right ) b \,c^{5} d^{3} e^{3} x^{2}-3 \ln \left (-c x +1\right ) b \,c^{4} d \,e^{5} x^{3}+9 \ln \left (-c x -1\right ) b \,c^{5} d^{4} e^{2} x +9 \ln \left (-c x -1\right ) b \,c^{4} d^{2} e^{4} x^{2}-18 \ln \left (e x +d \right ) b \,c^{5} d^{4} e^{2} x +9 \ln \left (-c x +1\right ) b \,c^{5} d^{4} e^{2} x -9 \ln \left (-c x +1\right ) b \,c^{4} d^{2} e^{4} x^{2}+9 \ln \left (-c x -1\right ) b \,c^{4} d^{3} e^{3} x +3 \ln \left (-c x -1\right ) b \,c^{3} d \,e^{5} x^{2}-6 \ln \left (e x +d \right ) b \,c^{3} d \,e^{5} x^{2}-9 \ln \left (-c x +1\right ) b \,c^{4} d^{3} e^{3} x +3 \ln \left (-c x +1\right ) b \,c^{3} d \,e^{5} x^{2}+3 \ln \left (-c x -1\right ) b \,c^{3} d^{2} e^{4} x -6 \ln \left (e x +d \right ) b \,c^{3} d^{2} e^{4} x +3 \ln \left (-c x +1\right ) b \,c^{3} d^{2} e^{4} x -b \,e^{6} \ln \left (-c x +1\right )}{6 \left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \left (e x +d \right )^{3} e}\)	\(1020\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 322, normalized size = 1.84 \begin {gather*} \frac {1}{6} \, {\left ({\left (\frac {c^{2} \log \left (c x + 1\right )}{c^{3} d^{3} e - 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} - e^{4}} - \frac {c^{2} \log \left (c x - 1\right )}{c^{3} d^{3} e + 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} + e^{4}} - \frac {2 \, {\left (3 \, c^{4} d^{2} + c^{2} e^{2}\right )} \log \left (x e + d\right )}{c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} - e^{6}} + \frac {4 \, c^{2} d x e + 5 \, c^{2} d^{2} - e^{2}}{c^{4} d^{6} - 2 \, c^{2} d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + d^{2} e^{4} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e}\right )} b - \frac {a}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4030 vs. \(2 (161) = 322\).
time = 0.63, size = 4030, normalized size = 23.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a*c^6*d^6 - 5*b*c^5*d^5*cosh(1) - (b*c*x + 2*a)*cosh(1)^6 - (b*c*x + 2*a)*sinh(1)^6 + (4*b*c^3*d*x^2 -
b*c*d)*cosh(1)^5 + (4*b*c^3*d*x^2 - b*c*d - 6*(b*c*x + 2*a)*cosh(1))*sinh(1)^5 + 2*(5*b*c^3*d^2*x + 3*a*c^2*d
^2)*cosh(1)^4 + (10*b*c^3*d^2*x + 6*a*c^2*d^2 - 15*(b*c*x + 2*a)*cosh(1)^2 + 5*(4*b*c^3*d*x^2 - b*c*d)*cosh(1)
)*sinh(1)^4 - 2*(2*b*c^5*d^3*x^2 - 3*b*c^3*d^3)*cosh(1)^3 - 2*(2*b*c^5*d^3*x^2 - 3*b*c^3*d^3 + 10*(b*c*x + 2*a
)*cosh(1)^3 - 5*(4*b*c^3*d*x^2 - b*c*d)*cosh(1)^2 - 4*(5*b*c^3*d^2*x + 3*a*c^2*d^2)*cosh(1))*sinh(1)^3 - 3*(3*
b*c^5*d^4*x + 2*a*c^4*d^4)*cosh(1)^2 - (9*b*c^5*d^4*x + 6*a*c^4*d^4 + 15*(b*c*x + 2*a)*cosh(1)^4 - 10*(4*b*c^3
*d*x^2 - b*c*d)*cosh(1)^3 - 12*(5*b*c^3*d^2*x + 3*a*c^2*d^2)*cosh(1)^2 + 6*(2*b*c^5*d^3*x^2 - 3*b*c^3*d^3)*cos
h(1))*sinh(1)^2 - (b*c^6*d^6 + b*c^3*x^3*cosh(1)^6 + b*c^3*x^3*sinh(1)^6 + 3*(b*c^4*d*x^3 + b*c^3*d*x^2)*cosh(
1)^5 + 3*(b*c^4*d*x^3 + 2*b*c^3*x^3*cosh(1) + b*c^3*d*x^2)*sinh(1)^5 + 3*(b*c^5*d^2*x^3 + 3*b*c^4*d^2*x^2 + b*
c^3*d^2*x)*cosh(1)^4 + 3*(b*c^5*d^2*x^3 + 3*b*c^4*d^2*x^2 + 5*b*c^3*x^3*cosh(1)^2 + b*c^3*d^2*x + 5*(b*c^4*d*x
^3 + b*c^3*d*x^2)*cosh(1))*sinh(1)^4 + (b*c^6*d^3*x^3 + 9*b*c^5*d^3*x^2 + 9*b*c^4*d^3*x + b*c^3*d^3)*cosh(1)^3
+ (b*c^6*d^3*x^3 + 9*b*c^5*d^3*x^2 + 20*b*c^3*x^3*cosh(1)^3 + 9*b*c^4*d^3*x + b*c^3*d^3 + 30*(b*c^4*d*x^3 + b
*c^3*d*x^2)*cosh(1)^2 + 12*(b*c^5*d^2*x^3 + 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1))*sinh(1)^3 + 3*(b*c^6*d^4*x
^2 + 3*b*c^5*d^4*x + b*c^4*d^4)*cosh(1)^2 + 3*(b*c^6*d^4*x^2 + 3*b*c^5*d^4*x + 5*b*c^3*x^3*cosh(1)^4 + b*c^4*d
^4 + 10*(b*c^4*d*x^3 + b*c^3*d*x^2)*cosh(1)^3 + 6*(b*c^5*d^2*x^3 + 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1)^2 +
(b*c^6*d^3*x^3 + 9*b*c^5*d^3*x^2 + 9*b*c^4*d^3*x + b*c^3*d^3)*cosh(1))*sinh(1)^2 + 3*(b*c^6*d^5*x + b*c^5*d^5)
*cosh(1) + 3*(b*c^6*d^5*x + 2*b*c^3*x^3*cosh(1)^5 + b*c^5*d^5 + 5*(b*c^4*d*x^3 + b*c^3*d*x^2)*cosh(1)^4 + 4*(b
*c^5*d^2*x^3 + 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1)^3 + (b*c^6*d^3*x^3 + 9*b*c^5*d^3*x^2 + 9*b*c^4*d^3*x + b
*c^3*d^3)*cosh(1)^2 + 2*(b*c^6*d^4*x^2 + 3*b*c^5*d^4*x + b*c^4*d^4)*cosh(1))*sinh(1))*log(c*x + 1) + (b*c^6*d^
6 - b*c^3*x^3*cosh(1)^6 - b*c^3*x^3*sinh(1)^6 + 3*(b*c^4*d*x^3 - b*c^3*d*x^2)*cosh(1)^5 + 3*(b*c^4*d*x^3 - 2*b
*c^3*x^3*cosh(1) - b*c^3*d*x^2)*sinh(1)^5 - 3*(b*c^5*d^2*x^3 - 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1)^4 - 3*(b
*c^5*d^2*x^3 - 3*b*c^4*d^2*x^2 + 5*b*c^3*x^3*cosh(1)^2 + b*c^3*d^2*x - 5*(b*c^4*d*x^3 - b*c^3*d*x^2)*cosh(1))*
sinh(1)^4 + (b*c^6*d^3*x^3 - 9*b*c^5*d^3*x^2 + 9*b*c^4*d^3*x - b*c^3*d^3)*cosh(1)^3 + (b*c^6*d^3*x^3 - 9*b*c^5
*d^3*x^2 - 20*b*c^3*x^3*cosh(1)^3 + 9*b*c^4*d^3*x - b*c^3*d^3 + 30*(b*c^4*d*x^3 - b*c^3*d*x^2)*cosh(1)^2 - 12*
(b*c^5*d^2*x^3 - 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1))*sinh(1)^3 + 3*(b*c^6*d^4*x^2 - 3*b*c^5*d^4*x + b*c^4*
d^4)*cosh(1)^2 + 3*(b*c^6*d^4*x^2 - 3*b*c^5*d^4*x - 5*b*c^3*x^3*cosh(1)^4 + b*c^4*d^4 + 10*(b*c^4*d*x^3 - b*c^
3*d*x^2)*cosh(1)^3 - 6*(b*c^5*d^2*x^3 - 3*b*c^4*d^2*x^2 + b*c^3*d^2*x)*cosh(1)^2 + (b*c^6*d^3*x^3 - 9*b*c^5*d^
3*x^2 + 9*b*c^4*d^3*x - b*c^3*d^3)*cosh(1))*sinh(1)^2 + 3*(b*c^6*d^5*x - b*c^5*d^5)*cosh(1) + 3*(b*c^6*d^5*x -
2*b*c^3*x^3*cosh(1)^5 - b*c^5*d^5 + 5*(b*c^4*d*x^3 - b*c^3*d*x^2)*cosh(1)^4 - 4*(b*c^5*d^2*x^3 - 3*b*c^4*d^2*
x^2 + b*c^3*d^2*x)*cosh(1)^3 + (b*c^6*d^3*x^3 - 9*b*c^5*d^3*x^2 + 9*b*c^4*d^3*x - b*c^3*d^3)*cosh(1)^2 + 2*(b*
c^6*d^4*x^2 - 3*b*c^5*d^4*x + b*c^4*d^4)*cosh(1))*sinh(1))*log(c*x - 1) + 2*(9*b*c^5*d^4*x*cosh(1)^2 + b*c^3*x
^3*cosh(1)^6 + b*c^3*x^3*sinh(1)^6 + 3*b*c^5*d^5*cosh(1) + 3*b*c^3*d*x^2*cosh(1)^5 + 3*(2*b*c^3*x^3*cosh(1) +
b*c^3*d*x^2)*sinh(1)^5 + 3*(b*c^5*d^2*x^3 + b*c^3*d^2*x)*cosh(1)^4 + 3*(b*c^5*d^2*x^3 + 5*b*c^3*x^3*cosh(1)^2
+ 5*b*c^3*d*x^2*cosh(1) + b*c^3*d^2*x)*sinh(1)^4 + (9*b*c^5*d^3*x^2 + b*c^3*d^3)*cosh(1)^3 + (9*b*c^5*d^3*x^2
+ 20*b*c^3*x^3*cosh(1)^3 + 30*b*c^3*d*x^2*cosh(1)^2 + b*c^3*d^3 + 12*(b*c^5*d^2*x^3 + b*c^3*d^2*x)*cosh(1))*si
nh(1)^3 + 3*(3*b*c^5*d^4*x + 5*b*c^3*x^3*cosh(1)^4 + 10*b*c^3*d*x^2*cosh(1)^3 + 6*(b*c^5*d^2*x^3 + b*c^3*d^2*x
)*cosh(1)^2 + (9*b*c^5*d^3*x^2 + b*c^3*d^3)*cosh(1))*sinh(1)^2 + 3*(6*b*c^5*d^4*x*cosh(1) + 2*b*c^3*x^3*cosh(1
)^5 + b*c^5*d^5 + 5*b*c^3*d*x^2*cosh(1)^4 + 4*(b*c^5*d^2*x^3 + b*c^3*d^2*x)*cosh(1)^3 + (9*b*c^5*d^3*x^2 + b*c
^3*d^3)*cosh(1)^2)*sinh(1))*log(x*cosh(1) + x*sinh(1) + d) + (b*c^6*d^6 - 3*b*c^4*d^4*cosh(1)^2 + 3*b*c^2*d^2*
cosh(1)^4 - b*cosh(1)^6 - 6*b*cosh(1)*sinh(1)^5 - b*sinh(1)^6 + 3*(b*c^2*d^2 - 5*b*cosh(1)^2)*sinh(1)^4 + 4*(3
*b*c^2*d^2*cosh(1) - 5*b*cosh(1)^3)*sinh(1)^3 - 3*(b*c^4*d^4 - 6*b*c^2*d^2*cosh(1)^2 + 5*b*cosh(1)^4)*sinh(1)^
2 - 6*(b*c^4*d^4*cosh(1) - 2*b*c^2*d^2*cosh(1)^3 + b*cosh(1)^5)*sinh(1))*log(-(c*x + 1)/(c*x - 1)) - (5*b*c^5*
d^5 + 6*(b*c*x + 2*a)*cosh(1)^5 - 5*(4*b*c^3*d*x^2 - b*c*d)*cosh(1)^4 - 8*(5*b*c^3*d^2*x + 3*a*c^2*d^2)*cosh(1
)^3 + 6*(2*b*c^5*d^3*x^2 - 3*b*c^3*d^3)*cosh(1)^2 + 6*(3*b*c^5*d^4*x + 2*a*c^4*d^4)*cosh(1))*sinh(1))/(3*c^6*d
^8*x*cosh(1)^2 + c^6*d^9*cosh(1) - x^3*cosh(1)^10 - x^3*sinh(1)^10 - 3*d*x^2*cosh(1)^9 - (10*x^3*cosh(1) + 3*d
*x^2)*sinh(1)^9 + 3*(c^2*d^2*x^3 - d^2*x)*cosh(...

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10946 vs. \(2 (151) = 302\).
time = 5.32, size = 10946, normalized size = 62.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x/d**4, Eq(c, 0) & Eq(e, 0)), (-a/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3), Eq(c,
0)), ((a*x + b*x*atanh(c*x) + b*log(x - 1/c)/c + b*atanh(c*x)/c)/d**4, Eq(e, 0)), (-24*a*d**3/(72*d**6*e + 21
6*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 21*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x
+ 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 10*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d
**3*e**4*x**3) - 9*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x*
*3) + 9*b*d**2*e*x/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d*e**2*x**2*at
anh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 3*b*d*e**2*x**2/(72*d**6*e
+ 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 3*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 216*d**
5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c, -e/d)), (-24*a*d**3/(72*d**6*e + 216*d**5*e**2*x + 2
16*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 21*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x
**2 + 72*d**3*e**4*x**3) - 10*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) +
9*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d**2*e*
x/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d*e**2*x**2*atanh(e*x/d)/(72*d*
*6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 3*b*d*e**2*x**2/(72*d**6*e + 216*d**5*e**2*
x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 3*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d*
*4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c, e/d)), (-2*a*c**6*d**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6
*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 1
8*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x
**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*a*c**4*d**4*e**2/(6*c**6*d**9*e + 18*c
**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 5
4*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x
**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*a*c**2*d**2*e
**4/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3
- 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**
6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**1
0*x**3) + 2*a*e**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*
c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 5
4*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**
9*x**2 - 6*e**10*x**3) + 6*b*c**6*d**5*e*x*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3
*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**
4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d*
*3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**6*d**4*e**2*x**2*atanh(c*x)/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4
*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*
e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**6*
d**3*e**3*x**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**
3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e
**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 1
8*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**5*d**5*e*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d*
*7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c
**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3
- 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**5*d**5*e*log(d/e + x)/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**
4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3
*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**5
*d**5*e*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1792 vs. \(2 (162) = 324\).
time = 0.44, size = 1792, normalized size = 10.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c*((3*b*c^4*d^2 + b*c^2*e^2)*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c^6*d^6 - 3
*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6) - (3*(c*x + 1)^2*b*c^4*d^2/(c*x - 1)^2 - 6*(c*x + 1)*b*c^4*d^2/(c*x - 1) +
3*b*c^4*d^2 + 6*(c*x + 1)^2*b*c^3*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c^3*d*e/(c*x - 1) + 3*(c*x + 1)^2*b*c^2*e^2
/(c*x - 1)^2 + b*c^2*e^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^6*d^6/(c*x - 1)^3 - 3*(c*x + 1)^2*c^6*d^6/(
c*x - 1)^2 + 3*(c*x + 1)*c^6*d^6/(c*x - 1) - c^6*d^6 + 6*(c*x + 1)^3*c^5*d^5*e/(c*x - 1)^3 - 12*(c*x + 1)^2*c^
5*d^5*e/(c*x - 1)^2 + 6*(c*x + 1)*c^5*d^5*e/(c*x - 1) + 15*(c*x + 1)^3*c^4*d^4*e^2/(c*x - 1)^3 - 15*(c*x + 1)^
2*c^4*d^4*e^2/(c*x - 1)^2 - 3*(c*x + 1)*c^4*d^4*e^2/(c*x - 1) + 3*c^4*d^4*e^2 + 20*(c*x + 1)^3*c^3*d^3*e^3/(c*
x - 1)^3 - 12*(c*x + 1)*c^3*d^3*e^3/(c*x - 1) + 15*(c*x + 1)^3*c^2*d^2*e^4/(c*x - 1)^3 + 15*(c*x + 1)^2*c^2*d^
2*e^4/(c*x - 1)^2 - 3*(c*x + 1)*c^2*d^2*e^4/(c*x - 1) - 3*c^2*d^2*e^4 + 6*(c*x + 1)^3*c*d*e^5/(c*x - 1)^3 + 12
*(c*x + 1)^2*c*d*e^5/(c*x - 1)^2 + 6*(c*x + 1)*c*d*e^5/(c*x - 1) + (c*x + 1)^3*e^6/(c*x - 1)^3 + 3*(c*x + 1)^2
*e^6/(c*x - 1)^2 + 3*(c*x + 1)*e^6/(c*x - 1) + e^6) - (3*b*c^4*d^2 + b*c^2*e^2)*log(-(c*x + 1)/(c*x - 1))/(c^6
*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6) - 2*(3*(c*x + 1)^2*a*c^6*d^4/(c*x - 1)^2 - 6*(c*x + 1)*a*c^6*d^4/(
c*x - 1) + 3*a*c^6*d^4 + 6*(c*x + 1)*a*c^5*d^3*e/(c*x - 1) - 6*a*c^5*d^3*e - 3*(c*x + 1)^2*b*c^5*d^3*e/(c*x -
1)^2 + 6*(c*x + 1)*b*c^5*d^3*e/(c*x - 1) - 3*b*c^5*d^3*e - 6*(c*x + 1)^2*a*c^4*d^2*e^2/(c*x - 1)^2 + 6*(c*x +
1)*a*c^4*d^2*e^2/(c*x - 1) + 4*a*c^4*d^2*e^2 - 5*(c*x + 1)^2*b*c^4*d^2*e^2/(c*x - 1)^2 - (c*x + 1)*b*c^4*d^2*e
^2/(c*x - 1) + 6*b*c^4*d^2*e^2 - 6*(c*x + 1)*a*c^3*d*e^3/(c*x - 1) - 2*a*c^3*d*e^3 - (c*x + 1)^2*b*c^3*d*e^3/(
c*x - 1)^2 - 6*(c*x + 1)*b*c^3*d*e^3/(c*x - 1) - 3*b*c^3*d*e^3 + 3*(c*x + 1)^2*a*c^2*e^4/(c*x - 1)^2 + a*c^2*e
^4 + (c*x + 1)^2*b*c^2*e^4/(c*x - 1)^2 + (c*x + 1)*b*c^2*e^4/(c*x - 1))/((c*x + 1)^3*c^8*d^8/(c*x - 1)^3 - 3*(
c*x + 1)^2*c^8*d^8/(c*x - 1)^2 + 3*(c*x + 1)*c^8*d^8/(c*x - 1) - c^8*d^8 + 4*(c*x + 1)^3*c^7*d^7*e/(c*x - 1)^3
- 6*(c*x + 1)^2*c^7*d^7*e/(c*x - 1)^2 + 2*c^7*d^7*e + 4*(c*x + 1)^3*c^6*d^6*e^2/(c*x - 1)^3 + 6*(c*x + 1)^2*c
^6*d^6*e^2/(c*x - 1)^2 - 12*(c*x + 1)*c^6*d^6*e^2/(c*x - 1) + 2*c^6*d^6*e^2 - 4*(c*x + 1)^3*c^5*d^5*e^3/(c*x -
1)^3 + 18*(c*x + 1)^2*c^5*d^5*e^3/(c*x - 1)^2 - 6*c^5*d^5*e^3 - 10*(c*x + 1)^3*c^4*d^4*e^4/(c*x - 1)^3 + 18*(
c*x + 1)*c^4*d^4*e^4/(c*x - 1) - 4*(c*x + 1)^3*c^3*d^3*e^5/(c*x - 1)^3 - 18*(c*x + 1)^2*c^3*d^3*e^5/(c*x - 1)^
2 + 6*c^3*d^3*e^5 + 4*(c*x + 1)^3*c^2*d^2*e^6/(c*x - 1)^3 - 6*(c*x + 1)^2*c^2*d^2*e^6/(c*x - 1)^2 - 12*(c*x +
1)*c^2*d^2*e^6/(c*x - 1) - 2*c^2*d^2*e^6 + 4*(c*x + 1)^3*c*d*e^7/(c*x - 1)^3 + 6*(c*x + 1)^2*c*d*e^7/(c*x - 1)
^2 - 2*c*d*e^7 + (c*x + 1)^3*e^8/(c*x - 1)^3 + 3*(c*x + 1)^2*e^8/(c*x - 1)^2 + 3*(c*x + 1)*e^8/(c*x - 1) + e^8
))

________________________________________________________________________________________

Mupad [B]
time = 2.29, size = 418, normalized size = 2.39 \begin {gather*} \ln \left (d+e\,x\right )\,\left (\frac {b\,c^3}{6\,e\,{\left (e+c\,d\right )}^3}+\frac {b\,c^3}{6\,e\,{\left (e-c\,d\right )}^3}\right )-\frac {\frac {2\,a\,c^4\,d^4-5\,b\,c^3\,d^3\,e-4\,a\,c^2\,d^2\,e^2+b\,c\,d\,e^3+2\,a\,e^4}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}+\frac {x\,\left (b\,c\,e^4-9\,b\,c^3\,d^2\,e^2\right )}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}-\frac {2\,b\,c^3\,d\,e^3\,x^2}{c^4\,d^4-2\,c^2\,d^2\,e^2+e^4}}{3\,d^3\,e+9\,d^2\,e^2\,x+9\,d\,e^3\,x^2+3\,e^4\,x^3}-\frac {b\,c^3\,\ln \left (c\,x-1\right )}{6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2+18\,c\,d\,e^3+6\,e^4}-\frac {b\,c^3\,\ln \left (c\,x+1\right )}{-6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2-18\,c\,d\,e^3+6\,e^4}-\frac {b\,\ln \left (c\,x+1\right )}{6\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )}+\frac {b\,\ln \left (1-c\,x\right )}{3\,e\,\left (2\,d^3+6\,d^2\,e\,x+6\,d\,e^2\,x^2+2\,e^3\,x^3\right )} \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]

log(d + e*x)*((b*c^3)/(6*e*(e + c*d)^3) + (b*c^3)/(6*e*(e - c*d)^3)) - ((2*a*e^4 + 2*a*c^4*d^4 - 4*a*c^2*d^2*e
^2 + b*c*d*e^3 - 5*b*c^3*d^3*e)/(2*(e^4 + c^4*d^4 - 2*c^2*d^2*e^2)) + (x*(b*c*e^4 - 9*b*c^3*d^2*e^2))/(2*(e^4
+ c^4*d^4 - 2*c^2*d^2*e^2)) - (2*b*c^3*d*e^3*x^2)/(e^4 + c^4*d^4 - 2*c^2*d^2*e^2))/(3*d^3*e + 3*e^4*x^3 + 9*d^
2*e^2*x + 9*d*e^3*x^2) - (b*c^3*log(c*x - 1))/(6*e^4 + 6*c^3*d^3*e + 18*c^2*d^2*e^2 + 18*c*d*e^3) - (b*c^3*log
(c*x + 1))/(6*e^4 - 6*c^3*d^3*e + 18*c^2*d^2*e^2 - 18*c*d*e^3) - (b*log(c*x + 1))/(6*e*(d^3 + e^3*x^3 + 3*d*e^
2*x^2 + 3*d^2*e*x)) + (b*log(1 - c*x))/(3*e*(2*d^3 + 2*e^3*x^3 + 6*d*e^2*x^2 + 6*d^2*e*x))

________________________________________________________________________________________