3.1.37 \(\int \frac {a+b \tanh ^{-1}(c+d x)}{(e+f x)^3} \, dx\) [37]

Optimal. Leaf size=167 \[ \frac {b d}{2 (d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac {a+b \tanh ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 \log (1-c-d x)}{4 f (d e+f-c f)^2}+\frac {b d^2 \log (1+c+d x)}{4 f (d e-f-c f)^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6244, 2007, 723, 814} \begin {gather*} -\frac {a+b \tanh ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 \log (-c-d x+1)}{4 f (-c f+d e+f)^2}+\frac {b d^2 \log (c+d x+1)}{4 f (-c f+d e-f)^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b d}{2 (e+f x) (-c f+d e+f) (d e-(c+1) f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2007

Int[(u_)^(m_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 6244

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*((a + b*ArcTanh[c + d*x])^p/(f*(m + 1))), x] - Dist[b*d*(p/(f*(m + 1))), Int[(e + f*x)^(m + 1)*((a + b*A
rcTanh[c + d*x])^(p - 1)/(1 - (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -
1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 174, normalized size = 1.04 \begin {gather*} \frac {1}{4} \left (-\frac {2 a}{f (e+f x)^2}+\frac {2 b d}{\left (d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2\right ) (e+f x)}-\frac {2 b \tanh ^{-1}(c+d x)}{f (e+f x)^2}-\frac {b d^2 \log (1-c-d x)}{f (d e+f-c f)^2}+\frac {b d^2 \log (1+c+d x)}{f (-d e+f+c f)^2}-\frac {4 b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.84, size = 266, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {b \,d^{3} \arctanh \left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}+\frac {b \,d^{3} \ln \left (d x +c +1\right )}{4 f \left (c f -d e +f \right )^{2}}-\frac {b \,d^{3}}{2 \left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {b \,d^{3} f \ln \left (c f -d e -f \left (d x +c \right )\right ) c}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {b \,d^{4} \ln \left (c f -d e -f \left (d x +c \right )\right ) e}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {b \,d^{3} \ln \left (d x +c -1\right )}{4 f \left (c f -d e -f \right )^{2}}}{d}\)	\(266\)
default	\(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {b \,d^{3} \arctanh \left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}+\frac {b \,d^{3} \ln \left (d x +c +1\right )}{4 f \left (c f -d e +f \right )^{2}}-\frac {b \,d^{3}}{2 \left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {b \,d^{3} f \ln \left (c f -d e -f \left (d x +c \right )\right ) c}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {b \,d^{4} \ln \left (c f -d e -f \left (d x +c \right )\right ) e}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {b \,d^{3} \ln \left (d x +c -1\right )}{4 f \left (c f -d e -f \right )^{2}}}{d}\)	\(266\)
risch	\(-\frac {b \ln \left (d x +c +1\right )}{4 f \left (f x +e \right )^{2}}-\frac {b \,d^{4} e^{2} f^{2} x^{2} \ln \left (-d x -c +1\right )+2 b \,d^{4} e^{3} f x \ln \left (-d x -c +1\right )+2 \ln \left (-d x -c +1\right ) b c \,d^{2} e^{2} f^{2}-4 \ln \left (-d x -c +1\right ) b c d e \,f^{3}+4 b c \,d^{2} e \,f^{3} x +4 \ln \left (-d x -c +1\right ) b \,c^{3} d e \,f^{3}-5 \ln \left (-d x -c +1\right ) b \,c^{2} d^{2} e^{2} f^{2}+2 \ln \left (-d x -c +1\right ) b c \,d^{3} e^{3} f -2 \ln \left (-d x -c +1\right ) b \,d^{3} e^{3} f +3 \ln \left (-d x -c +1\right ) b \,d^{2} e^{2} f^{2}-2 b \,c^{2} d \,f^{4} x -2 b \,d^{3} e^{2} f^{2} x -2 b \,c^{2} d e \,f^{3}+4 b c \,d^{2} e^{2} f^{2}-\ln \left (-d x -c +1\right ) b \,f^{4}+2 b d \,f^{4} x -\ln \left (-d x -c +1\right ) b \,c^{4} f^{4}+2 \ln \left (-d x -c +1\right ) b \,c^{2} f^{4}-2 b \,d^{3} e^{3} f +2 b d e \,f^{3}+2 a \,f^{4}+2 a \,c^{4} f^{4}+2 a \,d^{4} e^{4}-4 a \,c^{2} f^{4}-8 a \,c^{3} d e \,f^{3}+12 a \,c^{2} d^{2} e^{2} f^{2}-8 a c \,d^{3} e^{3} f +8 a c d e \,f^{3}-4 a \,d^{2} e^{2} f^{2}+8 \ln \left (f x +e \right ) b \,d^{3} e^{2} f^{2} x +2 \ln \left (-d x -c +1\right ) b \,d^{2} e \,f^{3} x +2 \ln \left (-d x -c -1\right ) b c \,d^{2} e^{2} f^{2}-2 \ln \left (-d x -c -1\right ) b \,d^{2} e \,f^{3} x -4 \ln \left (f x +e \right ) b c \,d^{2} e^{2} f^{2}-\ln \left (-d x -c -1\right ) b \,d^{4} e^{4}-\ln \left (-d x -c -1\right ) b \,d^{2} e^{2} f^{2}+\ln \left (-d x -c +1\right ) b \,d^{2} f^{4} x^{2}-\ln \left (-d x -c -1\right ) b \,d^{2} f^{4} x^{2}-2 \ln \left (-d x -c -1\right ) b \,d^{3} e^{3} f +4 \ln \left (f x +e \right ) b \,d^{3} e^{3} f +\ln \left (-d x -c +1\right ) b \,c^{2} d^{2} f^{4} x^{2}-\ln \left (-d x -c -1\right ) b \,c^{2} d^{2} f^{4} x^{2}-\ln \left (-d x -c -1\right ) b \,d^{4} e^{2} f^{2} x^{2}+2 \ln \left (-d x -c +1\right ) b c \,d^{2} f^{4} x^{2}-2 \ln \left (-d x -c +1\right ) b \,d^{3} e \,f^{3} x^{2}+2 \ln \left (-d x -c -1\right ) b c \,d^{2} f^{4} x^{2}-2 \ln \left (-d x -c -1\right ) b \,d^{4} e^{3} f x -2 \ln \left (-d x -c -1\right ) b \,d^{3} e \,f^{3} x^{2}-4 \ln \left (f x +e \right ) b c \,d^{2} f^{4} x^{2}+4 \ln \left (f x +e \right ) b \,d^{3} e \,f^{3} x^{2}-4 \ln \left (-d x -c +1\right ) b \,d^{3} e^{2} f^{2} x -\ln \left (-d x -c -1\right ) b \,c^{2} d^{2} e^{2} f^{2}+2 \ln \left (-d x -c -1\right ) b c \,d^{3} e^{3} f -4 \ln \left (-d x -c -1\right ) b \,d^{3} e^{2} f^{2} x -8 \ln \left (f x +e \right ) b c \,d^{2} e \,f^{3} x -2 \ln \left (-d x -c +1\right ) b c \,d^{3} e \,f^{3} x^{2}+2 \ln \left (-d x -c -1\right ) b c \,d^{3} e \,f^{3} x^{2}+2 \ln \left (-d x -c +1\right ) b \,c^{2} d^{2} e \,f^{3} x -4 \ln \left (-d x -c +1\right ) b c \,d^{3} e^{2} f^{2} x -2 \ln \left (-d x -c -1\right ) b \,c^{2} d^{2} e \,f^{3} x +4 \ln \left (-d x -c -1\right ) b c \,d^{3} e^{2} f^{2} x +4 \ln \left (-d x -c +1\right ) b c \,d^{2} e \,f^{3} x +4 \ln \left (-d x -c -1\right ) b c \,d^{2} e \,f^{3} x}{4 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+2 c \,f^{2}-2 d e f +f^{2}\right ) \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-2 c \,f^{2}+2 d e f +f^{2}\right ) f \left (f x +e \right )^{2}}\)	\(1282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 316, normalized size = 1.89 \begin {gather*} -\frac {1}{4} \, {\left (d {\left (\frac {d \log \left (d x + c + 1\right )}{2 \, {\left (c e + e\right )} d f^{2} - {\left (c^{2} + 2 \, c + 1\right )} f^{3} - d^{2} f e^{2}} - \frac {d \log \left (d x + c - 1\right )}{2 \, {\left (c e - e\right )} d f^{2} - {\left (c^{2} - 2 \, c + 1\right )} f^{3} - d^{2} f e^{2}} + \frac {4 \, {\left (c d f - d^{2} e\right )} \log \left (f x + e\right )}{4 \, c d^{3} f e^{3} - 2 \, {\left (3 \, c^{2} e^{2} - e^{2}\right )} d^{2} f^{2} + 4 \, {\left (c^{3} e - c e\right )} d f^{3} - {\left (c^{4} - 2 \, c^{2} + 1\right )} f^{4} - d^{4} e^{4}} + \frac {2}{2 \, c d f e^{2} - {\left (c^{2} e - e\right )} f^{2} - d^{2} e^{3} + {\left (2 \, c d f^{2} e - {\left (c^{2} - 1\right )} f^{3} - d^{2} f e^{2}\right )} x}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{f^{3} x^{2} + 2 \, f^{2} x e + f e^{2}}\right )} b - \frac {a}{2 \, {\left (f^{3} x^{2} + 2 \, f^{2} x e + f e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2443 vs. \(2 (171) = 342\).
time = 1.09, size = 2443, normalized size = 14.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*d^4*cosh(1)^4 + 2*a*d^4*sinh(1)^4 - 2*(4*a*c + b)*d^3*f*cosh(1)^3 - 2*(b*c^2 - b)*d*f^4*x + 2*(a*c^4
- 2*a*c^2 + a)*f^4 + 2*(4*a*d^4*cosh(1) - (4*a*c + b)*d^3*f)*sinh(1)^3 - 2*(b*d^3*f^2*x - 2*(3*a*c^2 + b*c -
a)*d^2*f^2)*cosh(1)^2 - 2*(b*d^3*f^2*x - 6*a*d^4*cosh(1)^2 + 3*(4*a*c + b)*d^3*f*cosh(1) - 2*(3*a*c^2 + b*c -
a)*d^2*f^2)*sinh(1)^2 + 2*(2*b*c*d^2*f^3*x - (4*a*c^3 + b*c^2 - 4*a*c - b)*d*f^3)*cosh(1) - ((b*c^2 - 2*b*c +
b)*d^2*f^4*x^2 + b*d^4*cosh(1)^4 + b*d^4*sinh(1)^4 + 2*(b*d^4*f*x - (b*c - b)*d^3*f)*cosh(1)^3 + 2*(b*d^4*f*x
+ 2*b*d^4*cosh(1) - (b*c - b)*d^3*f)*sinh(1)^3 + (b*d^4*f^2*x^2 - 4*(b*c - b)*d^3*f^2*x + (b*c^2 - 2*b*c + b)*
d^2*f^2)*cosh(1)^2 + (b*d^4*f^2*x^2 - 4*(b*c - b)*d^3*f^2*x + 6*b*d^4*cosh(1)^2 + (b*c^2 - 2*b*c + b)*d^2*f^2
+ 6*(b*d^4*f*x - (b*c - b)*d^3*f)*cosh(1))*sinh(1)^2 - 2*((b*c - b)*d^3*f^3*x^2 - (b*c^2 - 2*b*c + b)*d^2*f^3*
x)*cosh(1) - 2*((b*c - b)*d^3*f^3*x^2 - 2*b*d^4*cosh(1)^3 - (b*c^2 - 2*b*c + b)*d^2*f^3*x - 3*(b*d^4*f*x - (b*
c - b)*d^3*f)*cosh(1)^2 - (b*d^4*f^2*x^2 - 4*(b*c - b)*d^3*f^2*x + (b*c^2 - 2*b*c + b)*d^2*f^2)*cosh(1))*sinh(
1))*log(d*x + c + 1) + ((b*c^2 + 2*b*c + b)*d^2*f^4*x^2 + b*d^4*cosh(1)^4 + b*d^4*sinh(1)^4 + 2*(b*d^4*f*x - (
b*c + b)*d^3*f)*cosh(1)^3 + 2*(b*d^4*f*x + 2*b*d^4*cosh(1) - (b*c + b)*d^3*f)*sinh(1)^3 + (b*d^4*f^2*x^2 - 4*(
b*c + b)*d^3*f^2*x + (b*c^2 + 2*b*c + b)*d^2*f^2)*cosh(1)^2 + (b*d^4*f^2*x^2 - 4*(b*c + b)*d^3*f^2*x + 6*b*d^4
*cosh(1)^2 + (b*c^2 + 2*b*c + b)*d^2*f^2 + 6*(b*d^4*f*x - (b*c + b)*d^3*f)*cosh(1))*sinh(1)^2 - 2*((b*c + b)*d
^3*f^3*x^2 - (b*c^2 + 2*b*c + b)*d^2*f^3*x)*cosh(1) - 2*((b*c + b)*d^3*f^3*x^2 - 2*b*d^4*cosh(1)^3 - (b*c^2 +
2*b*c + b)*d^2*f^3*x - 3*(b*d^4*f*x - (b*c + b)*d^3*f)*cosh(1)^2 - (b*d^4*f^2*x^2 - 4*(b*c + b)*d^3*f^2*x + (b
*c^2 + 2*b*c + b)*d^2*f^2)*cosh(1))*sinh(1))*log(d*x + c - 1) - 4*(b*c*d^2*f^4*x^2 - b*d^3*f*cosh(1)^3 - b*d^3
*f*sinh(1)^3 - (2*b*d^3*f^2*x - b*c*d^2*f^2)*cosh(1)^2 - (2*b*d^3*f^2*x - b*c*d^2*f^2 + 3*b*d^3*f*cosh(1))*sin
h(1)^2 - (b*d^3*f^3*x^2 - 2*b*c*d^2*f^3*x)*cosh(1) - (b*d^3*f^3*x^2 - 2*b*c*d^2*f^3*x + 3*b*d^3*f*cosh(1)^2 +
2*(2*b*d^3*f^2*x - b*c*d^2*f^2)*cosh(1))*sinh(1))*log(f*x + cosh(1) + sinh(1)) - (4*b*c*d^3*f*cosh(1)^3 - b*d^
4*cosh(1)^4 - b*d^4*sinh(1)^4 - 2*(3*b*c^2 - b)*d^2*f^2*cosh(1)^2 + 4*(b*c^3 - b*c)*d*f^3*cosh(1) - (b*c^4 - 2
*b*c^2 + b)*f^4 + 4*(b*c*d^3*f - b*d^4*cosh(1))*sinh(1)^3 + 2*(6*b*c*d^3*f*cosh(1) - 3*b*d^4*cosh(1)^2 - (3*b*
c^2 - b)*d^2*f^2)*sinh(1)^2 + 4*(3*b*c*d^3*f*cosh(1)^2 - b*d^4*cosh(1)^3 - (3*b*c^2 - b)*d^2*f^2*cosh(1) + (b*
c^3 - b*c)*d*f^3)*sinh(1))*log(-(d*x + c + 1)/(d*x + c - 1)) + 2*(2*b*c*d^2*f^3*x + 4*a*d^4*cosh(1)^3 - 3*(4*a
*c + b)*d^3*f*cosh(1)^2 - (4*a*c^3 + b*c^2 - 4*a*c - b)*d*f^3 - 2*(b*d^3*f^2*x - 2*(3*a*c^2 + b*c - a)*d^2*f^2
)*cosh(1))*sinh(1))/(d^4*f*cosh(1)^6 + d^4*f*sinh(1)^6 + (c^4 - 2*c^2 + 1)*f^7*x^2 + 2*(d^4*f^2*x - 2*c*d^3*f^
2)*cosh(1)^5 + 2*(d^4*f^2*x - 2*c*d^3*f^2 + 3*d^4*f*cosh(1))*sinh(1)^5 + (d^4*f^3*x^2 - 8*c*d^3*f^3*x + 2*(3*c
^2 - 1)*d^2*f^3)*cosh(1)^4 + (d^4*f^3*x^2 - 8*c*d^3*f^3*x + 15*d^4*f*cosh(1)^2 + 2*(3*c^2 - 1)*d^2*f^3 + 10*(d
^4*f^2*x - 2*c*d^3*f^2)*cosh(1))*sinh(1)^4 - 4*(c*d^3*f^4*x^2 - (3*c^2 - 1)*d^2*f^4*x + (c^3 - c)*d*f^4)*cosh(
1)^3 - 4*(c*d^3*f^4*x^2 - (3*c^2 - 1)*d^2*f^4*x - 5*d^4*f*cosh(1)^3 + (c^3 - c)*d*f^4 - 5*(d^4*f^2*x - 2*c*d^3
*f^2)*cosh(1)^2 - (d^4*f^3*x^2 - 8*c*d^3*f^3*x + 2*(3*c^2 - 1)*d^2*f^3)*cosh(1))*sinh(1)^3 + (2*(3*c^2 - 1)*d^
2*f^5*x^2 - 8*(c^3 - c)*d*f^5*x + (c^4 - 2*c^2 + 1)*f^5)*cosh(1)^2 + (2*(3*c^2 - 1)*d^2*f^5*x^2 + 15*d^4*f*cos
h(1)^4 - 8*(c^3 - c)*d*f^5*x + (c^4 - 2*c^2 + 1)*f^5 + 20*(d^4*f^2*x - 2*c*d^3*f^2)*cosh(1)^3 + 6*(d^4*f^3*x^2
- 8*c*d^3*f^3*x + 2*(3*c^2 - 1)*d^2*f^3)*cosh(1)^2 - 12*(c*d^3*f^4*x^2 - (3*c^2 - 1)*d^2*f^4*x + (c^3 - c)*d*
f^4)*cosh(1))*sinh(1)^2 - 2*(2*(c^3 - c)*d*f^6*x^2 - (c^4 - 2*c^2 + 1)*f^6*x)*cosh(1) - 2*(2*(c^3 - c)*d*f^6*x
^2 - 3*d^4*f*cosh(1)^5 - (c^4 - 2*c^2 + 1)*f^6*x - 5*(d^4*f^2*x - 2*c*d^3*f^2)*cosh(1)^4 - 2*(d^4*f^3*x^2 - 8*
c*d^3*f^3*x + 2*(3*c^2 - 1)*d^2*f^3)*cosh(1)^3 + 6*(c*d^3*f^4*x^2 - (3*c^2 - 1)*d^2*f^4*x + (c^3 - c)*d*f^4)*c
osh(1)^2 - (2*(3*c^2 - 1)*d^2*f^5*x^2 - 8*(c^3 - c)*d*f^5*x + (c^4 - 2*c^2 + 1)*f^5)*cosh(1))*sinh(1))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19912 vs. \(2 (143) = 286\).
time = 12.57, size = 19912, normalized size = 119.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a*x + b*c*atanh(c + d*x)/d + b*x*atanh(c + d*x) + b*log(c/d + x + 1/d)/d - b*atanh(c + d*x)/d)/e**
3, Eq(f, 0)), (-(a + b*atanh(c))/(2*e**2*f + 4*e*f**2*x + 2*f**3*x**2), Eq(d, 0)), (-4*a*f**2/(8*e**2*f**3 + 1
6*e*f**4*x + 8*f**5*x**2) + b*d**2*e**2*atanh(d*e/f + d*x - 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) + 2*b
*d**2*e*f*x*atanh(d*e/f + d*x - 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) + b*d**2*f**2*x**2*atanh(d*e/f +
d*x - 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - b*d*e*f/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - b*d*f
**2*x/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - 4*b*f**2*atanh(d*e/f + d*x - 1)/(8*e**2*f**3 + 16*e*f**4*x +
8*f**5*x**2) - b*f**2/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2), Eq(c, (d*e - f)/f)), (-4*a*f**2/(8*e**2*f**3
+ 16*e*f**4*x + 8*f**5*x**2) + b*d**2*e**2*atanh(d*e/f + d*x + 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) +
2*b*d**2*e*f*x*atanh(d*e/f + d*x + 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) + b*d**2*f**2*x**2*atanh(d*e/
f + d*x + 1)/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - b*d*e*f/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - b
*d*f**2*x/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2) - 4*b*f**2*atanh(d*e/f + d*x + 1)/(8*e**2*f**3 + 16*e*f**4
*x + 8*f**5*x**2) + b*f**2/(8*e**2*f**3 + 16*e*f**4*x + 8*f**5*x**2), Eq(c, (d*e + f)/f)), (zoo*(a*x + b*c*ata
nh(c + d*x)/d + b*x*atanh(c + d*x) + b*log(c/d + x + 1/d)/d - b*atanh(c + d*x)/d), Eq(e, -f*x)), (-a*c**4*f**4
/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f**7*x**2 - 8*c**3*d*e**3*f**4 - 16*c**3*d*e**2*f**5*x - 8*c**3*
d*e*f**6*x**2 + 12*c**2*d**2*e**4*f**3 + 24*c**2*d**2*e**3*f**4*x + 12*c**2*d**2*e**2*f**5*x**2 - 4*c**2*e**2*
f**5 - 8*c**2*e*f**6*x - 4*c**2*f**7*x**2 - 8*c*d**3*e**5*f**2 - 16*c*d**3*e**4*f**3*x - 8*c*d**3*e**3*f**4*x*
*2 + 8*c*d*e**3*f**4 + 16*c*d*e**2*f**5*x + 8*c*d*e*f**6*x**2 + 2*d**4*e**6*f + 4*d**4*e**5*f**2*x + 2*d**4*e*
*4*f**3*x**2 - 4*d**2*e**4*f**3 - 8*d**2*e**3*f**4*x - 4*d**2*e**2*f**5*x**2 + 2*e**2*f**5 + 4*e*f**6*x + 2*f*
*7*x**2) + 4*a*c**3*d*e*f**3/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f**7*x**2 - 8*c**3*d*e**3*f**4 - 16*
c**3*d*e**2*f**5*x - 8*c**3*d*e*f**6*x**2 + 12*c**2*d**2*e**4*f**3 + 24*c**2*d**2*e**3*f**4*x + 12*c**2*d**2*e
**2*f**5*x**2 - 4*c**2*e**2*f**5 - 8*c**2*e*f**6*x - 4*c**2*f**7*x**2 - 8*c*d**3*e**5*f**2 - 16*c*d**3*e**4*f*
*3*x - 8*c*d**3*e**3*f**4*x**2 + 8*c*d*e**3*f**4 + 16*c*d*e**2*f**5*x + 8*c*d*e*f**6*x**2 + 2*d**4*e**6*f + 4*
d**4*e**5*f**2*x + 2*d**4*e**4*f**3*x**2 - 4*d**2*e**4*f**3 - 8*d**2*e**3*f**4*x - 4*d**2*e**2*f**5*x**2 + 2*e
**2*f**5 + 4*e*f**6*x + 2*f**7*x**2) - 6*a*c**2*d**2*e**2*f**2/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f*
*7*x**2 - 8*c**3*d*e**3*f**4 - 16*c**3*d*e**2*f**5*x - 8*c**3*d*e*f**6*x**2 + 12*c**2*d**2*e**4*f**3 + 24*c**2
*d**2*e**3*f**4*x + 12*c**2*d**2*e**2*f**5*x**2 - 4*c**2*e**2*f**5 - 8*c**2*e*f**6*x - 4*c**2*f**7*x**2 - 8*c*
d**3*e**5*f**2 - 16*c*d**3*e**4*f**3*x - 8*c*d**3*e**3*f**4*x**2 + 8*c*d*e**3*f**4 + 16*c*d*e**2*f**5*x + 8*c*
d*e*f**6*x**2 + 2*d**4*e**6*f + 4*d**4*e**5*f**2*x + 2*d**4*e**4*f**3*x**2 - 4*d**2*e**4*f**3 - 8*d**2*e**3*f*
*4*x - 4*d**2*e**2*f**5*x**2 + 2*e**2*f**5 + 4*e*f**6*x + 2*f**7*x**2) + 2*a*c**2*f**4/(2*c**4*e**2*f**5 + 4*c
**4*e*f**6*x + 2*c**4*f**7*x**2 - 8*c**3*d*e**3*f**4 - 16*c**3*d*e**2*f**5*x - 8*c**3*d*e*f**6*x**2 + 12*c**2*
d**2*e**4*f**3 + 24*c**2*d**2*e**3*f**4*x + 12*c**2*d**2*e**2*f**5*x**2 - 4*c**2*e**2*f**5 - 8*c**2*e*f**6*x -
4*c**2*f**7*x**2 - 8*c*d**3*e**5*f**2 - 16*c*d**3*e**4*f**3*x - 8*c*d**3*e**3*f**4*x**2 + 8*c*d*e**3*f**4 + 1
6*c*d*e**2*f**5*x + 8*c*d*e*f**6*x**2 + 2*d**4*e**6*f + 4*d**4*e**5*f**2*x + 2*d**4*e**4*f**3*x**2 - 4*d**2*e*
*4*f**3 - 8*d**2*e**3*f**4*x - 4*d**2*e**2*f**5*x**2 + 2*e**2*f**5 + 4*e*f**6*x + 2*f**7*x**2) + 4*a*c*d**3*e*
*3*f/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f**7*x**2 - 8*c**3*d*e**3*f**4 - 16*c**3*d*e**2*f**5*x - 8*c
**3*d*e*f**6*x**2 + 12*c**2*d**2*e**4*f**3 + 24*c**2*d**2*e**3*f**4*x + 12*c**2*d**2*e**2*f**5*x**2 - 4*c**2*e
**2*f**5 - 8*c**2*e*f**6*x - 4*c**2*f**7*x**2 - 8*c*d**3*e**5*f**2 - 16*c*d**3*e**4*f**3*x - 8*c*d**3*e**3*f**
4*x**2 + 8*c*d*e**3*f**4 + 16*c*d*e**2*f**5*x + 8*c*d*e*f**6*x**2 + 2*d**4*e**6*f + 4*d**4*e**5*f**2*x + 2*d**
4*e**4*f**3*x**2 - 4*d**2*e**4*f**3 - 8*d**2*e**3*f**4*x - 4*d**2*e**2*f**5*x**2 + 2*e**2*f**5 + 4*e*f**6*x +
2*f**7*x**2) - 4*a*c*d*e*f**3/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f**7*x**2 - 8*c**3*d*e**3*f**4 - 16
*c**3*d*e**2*f**5*x - 8*c**3*d*e*f**6*x**2 + 12*c**2*d**2*e**4*f**3 + 24*c**2*d**2*e**3*f**4*x + 12*c**2*d**2*
e**2*f**5*x**2 - 4*c**2*e**2*f**5 - 8*c**2*e*f**6*x - 4*c**2*f**7*x**2 - 8*c*d**3*e**5*f**2 - 16*c*d**3*e**4*f
**3*x - 8*c*d**3*e**3*f**4*x**2 + 8*c*d*e**3*f**4 + 16*c*d*e**2*f**5*x + 8*c*d*e*f**6*x**2 + 2*d**4*e**6*f + 4
*d**4*e**5*f**2*x + 2*d**4*e**4*f**3*x**2 - 4*d**2*e**4*f**3 - 8*d**2*e**3*f**4*x - 4*d**2*e**2*f**5*x**2 + 2*
e**2*f**5 + 4*e*f**6*x + 2*f**7*x**2) - a*d**4*e**4/(2*c**4*e**2*f**5 + 4*c**4*e*f**6*x + 2*c**4*f**7*x**2 - 8
*c**3*d*e**3*f**4 - 16*c**3*d*e**2*f**5*x - 8*c...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2567 vs. \(2 (160) = 320\).
time = 0.50, size = 2567, normalized size = 15.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 3.03, size = 417, normalized size = 2.50 \begin {gather*} \frac {b\,d^2\,\ln \left (c+d\,x+1\right )}{4\,c^2\,f^3-8\,c\,d\,e\,f^2+8\,c\,f^3+4\,d^2\,e^2\,f-8\,d\,e\,f^2+4\,f^3}-\frac {\ln \left (e+f\,x\right )\,\left (b\,d^3\,e-b\,c\,d^2\,f\right )}{c^4\,f^4-4\,c^3\,d\,e\,f^3+6\,c^2\,d^2\,e^2\,f^2-2\,c^2\,f^4-4\,c\,d^3\,e^3\,f+4\,c\,d\,e\,f^3+d^4\,e^4-2\,d^2\,e^2\,f^2+f^4}-\frac {b\,\ln \left (c+d\,x+1\right )}{4\,f\,\left (e^2+2\,e\,f\,x+f^2\,x^2\right )}-\frac {b\,d^2\,\ln \left (c+d\,x-1\right )}{4\,c^2\,f^3-8\,c\,d\,e\,f^2-8\,c\,f^3+4\,d^2\,e^2\,f+8\,d\,e\,f^2+4\,f^3}-\frac {\frac {-a\,c^2\,f^2+2\,a\,c\,d\,e\,f-a\,d^2\,e^2+b\,d\,e\,f+a\,f^2}{-c^2\,f^2+2\,c\,d\,e\,f-d^2\,e^2+f^2}+\frac {b\,d\,f^2\,x}{-c^2\,f^2+2\,c\,d\,e\,f-d^2\,e^2+f^2}}{2\,e^2\,f+4\,e\,f^2\,x+2\,f^3\,x^2}+\frac {b\,\ln \left (1-d\,x-c\right )}{2\,f\,\left (2\,e^2+4\,e\,f\,x+2\,f^2\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________