3.1.36 \(\int \frac {a+b \tanh ^{-1}(c+d x)}{(e+f x)^2} \, dx\) [36]

Optimal. Leaf size=115 \[ -\frac {a+b \tanh ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (1-c-d x)}{2 f (d e+f-c f)}+\frac {b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6244, 2007, 719, 31, 646} \begin {gather*} -\frac {a+b \tanh ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (-c-d x+1)}{2 f (-c f+d e+f)}+\frac {b d \log (c+d x+1)}{2 f (-c f+d e-f)}-\frac {b d \log (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)^2,x]

[Out]

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

Rule 31

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

Rule 646

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

Rule 719

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

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

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Mathematica [A]
time = 0.13, size = 125, normalized size = 1.09 \begin {gather*} \frac {1}{2} \left (-\frac {2 a}{f (e+f x)}-\frac {2 b \tanh ^{-1}(c+d x)}{f (e+f x)}+\frac {b d \log (1-c-d x)}{f (-d e+(-1+c) f)}-\frac {b d \log (1+c+d x)}{f (-d e+f+c f)}-\frac {2 b d \log (e+f x)}{d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.76, size = 170, normalized size = 1.48

method	result	size
derivativedivides	\(\frac {\frac {a \,d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {b \,d^{2} \arctanh \left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {b \,d^{2} \ln \left (d x +c +1\right )}{f \left (2 c f -2 d e +2 f \right )}-\frac {b \,d^{2} \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {b \,d^{2} \ln \left (d x +c -1\right )}{f \left (2 c f -2 d e -2 f \right )}}{d}\)	\(170\)
default	\(\frac {\frac {a \,d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {b \,d^{2} \arctanh \left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {b \,d^{2} \ln \left (d x +c +1\right )}{f \left (2 c f -2 d e +2 f \right )}-\frac {b \,d^{2} \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {b \,d^{2} \ln \left (d x +c -1\right )}{f \left (2 c f -2 d e -2 f \right )}}{d}\)	\(170\)
risch	\(-\frac {b \ln \left (d x +c +1\right )}{2 f \left (f x +e \right )}-\frac {\ln \left (d x +c +1\right ) b c d \,f^{2} x -\ln \left (d x +c +1\right ) b \,d^{2} e f x -\ln \left (-d x -c +1\right ) b c d \,f^{2} x +\ln \left (-d x -c +1\right ) b \,d^{2} e f x +\ln \left (d x +c +1\right ) b c d e f -\ln \left (d x +c +1\right ) b \,d^{2} e^{2}-\ln \left (d x +c +1\right ) b d \,f^{2} x +\ln \left (-d x -c +1\right ) b c d e f -\ln \left (-d x -c +1\right ) b d \,f^{2} x +2 \ln \left (-f x -e \right ) b d \,f^{2} x -b \,c^{2} f^{2} \ln \left (-d x -c +1\right )-\ln \left (d x +c +1\right ) b d e f -\ln \left (-d x -c +1\right ) b d e f +2 \ln \left (-f x -e \right ) b d e f +2 a \,c^{2} f^{2}-4 a c d e f +2 d^{2} e^{2} a +b \,f^{2} \ln \left (-d x -c +1\right )-2 a \,f^{2}}{2 \left (c f -d e +f \right ) \left (c f -d e -f \right ) \left (f x +e \right ) f}\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 128, normalized size = 1.11 \begin {gather*} -\frac {1}{2} \, {\left (d {\left (\frac {\log \left (d x + c + 1\right )}{{\left (c + 1\right )} f^{2} - d f e} - \frac {\log \left (d x + c - 1\right )}{{\left (c - 1\right )} f^{2} - d f e} - \frac {2 \, \log \left (f x + e\right )}{2 \, c d f e - {\left (c^{2} - 1\right )} f^{2} - d^{2} e^{2}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{f^{2} x + f e}\right )} b - \frac {a}{f^{2} x + f e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (118) = 236\).
time = 0.50, size = 517, normalized size = 4.50 \begin {gather*} \frac {4 \, a c d f \cosh \left (1\right ) - 2 \, a d^{2} \cosh \left (1\right )^{2} - 2 \, a d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (a c^{2} - a\right )} f^{2} - {\left ({\left (b c - b\right )} d f^{2} x - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} - {\left (b d^{2} f x - {\left (b c - b\right )} d f\right )} \cosh \left (1\right ) - {\left (b d^{2} f x + 2 \, b d^{2} \cosh \left (1\right ) - {\left (b c - b\right )} d f\right )} \sinh \left (1\right )\right )} \log \left (d x + c + 1\right ) + {\left ({\left (b c + b\right )} d f^{2} x - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} - {\left (b d^{2} f x - {\left (b c + b\right )} d f\right )} \cosh \left (1\right ) - {\left (b d^{2} f x + 2 \, b d^{2} \cosh \left (1\right ) - {\left (b c + b\right )} d f\right )} \sinh \left (1\right )\right )} \log \left (d x + c - 1\right ) - 2 \, {\left (b d f^{2} x + b d f \cosh \left (1\right ) + b d f \sinh \left (1\right )\right )} \log \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} - {\left (b c^{2} - b\right )} f^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 4 \, {\left (a c d f - a d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )}{2 \, {\left ({\left (c^{2} - 1\right )} f^{4} x + d^{2} f \cosh \left (1\right )^{3} + d^{2} f \sinh \left (1\right )^{3} + {\left (d^{2} f^{2} x - 2 \, c d f^{2}\right )} \cosh \left (1\right )^{2} + {\left (d^{2} f^{2} x - 2 \, c d f^{2} + 3 \, d^{2} f \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - {\left (2 \, c d f^{3} x - {\left (c^{2} - 1\right )} f^{3}\right )} \cosh \left (1\right ) - {\left (2 \, c d f^{3} x - 3 \, d^{2} f \cosh \left (1\right )^{2} - {\left (c^{2} - 1\right )} f^{3} - 2 \, {\left (d^{2} f^{2} x - 2 \, c d f^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (92) = 184\).
time = 4.62, size = 1658, normalized size = 14.42 \begin {gather*} \begin {cases} - \frac {a + b \operatorname {atanh}{\left (c \right )}}{e f + f^{2} x} & \text {for}\: d = 0 \\- \frac {2 a f}{2 e f^{2} + 2 f^{3} x} + \frac {b d e \operatorname {atanh}{\left (\frac {d e}{f} + d x - 1 \right )}}{2 e f^{2} + 2 f^{3} x} + \frac {b d f x \operatorname {atanh}{\left (\frac {d e}{f} + d x - 1 \right )}}{2 e f^{2} + 2 f^{3} x} - \frac {2 b f \operatorname {atanh}{\left (\frac {d e}{f} + d x - 1 \right )}}{2 e f^{2} + 2 f^{3} x} - \frac {b f}{2 e f^{2} + 2 f^{3} x} & \text {for}\: c = \frac {d e - f}{f} \\- \frac {2 a f}{2 e f^{2} + 2 f^{3} x} - \frac {b d e \operatorname {atanh}{\left (\frac {d e}{f} + d x + 1 \right )}}{2 e f^{2} + 2 f^{3} x} - \frac {b d f x \operatorname {atanh}{\left (\frac {d e}{f} + d x + 1 \right )}}{2 e f^{2} + 2 f^{3} x} - \frac {2 b f \operatorname {atanh}{\left (\frac {d e}{f} + d x + 1 \right )}}{2 e f^{2} + 2 f^{3} x} + \frac {b f}{2 e f^{2} + 2 f^{3} x} & \text {for}\: c = \frac {d e + f}{f} \\\tilde {\infty } \left (a x + \frac {b c \operatorname {atanh}{\left (c + d x \right )}}{d} + b x \operatorname {atanh}{\left (c + d x \right )} + \frac {b \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b \operatorname {atanh}{\left (c + d x \right )}}{d}\right ) & \text {for}\: e = - f x \\\frac {a x + \frac {b c \operatorname {atanh}{\left (c + d x \right )}}{d} + b x \operatorname {atanh}{\left (c + d x \right )} + \frac {b \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b \operatorname {atanh}{\left (c + d x \right )}}{d}}{e^{2}} & \text {for}\: f = 0 \\- \frac {a c^{2} f^{2}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {2 a c d e f}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {a d^{2} e^{2}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {a f^{2}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b c^{2} f^{2} \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {b c d e f \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b c d f^{2} x \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {b d^{2} e f x \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b d e f \log {\left (\frac {e}{f} + x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {b d e f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b d e f \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b d f^{2} x \log {\left (\frac {e}{f} + x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {b d f^{2} x \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} - \frac {b d f^{2} x \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} + \frac {b f^{2} \operatorname {atanh}{\left (c + d x \right )}}{c^{2} e f^{3} + c^{2} f^{4} x - 2 c d e^{2} f^{2} - 2 c d e f^{3} x + d^{2} e^{3} f + d^{2} e^{2} f^{2} x - e f^{3} - f^{4} x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(a + b*atanh(c))/(e*f + f**2*x), Eq(d, 0)), (-2*a*f/(2*e*f**2 + 2*f**3*x) + b*d*e*atanh(d*e/f + d*
x - 1)/(2*e*f**2 + 2*f**3*x) + b*d*f*x*atanh(d*e/f + d*x - 1)/(2*e*f**2 + 2*f**3*x) - 2*b*f*atanh(d*e/f + d*x
- 1)/(2*e*f**2 + 2*f**3*x) - b*f/(2*e*f**2 + 2*f**3*x), Eq(c, (d*e - f)/f)), (-2*a*f/(2*e*f**2 + 2*f**3*x) - b
*d*e*atanh(d*e/f + d*x + 1)/(2*e*f**2 + 2*f**3*x) - b*d*f*x*atanh(d*e/f + d*x + 1)/(2*e*f**2 + 2*f**3*x) - 2*b
*f*atanh(d*e/f + d*x + 1)/(2*e*f**2 + 2*f**3*x) + b*f/(2*e*f**2 + 2*f**3*x), Eq(c, (d*e + f)/f)), (zoo*(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), Eq(e, -f*x)), ((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**2, Eq(f, 0)), (-
a*c**2*f**2/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e
*f**3 - f**4*x) + 2*a*c*d*e*f/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d*
*2*e**2*f**2*x - e*f**3 - f**4*x) - a*d**2*e**2/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x
+ d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) + a*f**2/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*
c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) - b*c**2*f**2*atanh(c + d*x)/(c**2*e*f**3 + c
**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) + b*c*d*e*f*
atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x
- e*f**3 - f**4*x) - b*c*d*f**2*x*atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x
+ d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) + b*d**2*e*f*x*atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x
- 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) - b*d*e*f*log(e/f + x)/
(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4
*x) + b*d*e*f*log(c/d + x + 1/d)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f +
d**2*e**2*f**2*x - e*f**3 - f**4*x) - b*d*e*f*atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2
*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) - b*d*f**2*x*log(e/f + x)/(c**2*e*f**3 + c**
2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) + b*d*f**2*x*l
og(c/d + x + 1/d)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2
*x - e*f**3 - f**4*x) - b*d*f**2*x*atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x - 2*c*d*e**2*f**2 - 2*c*d*e*f**3*
x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x) + b*f**2*atanh(c + d*x)/(c**2*e*f**3 + c**2*f**4*x - 2*c
*d*e**2*f**2 - 2*c*d*e*f**3*x + d**2*e**3*f + d**2*e**2*f**2*x - e*f**3 - f**4*x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (112) = 224\).
time = 0.43, size = 474, normalized size = 4.12 \begin {gather*} -\frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {b \log \left (-\frac {{\left (d x + c + 1\right )} d e}{d x + c - 1} + d e + \frac {{\left (d x + c + 1\right )} c f}{d x + c - 1} - c f - \frac {{\left (d x + c + 1\right )} f}{d x + c - 1} - f\right )}{d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - f^{2}} - \frac {b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )} d^{2} e^{2}}{d x + c - 1} - d^{2} e^{2} - \frac {2 \, {\left (d x + c + 1\right )} c d e f}{d x + c - 1} + 2 \, c d e f + \frac {{\left (d x + c + 1\right )} c^{2} f^{2}}{d x + c - 1} - c^{2} f^{2} + \frac {2 \, {\left (d x + c + 1\right )} d e f}{d x + c - 1} - \frac {2 \, {\left (d x + c + 1\right )} c f^{2}}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} f^{2}}{d x + c - 1} + f^{2}} - \frac {b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - f^{2}} - \frac {2 \, a}{\frac {{\left (d x + c + 1\right )} d^{2} e^{2}}{d x + c - 1} - d^{2} e^{2} - \frac {2 \, {\left (d x + c + 1\right )} c d e f}{d x + c - 1} + 2 \, c d e f + \frac {{\left (d x + c + 1\right )} c^{2} f^{2}}{d x + c - 1} - c^{2} f^{2} + \frac {2 \, {\left (d x + c + 1\right )} d e f}{d x + c - 1} - \frac {2 \, {\left (d x + c + 1\right )} c f^{2}}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} f^{2}}{d x + c - 1} + f^{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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.64, size = 170, normalized size = 1.48 \begin {gather*} \ln \left (e+f\,x\right )\,\left (\frac {b\,\left (c-1\right )}{2\,e\,\left (d\,e-f\,\left (c-1\right )\right )}-\frac {b\,\left (c+1\right )}{2\,e\,\left (d\,e-f\,\left (c+1\right )\right )}\right )-\frac {a}{x\,f^2+e\,f}+\frac {b\,\ln \left (1-d\,x-c\right )}{f\,\left (2\,e+2\,f\,x\right )}-\frac {b\,\ln \left (c+d\,x+1\right )}{2\,f\,\left (e+f\,x\right )}-\frac {b\,d\,\ln \left (c+d\,x-1\right )}{2\,f^2-2\,c\,f^2+2\,d\,e\,f}-\frac {b\,d\,\ln \left (c+d\,x+1\right )}{2\,c\,f^2+2\,f^2-2\,d\,e\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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