Optimal. Leaf size=63 \[ -\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left (1-(c+d x)^2\right )}{2 d e^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6242, 12, 6037,
272, 36, 31, 29} \begin {gather*} -\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left (1-(c+d x)^2\right )}{2 d e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 272
Rule 6037
Rule 6242
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,(c+d x)^2\right )}{2 d e^2}\\ &=-\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,(c+d x)^2\right )}{2 d e^2}+\frac {b \text {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^2\right )}{2 d e^2}\\ &=-\frac {a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left (1-(c+d x)^2\right )}{2 d e^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 69, normalized size = 1.10 \begin {gather*} -\frac {\frac {2 a}{c+d x}+\frac {2 b \tanh ^{-1}(c+d x)}{c+d x}-2 b \log (c+d x)+b \log \left (1-c^2-2 c d x-d^2 x^2\right )}{2 d e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 75, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}-\frac {b \arctanh \left (d x +c \right )}{e^{2} \left (d x +c \right )}-\frac {b \ln \left (d x +c +1\right )}{2 e^{2}}+\frac {b \ln \left (d x +c \right )}{e^{2}}-\frac {b \ln \left (d x +c -1\right )}{2 e^{2}}}{d}\) | \(75\) |
default | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}-\frac {b \arctanh \left (d x +c \right )}{e^{2} \left (d x +c \right )}-\frac {b \ln \left (d x +c +1\right )}{2 e^{2}}+\frac {b \ln \left (d x +c \right )}{e^{2}}-\frac {b \ln \left (d x +c -1\right )}{2 e^{2}}}{d}\) | \(75\) |
risch | \(-\frac {b \ln \left (d x +c +1\right )}{2 d \left (d x +c \right ) e^{2}}-\frac {\ln \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) b d x -2 \ln \left (-d x -c \right ) b d x +\ln \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) b c -2 \ln \left (-d x -c \right ) b c -b \ln \left (-d x -c +1\right )+2 a}{2 e^{2} \left (d x +c \right ) d}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 88, normalized size = 1.40 \begin {gather*} -\frac {1}{2} \, {\left (d {\left (\frac {e^{\left (-2\right )} \log \left (d x + c + 1\right )}{d^{2}} - \frac {2 \, e^{\left (-2\right )} \log \left (d x + c\right )}{d^{2}} + \frac {e^{\left (-2\right )} \log \left (d x + c - 1\right )}{d^{2}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{d^{2} x e^{2} + c d e^{2}}\right )} b - \frac {a}{d^{2} x e^{2} + c d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 114, normalized size = 1.81 \begin {gather*} -\frac {{\left (b d x + b c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right ) - 2 \, {\left (b d x + b c\right )} \log \left (d x + c\right ) + b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 2 \, a}{2 \, {\left ({\left (d^{2} x + c d\right )} \cosh \left (1\right )^{2} + 2 \, {\left (d^{2} x + c d\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (d^{2} x + c d\right )} \sinh \left (1\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs.
\(2 (53) = 106\).
time = 1.20, size = 235, normalized size = 3.73 \begin {gather*} \begin {cases} \frac {\tilde {\infty } a}{e^{2} x} & \text {for}\: c = 0 \wedge d = 0 \\\frac {x \left (a + b \operatorname {atanh}{\left (c \right )}\right )}{c^{2} e^{2}} & \text {for}\: d = 0 \\\tilde {\infty } a x & \text {for}\: c = - d x \\- \frac {a}{c d e^{2} + d^{2} e^{2} x} + \frac {b c \log {\left (\frac {c}{d} + x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b c \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {b c \operatorname {atanh}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {b d x \log {\left (\frac {c}{d} + x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b d x \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {b d x \operatorname {atanh}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b \operatorname {atanh}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{2} x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (61) = 122\).
time = 0.41, size = 152, normalized size = 2.41 \begin {gather*} \frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\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 \, a}{\frac {{\left (d x + c + 1\right )} d^{2} e^{2}}{d x + c - 1} + d^{2} e^{2}} + \frac {b \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{d^{2} e^{2}} - \frac {b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2} e^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 122, normalized size = 1.94 \begin {gather*} \frac {b\,\ln \left (1-d\,x-c\right )}{2\,x\,d^2\,e^2+2\,c\,d\,e^2}-\frac {b\,\ln \left (c+d\,x+1\right )}{2\,\left (x\,d^2\,e^2+c\,d\,e^2\right )}-\frac {a}{x\,d^2\,e^2+c\,d\,e^2}-\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2\,d\,e^2}+\frac {b\,\ln \left (c+d\,x\right )}{d\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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