Optimal. Leaf size=63 \[ -\frac {a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b}{2 d e^3 (c+d x)}+\frac {b \tanh ^{-1}(c+d x)}{2 d e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6107, 12, 5916, 325, 206} \[ -\frac {a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b}{2 d e^3 (c+d x)}+\frac {b \tanh ^{-1}(c+d x)}{2 d e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 325
Rule 5916
Rule 6107
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}-\frac {a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}+\frac {b \tanh ^{-1}(c+d x)}{2 d e^3}-\frac {a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 100, normalized size = 1.59 \[ -\frac {a}{2 d e^3 (c+d x)^2}-\frac {b}{2 d e^3 (c+d x)}-\frac {b \log (-c-d x+1)}{4 d e^3}+\frac {b \log (c+d x+1)}{4 d e^3}-\frac {b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 88, normalized size = 1.40 \[ -\frac {2 \, b d x + 2 \, b c - {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - b\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 2 \, a}{4 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 134, normalized size = 2.13 \[ \frac {{\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {{\left (d x + c + 1\right )} b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} + \frac {2 \, {\left (d x + c + 1\right )} a}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b}{d x + c - 1} + b\right )}}{2 \, {\left (\frac {{\left (d x + c + 1\right )}^{2} d^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} d^{2} e^{3}}{d x + c - 1} + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 1.40 \[ -\frac {a}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b \arctanh \left (d x +c \right )}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b}{2 d \,e^{3} \left (d x +c \right )}-\frac {b \ln \left (d x +c -1\right )}{4 d \,e^{3}}+\frac {b \ln \left (d x +c +1\right )}{4 d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 131, normalized size = 2.08 \[ -\frac {1}{4} \, {\left (d {\left (\frac {2}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c + 1\right )}{d^{2} e^{3}} + \frac {\log \left (d x + c - 1\right )}{d^{2} e^{3}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} b - \frac {a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 67, normalized size = 1.06 \[ \frac {b\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d\,e^3}-\frac {\frac {a}{2}+\frac {b\,c}{2}+\frac {b\,\ln \left (c+d\,x+1\right )}{4}-\frac {b\,\ln \left (1-d\,x-c\right )}{4}+\frac {b\,d\,x}{2}}{d\,e^3\,{\left (c+d\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.55, size = 313, normalized size = 4.97 \[ \begin {cases} - \frac {a}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} + \frac {b c^{2} \operatorname {atanh}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} + \frac {2 b c d x \operatorname {atanh}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b c}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} + \frac {b d^{2} x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b d x}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b \operatorname {atanh}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} & \text {for}\: d \neq 0 \\\frac {x \left (a + b \operatorname {atanh}{\relax (c )}\right )}{c^{3} e^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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