Optimal. Leaf size=84 \[ \frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {b (c d-e)^2 \log (c x+1)}{4 c^2 e}+\frac {b (c d+e)^2 \log (1-c x)}{4 c^2 e}+\frac {b e x}{2 c} \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5926, 702, 633, 31} \[ \frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {b (c d-e)^2 \log (c x+1)}{4 c^2 e}+\frac {b (c d+e)^2 \log (1-c x)}{4 c^2 e}+\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 5926
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{1-c^2 x^2} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \left (-\frac {e^2}{c^2}+\frac {c^2 d^2+e^2+2 c^2 d e x}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {b e x}{2 c}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {c^2 d^2+e^2+2 c^2 d e x}{1-c^2 x^2} \, dx}{2 c e}\\ &=\frac {b e x}{2 c}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}+\frac {\left (b (c d-e)^2\right ) \int \frac {1}{-c-c^2 x} \, dx}{4 e}-\frac {\left (b (c d+e)^2\right ) \int \frac {1}{c-c^2 x} \, dx}{4 e}\\ &=\frac {b e x}{2 c}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}+\frac {b (c d+e)^2 \log (1-c x)}{4 c^2 e}-\frac {b (c d-e)^2 \log (1+c x)}{4 c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 96, normalized size = 1.14 \[ a d x+\frac {1}{2} a e x^2+\frac {b d \log \left (1-c^2 x^2\right )}{2 c}+\frac {b e \log (1-c x)}{4 c^2}-\frac {b e \log (c x+1)}{4 c^2}+b d x \tanh ^{-1}(c x)+\frac {1}{2} b e x^2 \tanh ^{-1}(c x)+\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 98, normalized size = 1.17 \[ \frac {2 \, a c^{2} e x^{2} + 2 \, {\left (2 \, a c^{2} d + b c e\right )} x + {\left (2 \, b c d - b e\right )} \log \left (c x + 1\right ) + {\left (2 \, b c d + b e\right )} \log \left (c x - 1\right ) + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 292, normalized size = 3.48 \[ -\frac {{\left (\frac {{\left (c x + 1\right )}^{2} b c d \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{{\left (c x - 1\right )}^{2}} - \frac {2 \, {\left (c x + 1\right )} b c d \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c x - 1} + b c d \log \left (-\frac {c x + 1}{c x - 1} + 1\right ) - \frac {{\left (c x + 1\right )}^{2} b c d \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x - 1\right )}^{2}} + \frac {{\left (c x + 1\right )} b c d \log \left (-\frac {c x + 1}{c x - 1}\right )}{c x - 1} - \frac {2 \, {\left (c x + 1\right )} a c d}{c x - 1} + 2 \, a c d - \frac {{\left (c x + 1\right )} b e \log \left (-\frac {c x + 1}{c x - 1}\right )}{c x - 1} - \frac {2 \, {\left (c x + 1\right )} a e}{c x - 1} - \frac {{\left (c x + 1\right )} b e}{c x - 1} + b e\right )} c}{\frac {{\left (c x + 1\right )}^{2} c^{3}}{{\left (c x - 1\right )}^{2}} - \frac {2 \, {\left (c x + 1\right )} c^{3}}{c x - 1} + c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 92, normalized size = 1.10 \[ \frac {a \,x^{2} e}{2}+a d x +\frac {b \arctanh \left (c x \right ) x^{2} e}{2}+b \arctanh \left (c x \right ) x d +\frac {b e x}{2 c}+\frac {b \ln \left (c x -1\right ) d}{2 c}+\frac {b \ln \left (c x -1\right ) e}{4 c^{2}}+\frac {b \ln \left (c x +1\right ) d}{2 c}-\frac {b \ln \left (c x +1\right ) e}{4 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 83, normalized size = 0.99 \[ \frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 67, normalized size = 0.80 \[ a\,d\,x+\frac {a\,e\,x^2}{2}+b\,d\,x\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,e\,x}{2\,c}-\frac {b\,e\,\mathrm {atanh}\left (c\,x\right )}{2\,c^2}+\frac {b\,e\,x^2\,\mathrm {atanh}\left (c\,x\right )}{2}+\frac {b\,d\,\ln \left (c^2\,x^2-1\right )}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 92, normalized size = 1.10 \[ \begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {atanh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {b d \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d \operatorname {atanh}{\left (c x \right )}}{c} + \frac {b e x}{2 c} - \frac {b e \operatorname {atanh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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