Optimal. Leaf size=63 \[ \frac {a \log (c+d x)}{d e}+\frac {i b \text {Li}_2(-i (c+d x))}{2 d e}-\frac {i b \text {Li}_2(i (c+d x))}{2 d e} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5043, 12, 4848, 2391} \[ \frac {i b \text {PolyLog}(2,-i (c+d x))}{2 d e}-\frac {i b \text {PolyLog}(2,i (c+d x))}{2 d e}+\frac {a \log (c+d x)}{d e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2391
Rule 4848
Rule 5043
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {a \log (c+d x)}{d e}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,c+d x\right )}{2 d e}-\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,c+d x\right )}{2 d e}\\ &=\frac {a \log (c+d x)}{d e}+\frac {i b \text {Li}_2(-i (c+d x))}{2 d e}-\frac {i b \text {Li}_2(i (c+d x))}{2 d e}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.83 \[ \frac {a \log (c+d x)+\frac {1}{2} i b \text {Li}_2(-i (c+d x))-\frac {1}{2} i b \text {Li}_2(i (c+d x))}{d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (d x + c\right ) + a}{d e x + c e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 132, normalized size = 2.10 \[ \frac {a \ln \left (d x +c \right )}{d e}+\frac {b \ln \left (d x +c \right ) \arctan \left (d x +c \right )}{d e}+\frac {i b \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2 d e}-\frac {i b \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2 d e}+\frac {i b \dilog \left (1+i \left (d x +c \right )\right )}{2 d e}-\frac {i b \dilog \left (1-i \left (d x +c \right )\right )}{2 d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\arctan \left (d x + c\right )}{2 \, {\left (d e x + c e\right )}}\,{d x} + \frac {a \log \left (d e x + c e\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {atan}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c + d x}\, dx + \int \frac {b \operatorname {atan}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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