Optimal. Leaf size=151 \[ -\frac {i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^2 (h+i x)}+\frac {b f \log (h+i x)}{d (f h-e i)^2}-\frac {f (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {b f \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2458, 12, 2389,
2379, 2438, 2351, 31} \begin {gather*} \frac {b f \text {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}-\frac {f \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}-\frac {i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^2}+\frac {b f \log (h+i x)}{d (f h-e i)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rubi steps
\begin {align*} \int \frac {a+b \log (c (e+f x))}{(h+181 x)^2 (d e+d f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{d x \left (\frac {-181 e+f h}{f}+\frac {181 x}{f}\right )^2} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-181 e+f h}{f}+\frac {181 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f}\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {-181 e+f h}{f}+\frac {181 x}{f}\right )} \, dx,x,e+f x\right )}{d (181 e-f h)}+\frac {181 \text {Subst}\left (\int \frac {a+b \log (c x)}{\left (\frac {-181 e+f h}{f}+\frac {181 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f (181 e-f h)}\\ &=-\frac {181 (e+f x) (a+b \log (c (e+f x)))}{d (181 e-f h)^2 (h+181 x)}-\frac {181 \text {Subst}\left (\int \frac {a+b \log (c x)}{\frac {-181 e+f h}{f}+\frac {181 x}{f}} \, dx,x,e+f x\right )}{d (181 e-f h)^2}+\frac {(181 b) \text {Subst}\left (\int \frac {1}{\frac {-181 e+f h}{f}+\frac {181 x}{f}} \, dx,x,e+f x\right )}{d (181 e-f h)^2}+\frac {f \text {Subst}\left (\int \frac {a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d (181 e-f h)^2}\\ &=\frac {b f \log (h+181 x)}{d (181 e-f h)^2}-\frac {181 (e+f x) (a+b \log (c (e+f x)))}{d (181 e-f h)^2 (h+181 x)}-\frac {f \log \left (-\frac {f (h+181 x)}{181 e-f h}\right ) (a+b \log (c (e+f x)))}{d (181 e-f h)^2}+\frac {f (a+b \log (c (e+f x)))^2}{2 b d (181 e-f h)^2}+\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {181 x}{-181 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (181 e-f h)^2}\\ &=\frac {b f \log (h+181 x)}{d (181 e-f h)^2}-\frac {181 (e+f x) (a+b \log (c (e+f x)))}{d (181 e-f h)^2 (h+181 x)}-\frac {f \log \left (-\frac {f (h+181 x)}{181 e-f h}\right ) (a+b \log (c (e+f x)))}{d (181 e-f h)^2}+\frac {f (a+b \log (c (e+f x)))^2}{2 b d (181 e-f h)^2}-\frac {b f \text {Li}_2\left (\frac {181 (e+f x)}{181 e-f h}\right )}{d (181 e-f h)^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 141, normalized size = 0.93 \begin {gather*} \frac {\frac {2 (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f (a+b \log (c (e+f x)))^2}{b}-2 b f (\log (e+f x)-\log (h+i x))-2 f (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs.
\(2(150)=300\).
time = 1.18, size = 372, normalized size = 2.46
method | result | size |
risch | \(\frac {a f \ln \left (f x +e \right )}{d \left (e i -f h \right )^{2}}-\frac {a}{d \left (e i -f h \right ) \left (i x +h \right )}-\frac {a f \ln \left (i x +h \right )}{d \left (e i -f h \right )^{2}}+\frac {b f \ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}-\frac {b c \,f^{2} i \ln \left (c f x +c e \right ) x}{d \left (e i -f h \right )^{2} \left (c f i x +h c f \right )}-\frac {b c f i \ln \left (c f x +c e \right ) e}{d \left (e i -f h \right )^{2} \left (c f i x +h c f \right )}+\frac {b f \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{2}}-\frac {b f \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}-\frac {b f \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}\) | \(330\) |
derivativedivides | \(\frac {\frac {c^{2} f^{2} a}{d \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {c \,f^{2} a \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} a \ln \left (c f x +c e \right )}{d \left (e i -f h \right )^{2}}-\frac {c \,f^{2} b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}-\frac {c \,f^{2} b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} b \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} b i \ln \left (c f x +c e \right ) \left (c f x +c e \right )}{d \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {c \,f^{2} b \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{2}}}{c f}\) | \(372\) |
default | \(\frac {\frac {c^{2} f^{2} a}{d \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {c \,f^{2} a \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} a \ln \left (c f x +c e \right )}{d \left (e i -f h \right )^{2}}-\frac {c \,f^{2} b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}-\frac {c \,f^{2} b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} b \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}+\frac {c \,f^{2} b i \ln \left (c f x +c e \right ) \left (c f x +c e \right )}{d \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {c \,f^{2} b \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{2}}}{c f}\) | \(372\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,\left (e+f\,x\right )\right )}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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