Optimal. Leaf size=142 \[ -\frac {(a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}+\frac {2 b (a+b \log (c (e+f x))) \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}+\frac {2 b^2 \text {Li}_3\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)} \]
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Rubi [A]
time = 0.21, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2458, 12, 2379,
2421, 6724} \begin {gather*} \frac {2 b \text {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)}+\frac {2 b^2 \text {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}-\frac {\log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2379
Rule 2421
Rule 2458
Rule 6724
Rubi steps
\begin {align*} \int \frac {(a+b \log (c (e+f x)))^2}{(h+188 x) (d e+d f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{d x \left (\frac {-188 e+f h}{f}+\frac {188 x}{f}\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x \left (\frac {-188 e+f h}{f}+\frac {188 x}{f}\right )} \, dx,x,e+f x\right )}{d f}\\ &=-\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}+\frac {188 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{\frac {-188 e+f h}{f}+\frac {188 x}{f}} \, dx,x,e+f x\right )}{d f (188 e-f h)}\\ &=\frac {\log \left (-\frac {f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac {\text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d (188 e-f h)}-\frac {(2 b) \text {Subst}\left (\int \frac {(a+b \log (c x)) \log \left (1+\frac {188 x}{-188 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}\\ &=\frac {\log \left (-\frac {f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac {(a+b \log (c (e+f x)))^3}{3 b d (188 e-f h)}+\frac {2 b (a+b \log (c (e+f x))) \text {Li}_2\left (\frac {188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {188 x}{-188 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}\\ &=\frac {\log \left (-\frac {f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac {(a+b \log (c (e+f x)))^3}{3 b d (188 e-f h)}+\frac {2 b (a+b \log (c (e+f x))) \text {Li}_2\left (\frac {188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}-\frac {2 b^2 \text {Li}_3\left (\frac {188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 189, normalized size = 1.33 \begin {gather*} \frac {3 a^2 \log (e+f x)+3 a b \log ^2(c (e+f x))+b^2 \log ^3(c (e+f x))-3 a^2 \log (h+i x)-6 a b \log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )-3 b^2 \log ^2(c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )-6 b (a+b \log (c (e+f x))) \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )+6 b^2 \text {Li}_3\left (\frac {i (e+f x)}{-f h+e i}\right )}{3 d (f h-e i)} \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. \(408\) vs.
\(2(140)=280\).
time = 1.18, size = 409, normalized size = 2.88
method | result | size |
risch | \(-\frac {a^{2} \ln \left (f x +e \right )}{d \left (e i -f h \right )}+\frac {a^{2} \ln \left (i x +h \right )}{d \left (e i -f h \right )}-\frac {b^{2} \ln \left (c f x +c e \right )^{3}}{3 d \left (e i -f h \right )}+\frac {b^{2} \ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 b^{2} \ln \left (c f x +c e \right ) \polylog \left (2, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {2 b^{2} \polylog \left (3, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {b a \ln \left (c f x +c e \right )^{2}}{d \left (e i -f h \right )}+\frac {2 b a \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 b a \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}\) | \(365\) |
derivativedivides | \(\frac {-\frac {c f \,a^{2} \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f \,a^{2} \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c f \,b^{2} \ln \left (c f x +c e \right )^{3}}{3 d \left (e i -f h \right )}+\frac {c f \,b^{2} \ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 c f \,b^{2} \ln \left (c f x +c e \right ) \polylog \left (2, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {2 c f \,b^{2} \polylog \left (3, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {c f a b \ln \left (c f x +c e \right )^{2}}{d \left (e i -f h \right )}+\frac {2 c f a b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 c f a b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}}{c f}\) | \(409\) |
default | \(\frac {-\frac {c f \,a^{2} \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f \,a^{2} \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c f \,b^{2} \ln \left (c f x +c e \right )^{3}}{3 d \left (e i -f h \right )}+\frac {c f \,b^{2} \ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 c f \,b^{2} \ln \left (c f x +c e \right ) \polylog \left (2, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {2 c f \,b^{2} \polylog \left (3, -\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {c f a b \ln \left (c f x +c e \right )^{2}}{d \left (e i -f h \right )}+\frac {2 c f a b \dilog \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 c f a b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}}{c f}\) | \(409\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 324 vs. \(2 (134) = 268\).
time = 0.35, size = 324, normalized size = 2.28 \begin {gather*} a^{2} {\left (\frac {\log \left (f x + e\right )}{d f h - i \, d e} - \frac {\log \left (h + i \, x\right )}{d f h - i \, d e}\right )} - \frac {i \, {\left (\log \left (f x + e\right )^{2} \log \left (-\frac {f x + e}{i \, f h + e} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {f x + e}{i \, f h + e}\right ) \log \left (f x + e\right ) - 2 \, {\rm Li}_{3}(\frac {f x + e}{i \, f h + e})\right )} b^{2}}{i \, d f h + d e} - \frac {2 \, {\left (i \, b^{2} \log \left (c\right ) + i \, a b\right )} {\left (\log \left (f x + e\right ) \log \left (-\frac {f x + e}{i \, f h + e} + 1\right ) + {\rm Li}_2\left (\frac {f x + e}{i \, f h + e}\right )\right )}}{i \, d f h + d e} + \frac {{\left (-i \, b^{2} \log \left (c\right )^{2} - 2 i \, a b \log \left (c\right )\right )} \log \left (-i \, h + x\right )}{i \, d f h + d e} + \frac {i \, b^{2} \log \left (f x + e\right )^{3} - 3 \, {\left (-i \, b^{2} \log \left (c\right ) - i \, a b\right )} \log \left (f x + e\right )^{2} - 3 \, {\left (-i \, b^{2} \log \left (c\right )^{2} - 2 i \, a b \log \left (c\right )\right )} \log \left (f x + e\right )}{3 i \, d f h + 3 \, d e} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{e h + e i x + f h x + f i x^{2}}\, dx + \int \frac {b^{2} \log {\left (c e + c f x \right )}^{2}}{e h + e i x + f h x + f i x^{2}}\, dx + \int \frac {2 a b \log {\left (c e + c f x \right )}}{e h + e i x + f h x + f i x^{2}}\, dx}{d} \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 {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^2}{\left (h+i\,x\right )\,\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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