Optimal. Leaf size=20 \[ \frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
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Rubi [A]
time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2641, 2624}
\begin {gather*} \frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 2624
Rule 2641
Rubi steps
\begin {align*} \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx &=\int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )}{x^2} \, dx\\ &=\int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )}{x} \, dx\\ &=\frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.90 \begin {gather*} \frac {a^2 x^2}{2}+a b x \log ^2\left (c x^n\right )+\frac {1}{2} b^2 \log ^4\left (c x^n\right ) \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. \(62\) vs.
\(2(18)=36\).
time = 0.24, size = 63, normalized size = 3.15
method | result | size |
default | \(\frac {a^{2} x^{2}}{2}+a b x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-2 a b n x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+\frac {b^{2} \ln \left (c \,x^{n}\right )^{4}}{2}+2 \ln \left (c \,x^{n}\right ) a b n x\) | \(63\) |
risch | \(\text {Expression too large to display}\) | \(2698\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (18) = 36\).
time = 0.28, size = 74, normalized size = 3.70 \begin {gather*} \frac {1}{2} \, b^{2} \log \left (c x^{n}\right )^{4} - 2 \, a b n^{2} x + 2 \, a b n x \log \left (c x^{n}\right ) + a b x \log \left (c x^{n}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs.
\(2 (18) = 36\).
time = 0.37, size = 89, normalized size = 4.45 \begin {gather*} \frac {1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + a b x \log \left (c\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} + 2 \, {\left (b^{2} n \log \left (c\right )^{3} + a b n x \log \left (c\right )\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.77, size = 51, normalized size = 2.55 \begin {gather*} \frac {a^{2} x^{2}}{2} + a b x \log {\left (c x^{n} \right )}^{2} - 2 b^{2} n \left (\begin {cases} - \log {\left (c \right )}^{3} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (18) = 36\).
time = 2.75, size = 90, normalized size = 4.50 \begin {gather*} \frac {1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + 2 \, b^{2} n \log \left (c\right )^{3} \log \left (x\right ) + 2 \, a b n x \log \left (c\right ) \log \left (x\right ) + a b x \log \left (c\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 18, normalized size = 0.90 \begin {gather*} \frac {{\left (b\,{\ln \left (c\,x^n\right )}^2+a\,x\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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