Integrand size = 20, antiderivative size = 78 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=-2 a b p q x+2 b^2 p^2 q^2 x-\frac {2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f} \]
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Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2333, 2332, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-2 a b p q x-\frac {2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+2 b^2 p^2 q^2 x \]
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Rule 2332
Rule 2333
Rule 2436
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -2 a b p q x+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\text {Subst}\left (\frac {\left (2 b^2 p q\right ) \text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -2 a b p q x+2 b^2 p^2 q^2 x-\frac {2 b^2 p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-2 b p q \left (a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(78)=156\).
Time = 0.42 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.12
method | result | size |
parallelrisch | \(\frac {-2 \ln \left (f x +e \right ) b^{2} e^{2} p^{2} q^{2}+2 x \,b^{2} e f \,p^{2} q^{2}-2 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e f p q +2 \ln \left (f x +e \right ) a b \,e^{2} p q +x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e f -2 x a b e f p q +2 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a b e f +{\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e^{2}+e \,a^{2} f x}{e f}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (78) = 156\).
Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.96 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {b^{2} f q^{2} x \log \left (d\right )^{2} + b^{2} f x \log \left (c\right )^{2} + {\left (b^{2} f p^{2} q^{2} x + b^{2} e p^{2} q^{2}\right )} \log \left (f x + e\right )^{2} - 2 \, {\left (b^{2} f p q - a b f\right )} x \log \left (c\right ) + {\left (2 \, b^{2} f p^{2} q^{2} - 2 \, a b f p q + a^{2} f\right )} x - 2 \, {\left (b^{2} e p^{2} q^{2} - a b e p q + {\left (b^{2} f p^{2} q^{2} - a b f p q\right )} x - {\left (b^{2} f p q x + b^{2} e p q\right )} \log \left (c\right ) - {\left (b^{2} f p q^{2} x + b^{2} e p q^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 2 \, {\left (b^{2} f q x \log \left (c\right ) - {\left (b^{2} f p q^{2} - a b f q\right )} x\right )} \log \left (d\right )}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (76) = 152\).
Time = 0.47 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.28 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\begin {cases} a^{2} x + \frac {2 a b e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - 2 a b p q x + 2 a b x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {2 b^{2} e p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {b^{2} e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} + 2 b^{2} p^{2} q^{2} x - 2 b^{2} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + b^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.90 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=-2 \, a b f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b^{2} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, a b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - {\left (2 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f}\right )} b^{2} + a^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (78) = 156\).
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.63 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {{\left (f x + e\right )} b^{2} p^{2} q^{2} \log \left (f x + e\right )^{2}}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} p^{2} q^{2} \log \left (f x + e\right )}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} p q^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} p^{2} q^{2}}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} p q \log \left (f x + e\right ) \log \left (c\right )}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} p q^{2} \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} b^{2} q^{2} \log \left (d\right )^{2}}{f} + \frac {2 \, {\left (f x + e\right )} a b p q \log \left (f x + e\right )}{f} - \frac {2 \, {\left (f x + e\right )} b^{2} p q \log \left (c\right )}{f} + \frac {2 \, {\left (f x + e\right )} b^{2} q \log \left (c\right ) \log \left (d\right )}{f} - \frac {2 \, {\left (f x + e\right )} a b p q}{f} + \frac {{\left (f x + e\right )} b^{2} \log \left (c\right )^{2}}{f} + \frac {2 \, {\left (f x + e\right )} a b q \log \left (d\right )}{f} + \frac {2 \, {\left (f x + e\right )} a b \log \left (c\right )}{f} + \frac {{\left (f x + e\right )} a^{2}}{f} \]
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Time = 1.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.42 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (b^2\,x+\frac {b^2\,e}{f}\right )+x\,\left (a^2-2\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )-\frac {\ln \left (e+f\,x\right )\,\left (2\,b^2\,e\,p^2\,q^2-2\,a\,b\,e\,p\,q\right )}{f}+2\,b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (a-b\,p\,q\right ) \]
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