Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2},x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac {a^2}{(d+e x)^2}+\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{(d+e x)^2}+\frac {b^2 \tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2}\right ) \, dx\\ &=-\frac {a^2}{e (d+e x)}+(2 a b) \int \frac {\tanh ^{-1}\left (c x^2\right )}{(d+e x)^2} \, dx+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b) \int \frac {2 c x}{(d+e x) \left (1-c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \frac {x}{(d+e x) \left (1-c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \left (-\frac {d e^3}{\left (-c^2 d^4+e^4\right ) (d+e x)}+\frac {e-c d x}{2 \left (c d^2-e^2\right ) \left (-1+c x^2\right )}+\frac {e+c d x}{2 \left (c d^2+e^2\right ) \left (1+c x^2\right )}\right ) \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b c) \int \frac {e-c d x}{-1+c x^2} \, dx}{e \left (c d^2-e^2\right )}+\frac {(2 a b c) \int \frac {e+c d x}{1+c x^2} \, dx}{e \left (c d^2+e^2\right )}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b c) \int \frac {1}{-1+c x^2} \, dx}{c d^2-e^2}-\frac {\left (2 a b c^2 d\right ) \int \frac {x}{-1+c x^2} \, dx}{e \left (c d^2-e^2\right )}+\frac {(2 a b c) \int \frac {1}{1+c x^2} \, dx}{c d^2+e^2}+\frac {\left (2 a b c^2 d\right ) \int \frac {x}{1+c x^2} \, dx}{e \left (c d^2+e^2\right )}\\ &=-\frac {a^2}{e (d+e x)}+\frac {2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )}{c d^2+e^2}-\frac {2 a b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )}{c d^2-e^2}-\frac {2 a b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {4 a b c d e \log (d+e x)}{c^2 d^4-e^4}-\frac {a b c d \log \left (1-c x^2\right )}{e \left (c d^2-e^2\right )}+\frac {a b c d \log \left (1+c x^2\right )}{e \left (c d^2+e^2\right )}+b^2 \int \frac {\tanh ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 25.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{\left (e x +d \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
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 [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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