Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{(d+e x)^2},x\right ) \]
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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 \text {ArcTan}\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 \tan ^{-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 \tan ^{-1}\left (c x^2\right )}{(d+e x)^2}+\frac {b^2 \tan ^{-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 {\tan ^{-1}\left (c x^2\right )}{(d+e x)^2} \, dx+b^2 \int \frac {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tan ^{-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 \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tan ^{-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 \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+b^2 \int \frac {\tan ^{-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^3+c^2 d^3 x-c^2 d^2 e x^2+c^2 d e^2 x^3}{\left (c^2 d^4+e^4\right ) \left (1+c^2 x^4\right )}\right ) \, dx}{e}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-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 {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \frac {e^3+c^2 d^3 x-c^2 d^2 e x^2+c^2 d e^2 x^3}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-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 {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \left (\frac {e^3-c^2 d^2 e x^2}{1+c^2 x^4}+\frac {x \left (c^2 d^3+c^2 d e^2 x^2\right )}{1+c^2 x^4}\right ) \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-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 {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(4 a b c) \int \frac {e^3-c^2 d^2 e x^2}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}+\frac {(4 a b c) \int \frac {x \left (c^2 d^3+c^2 d e^2 x^2\right )}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-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 {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {(2 a b c) \text {Subst}\left (\int \frac {c^2 d^3+c^2 d e^2 x}{1+c^2 x^2} \, dx,x,x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {\left (2 a b \left (c d^2-e^2\right )\right ) \int \frac {c+c^2 x^2}{1+c^2 x^4} \, dx}{c^2 d^4+e^4}+\frac {\left (2 a b \left (c d^2+e^2\right )\right ) \int \frac {c-c^2 x^2}{1+c^2 x^4} \, dx}{c^2 d^4+e^4}\\ &=-\frac {a^2}{e (d+e x)}-\frac {2 a b \tan ^{-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 {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx+\frac {\left (2 a b c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )}{e \left (c^2 d^4+e^4\right )}+\frac {\left (2 a b c^3 d e\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,x^2\right )}{c^2 d^4+e^4}-\frac {\left (a b \left (c d^2-e^2\right )\right ) \int \frac {1}{\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{c^2 d^4+e^4}-\frac {\left (a b \left (c d^2-e^2\right )\right ) \int \frac {1}{\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{c^2 d^4+e^4}-\frac {\left (a b \sqrt {c} \left (c d^2+e^2\right )\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}+2 x}{-\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {\left (a b \sqrt {c} \left (c d^2+e^2\right )\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}-2 x}{-\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{\sqrt {2} \left (c^2 d^4+e^4\right )}\\ &=-\frac {a^2}{e (d+e x)}+\frac {2 a b c^2 d^3 \tan ^{-1}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {2 a b \tan ^{-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 \sqrt {c} \left (c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {a b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {a b c d e \log \left (1+c^2 x^4\right )}{c^2 d^4+e^4}+b^2 \int \frac {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx-\frac {\left (\sqrt {2} a b \sqrt {c} \left (c d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {c} x\right )}{c^2 d^4+e^4}+\frac {\left (\sqrt {2} a b \sqrt {c} \left (c d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {c} x\right )}{c^2 d^4+e^4}\\ &=-\frac {a^2}{e (d+e x)}+\frac {2 a b c^2 d^3 \tan ^{-1}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {2 a b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {\sqrt {2} a b \sqrt {c} \left (c d^2-e^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{c^2 d^4+e^4}-\frac {\sqrt {2} a b \sqrt {c} \left (c d^2-e^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {c} x\right )}{c^2 d^4+e^4}-\frac {4 a b c d e \log (d+e x)}{c^2 d^4+e^4}-\frac {a b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {a b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {a b c d e \log \left (1+c^2 x^4\right )}{c^2 d^4+e^4}+b^2 \int \frac {\tan ^{-1}\left (c x^2\right )^2}{(d+e x)^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 44.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \text {ArcTan}\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 = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{\left (e x +d \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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 [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 {atan}\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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