Optimal. Leaf size=43 \[ \frac{\log (d+e x)}{2 e}-\frac{1}{2} \text{Unintegrable}\left (\frac{\cos \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right ) \]
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Rubi [A] time = 0.037116, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx &=\int \left (\frac{1}{2 (d+e x)}-\frac{\cos \left (2 a+2 b x+2 c x^2\right )}{2 (d+e x)}\right ) \, dx\\ &=\frac{\log (d+e x)}{2 e}-\frac{1}{2} \int \frac{\cos \left (2 a+2 b x+2 c x^2\right )}{d+e x} \, dx\\ \end{align*}
Mathematica [A] time = 7.40887, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( c{x}^{2}+bx+a \right ) \right ) ^{2}}{ex+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{1}{2} \, e \int \frac{\cos \left (2 \, c x^{2} + 2 \, b x\right ) \cos \left (2 \, a\right ) - \sin \left (2 \, c x^{2} + 2 \, b x\right ) \sin \left (2 \, a\right )}{{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} e x +{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} d}\,{d x} - \frac{1}{2} \, e \int \frac{\cos \left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}{e x + d}\,{d x} + \log \left (e x + d\right )}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (c x^{2} + b x + a\right )^{2} - 1}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (a + b x + c x^{2} \right )}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (c x^{2} + b x + a\right )^{2}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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