Optimal. Leaf size=74 \[ \frac{1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \sin (c) \text{CosIntegral}\left (d x^2\right )+a b \cos (c) \text{Si}\left (d x^2\right )-\frac{1}{4} b^2 \cos (2 c) \text{CosIntegral}\left (2 d x^2\right )+\frac{1}{4} b^2 \sin (2 c) \text{Si}\left (2 d x^2\right ) \]
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Rubi [A] time = 0.105475, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3403, 6, 3378, 3376, 3375, 3377} \[ \frac{1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \sin (c) \text{CosIntegral}\left (d x^2\right )+a b \cos (c) \text{Si}\left (d x^2\right )-\frac{1}{4} b^2 \cos (2 c) \text{CosIntegral}\left (2 d x^2\right )+\frac{1}{4} b^2 \sin (2 c) \text{Si}\left (2 d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3378
Rule 3376
Rule 3375
Rule 3377
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin \left (c+d x^2\right )\right )^2}{x} \, dx &=\int \left (\frac{a^2}{x}+\frac{b^2}{2 x}-\frac{b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+\frac{2 a b \sin \left (c+d x^2\right )}{x}\right ) \, dx\\ &=\int \left (\frac{a^2+\frac{b^2}{2}}{x}-\frac{b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+\frac{2 a b \sin \left (c+d x^2\right )}{x}\right ) \, dx\\ &=\frac{1}{2} \left (2 a^2+b^2\right ) \log (x)+(2 a b) \int \frac{\sin \left (c+d x^2\right )}{x} \, dx-\frac{1}{2} b^2 \int \frac{\cos \left (2 c+2 d x^2\right )}{x} \, dx\\ &=\frac{1}{2} \left (2 a^2+b^2\right ) \log (x)+(2 a b \cos (c)) \int \frac{\sin \left (d x^2\right )}{x} \, dx-\frac{1}{2} \left (b^2 \cos (2 c)\right ) \int \frac{\cos \left (2 d x^2\right )}{x} \, dx+(2 a b \sin (c)) \int \frac{\cos \left (d x^2\right )}{x} \, dx+\frac{1}{2} \left (b^2 \sin (2 c)\right ) \int \frac{\sin \left (2 d x^2\right )}{x} \, dx\\ &=-\frac{1}{4} b^2 \cos (2 c) \text{Ci}\left (2 d x^2\right )+\frac{1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \text{Ci}\left (d x^2\right ) \sin (c)+a b \cos (c) \text{Si}\left (d x^2\right )+\frac{1}{4} b^2 \sin (2 c) \text{Si}\left (2 d x^2\right )\\ \end{align*}
Mathematica [A] time = 0.1655, size = 71, normalized size = 0.96 \[ \frac{1}{2} \left (2 a^2+b^2\right ) \log (x)-\frac{1}{4} b \left (-4 a \sin (c) \text{CosIntegral}\left (d x^2\right )-4 a \cos (c) \text{Si}\left (d x^2\right )+b \cos (2 c) \text{CosIntegral}\left (2 d x^2\right )-b \sin (2 c) \text{Si}\left (2 d x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 69, normalized size = 0.9 \begin{align*} \ln \left ( x \right ){a}^{2}+{\frac{\ln \left ( x \right ){b}^{2}}{2}}+{\frac{{b}^{2}{\it Si} \left ( 2\,d{x}^{2} \right ) \sin \left ( 2\,c \right ) }{4}}-{\frac{{b}^{2}{\it Ci} \left ( 2\,d{x}^{2} \right ) \cos \left ( 2\,c \right ) }{4}}+ab\cos \left ( c \right ){\it Si} \left ( d{x}^{2} \right ) +ab{\it Ci} \left ( d{x}^{2} \right ) \sin \left ( c \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.21379, size = 146, normalized size = 1.97 \begin{align*} -\frac{1}{2} \,{\left ({\left (i \,{\rm Ei}\left (i \, d x^{2}\right ) - i \,{\rm Ei}\left (-i \, d x^{2}\right )\right )} \cos \left (c\right ) -{\left ({\rm Ei}\left (i \, d x^{2}\right ) +{\rm Ei}\left (-i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} a b - \frac{1}{8} \,{\left ({\left ({\rm Ei}\left (2 i \, d x^{2}\right ) +{\rm Ei}\left (-2 i \, d x^{2}\right )\right )} \cos \left (2 \, c\right ) -{\left (-i \,{\rm Ei}\left (2 i \, d x^{2}\right ) + i \,{\rm Ei}\left (-2 i \, d x^{2}\right )\right )} \sin \left (2 \, c\right ) - 4 \, \log \left (x\right )\right )} b^{2} + a^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04536, size = 321, normalized size = 4.34 \begin{align*} \frac{1}{4} \, b^{2} \sin \left (2 \, c\right ) \operatorname{Si}\left (2 \, d x^{2}\right ) + a b \cos \left (c\right ) \operatorname{Si}\left (d x^{2}\right ) - \frac{1}{8} \,{\left (b^{2} \operatorname{Ci}\left (2 \, d x^{2}\right ) + b^{2} \operatorname{Ci}\left (-2 \, d x^{2}\right )\right )} \cos \left (2 \, c\right ) + \frac{1}{2} \,{\left (2 \, a^{2} + b^{2}\right )} \log \left (x\right ) + \frac{1}{2} \,{\left (a b \operatorname{Ci}\left (d x^{2}\right ) + a b \operatorname{Ci}\left (-d x^{2}\right )\right )} \sin \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sin{\left (c + d x^{2} \right )}\right )^{2}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15248, size = 104, normalized size = 1.41 \begin{align*} -\frac{1}{4} \, b^{2} \cos \left (2 \, c\right ) \operatorname{Ci}\left (2 \, d x^{2}\right ) + a b \operatorname{Ci}\left (d x^{2}\right ) \sin \left (c\right ) + a b \cos \left (c\right ) \operatorname{Si}\left (d x^{2}\right ) - \frac{1}{4} \, b^{2} \sin \left (2 \, c\right ) \operatorname{Si}\left (-2 \, d x^{2}\right ) + \frac{1}{2} \, a^{2} \log \left (d x^{2}\right ) + \frac{1}{4} \, b^{2} \log \left (d x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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