Optimal. Leaf size=239 \[ -\frac{2 a^2+b^2}{6 x^3}-\frac{4}{3} \sqrt{2 \pi } a b d^{3/2} \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )-\frac{4}{3} \sqrt{2 \pi } a b d^{3/2} \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}-\frac{4 a b d \cos \left (c+d x^2\right )}{3 x}+\frac{4}{3} \sqrt{\pi } b^2 d^{3/2} \cos (2 c) \text{FresnelC}\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )-\frac{4}{3} \sqrt{\pi } b^2 d^{3/2} \sin (2 c) S\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )-\frac{2 b^2 d \sin \left (2 c+2 d x^2\right )}{3 x}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3} \]
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Rubi [A] time = 0.19728, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3403, 6, 3388, 3387, 3354, 3352, 3351, 3353} \[ -\frac{2 a^2+b^2}{6 x^3}-\frac{4}{3} \sqrt{2 \pi } a b d^{3/2} \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )-\frac{4}{3} \sqrt{2 \pi } a b d^{3/2} \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}-\frac{4 a b d \cos \left (c+d x^2\right )}{3 x}+\frac{4}{3} \sqrt{\pi } b^2 d^{3/2} \cos (2 c) \text{FresnelC}\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )-\frac{4}{3} \sqrt{\pi } b^2 d^{3/2} \sin (2 c) S\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )-\frac{2 b^2 d \sin \left (2 c+2 d x^2\right )}{3 x}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3388
Rule 3387
Rule 3354
Rule 3352
Rule 3351
Rule 3353
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin \left (c+d x^2\right )\right )^2}{x^4} \, dx &=\int \left (\frac{a^2}{x^4}+\frac{b^2}{2 x^4}-\frac{b^2 \cos \left (2 c+2 d x^2\right )}{2 x^4}+\frac{2 a b \sin \left (c+d x^2\right )}{x^4}\right ) \, dx\\ &=\int \left (\frac{a^2+\frac{b^2}{2}}{x^4}-\frac{b^2 \cos \left (2 c+2 d x^2\right )}{2 x^4}+\frac{2 a b \sin \left (c+d x^2\right )}{x^4}\right ) \, dx\\ &=-\frac{2 a^2+b^2}{6 x^3}+(2 a b) \int \frac{\sin \left (c+d x^2\right )}{x^4} \, dx-\frac{1}{2} b^2 \int \frac{\cos \left (2 c+2 d x^2\right )}{x^4} \, dx\\ &=-\frac{2 a^2+b^2}{6 x^3}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3}-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}+\frac{1}{3} (4 a b d) \int \frac{\cos \left (c+d x^2\right )}{x^2} \, dx+\frac{1}{3} \left (2 b^2 d\right ) \int \frac{\sin \left (2 c+2 d x^2\right )}{x^2} \, dx\\ &=-\frac{2 a^2+b^2}{6 x^3}-\frac{4 a b d \cos \left (c+d x^2\right )}{3 x}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3}-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}-\frac{2 b^2 d \sin \left (2 c+2 d x^2\right )}{3 x}-\frac{1}{3} \left (8 a b d^2\right ) \int \sin \left (c+d x^2\right ) \, dx+\frac{1}{3} \left (8 b^2 d^2\right ) \int \cos \left (2 c+2 d x^2\right ) \, dx\\ &=-\frac{2 a^2+b^2}{6 x^3}-\frac{4 a b d \cos \left (c+d x^2\right )}{3 x}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3}-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}-\frac{2 b^2 d \sin \left (2 c+2 d x^2\right )}{3 x}-\frac{1}{3} \left (8 a b d^2 \cos (c)\right ) \int \sin \left (d x^2\right ) \, dx+\frac{1}{3} \left (8 b^2 d^2 \cos (2 c)\right ) \int \cos \left (2 d x^2\right ) \, dx-\frac{1}{3} \left (8 a b d^2 \sin (c)\right ) \int \cos \left (d x^2\right ) \, dx-\frac{1}{3} \left (8 b^2 d^2 \sin (2 c)\right ) \int \sin \left (2 d x^2\right ) \, dx\\ &=-\frac{2 a^2+b^2}{6 x^3}-\frac{4 a b d \cos \left (c+d x^2\right )}{3 x}+\frac{b^2 \cos \left (2 c+2 d x^2\right )}{6 x^3}+\frac{4}{3} b^2 d^{3/2} \sqrt{\pi } \cos (2 c) C\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )-\frac{4}{3} a b d^{3/2} \sqrt{2 \pi } \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )-\frac{4}{3} a b d^{3/2} \sqrt{2 \pi } C\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right ) \sin (c)-\frac{4}{3} b^2 d^{3/2} \sqrt{\pi } S\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right ) \sin (2 c)-\frac{2 a b \sin \left (c+d x^2\right )}{3 x^3}-\frac{2 b^2 d \sin \left (2 c+2 d x^2\right )}{3 x}\\ \end{align*}
Mathematica [A] time = 0.663367, size = 226, normalized size = 0.95 \[ -\frac{2 a^2+8 \sqrt{2 \pi } a b d^{3/2} x^3 \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )+8 \sqrt{2 \pi } a b d^{3/2} x^3 \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )+4 a b \sin \left (c+d x^2\right )+8 a b d x^2 \cos \left (c+d x^2\right )-8 \sqrt{\pi } b^2 d^{3/2} x^3 \cos (2 c) \text{FresnelC}\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )+8 \sqrt{\pi } b^2 d^{3/2} x^3 \sin (2 c) S\left (\frac{2 \sqrt{d} x}{\sqrt{\pi }}\right )+4 b^2 d x^2 \sin \left (2 \left (c+d x^2\right )\right )-b^2 \cos \left (2 \left (c+d x^2\right )\right )+b^2}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 175, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,{x}^{3}} \left ({a}^{2}+{\frac{{b}^{2}}{2}} \right ) }-{\frac{{b}^{2}}{2} \left ( -{\frac{\cos \left ( 2\,d{x}^{2}+2\,c \right ) }{3\,{x}^{3}}}-{\frac{4\,d}{3} \left ( -{\frac{\sin \left ( 2\,d{x}^{2}+2\,c \right ) }{x}}+2\,\sqrt{d}\sqrt{\pi } \left ( \cos \left ( 2\,c \right ){\it FresnelC} \left ( 2\,{\frac{x\sqrt{d}}{\sqrt{\pi }}} \right ) -\sin \left ( 2\,c \right ){\it FresnelS} \left ( 2\,{\frac{x\sqrt{d}}{\sqrt{\pi }}} \right ) \right ) \right ) } \right ) }+2\,ab \left ( -1/3\,{\frac{\sin \left ( d{x}^{2}+c \right ) }{{x}^{3}}}+2/3\,d \left ( -{\frac{\cos \left ( d{x}^{2}+c \right ) }{x}}-\sqrt{d}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( c \right ){\it FresnelS} \left ({\frac{x\sqrt{d}\sqrt{2}}{\sqrt{\pi }}} \right ) +\sin \left ( c \right ){\it FresnelC} \left ({\frac{x\sqrt{d}\sqrt{2}}{\sqrt{\pi }}} \right ) \right ) \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.25199, size = 756, normalized size = 3.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44602, size = 556, normalized size = 2.33 \begin{align*} -\frac{4 \, \sqrt{2} \pi a b d x^{3} \sqrt{\frac{d}{\pi }} \cos \left (c\right ) \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) + 4 \, \sqrt{2} \pi a b d x^{3} \sqrt{\frac{d}{\pi }} \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) \sin \left (c\right ) - 4 \, \pi b^{2} d x^{3} \sqrt{\frac{d}{\pi }} \cos \left (2 \, c\right ) \operatorname{C}\left (2 \, x \sqrt{\frac{d}{\pi }}\right ) + 4 \, \pi b^{2} d x^{3} \sqrt{\frac{d}{\pi }} \operatorname{S}\left (2 \, x \sqrt{\frac{d}{\pi }}\right ) \sin \left (2 \, c\right ) + 4 \, a b d x^{2} \cos \left (d x^{2} + c\right ) - b^{2} \cos \left (d x^{2} + c\right )^{2} + a^{2} + b^{2} + 2 \,{\left (2 \, b^{2} d x^{2} \cos \left (d x^{2} + c\right ) + a b\right )} \sin \left (d x^{2} + c\right )}{3 \, x^{3}} \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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