Optimal. Leaf size=130 \[ -\frac{\sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{c+d x}}{d} \]
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Rubi [A] time = 0.234267, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\sin ^2(a+b x)}{\sqrt{c+d x}} \, dx &=\int \left (\frac{1}{2 \sqrt{c+d x}}-\frac{\cos (2 a+2 b x)}{2 \sqrt{c+d x}}\right ) \, dx\\ &=\frac{\sqrt{c+d x}}{d}-\frac{1}{2} \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx\\ &=\frac{\sqrt{c+d x}}{d}-\frac{1}{2} \cos \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx+\frac{1}{2} \sin \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx\\ &=\frac{\sqrt{c+d x}}{d}-\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{\sqrt{c+d x}}{d}-\frac{\sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{2 \sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.225902, size = 126, normalized size = 0.97 \[ \frac{\sqrt{\frac{b}{d}} \left (-\sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+\sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+2 \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 108, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{d} \left ( 1/2\,\sqrt{dx+c}-1/4\,{\sqrt{\pi } \left ( \cos \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.85129, size = 747, normalized size = 5.75 \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 = 2.05816, size = 281, normalized size = 2.16 \begin{align*} -\frac{\pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - \pi d \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \sqrt{d x + c} b}{2 \, b d} \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{\sin ^{2}{\left (a + b x \right )}}{\sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.18342, size = 220, normalized size = 1.69 \begin{align*} \frac{\frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{2 i \, b c - 2 i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}} + \frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{-2 i \, b c + 2 i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}} + 4 \, \sqrt{d x + c}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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