Optimal. Leaf size=142 \[ \frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b} \]
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Rubi [A] time = 0.22096, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4406, 12, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 12
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \sqrt{c+d x} \cos (a+b x) \sin (a+b x) \, dx &=\int \frac{1}{2} \sqrt{c+d x} \sin (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int \sqrt{c+d x} \sin (2 a+2 b x) \, dx\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}+\frac{d \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{8 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}+\frac{\left (d \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{8 b}-\frac{\left (d \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{8 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}+\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 )}{4 b}-\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 )}{4 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}+\frac{\sqrt{d} \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 )}{8 b^{3/2}}-\frac{\sqrt{d} \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 )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.026048, size = 134, normalized size = 0.94 \[ \frac{\sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-\sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))}{8 b \sqrt{\frac{b}{d}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 142, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( -1/8\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+1/16\,{\frac{d\sqrt{\pi }}{b} \left ( \cos \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\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 = 1.97827, size = 788, normalized size = 5.55 \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 = 0.519941, size = 308, normalized size = 2.17 \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 \,{\left (2 \, b \cos \left (b x + a\right )^{2} - b\right )} \sqrt{d x + c}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.84613, size = 389, normalized size = 2.74 \begin{align*} - \frac{b^{\frac{3}{2}} \sqrt{\frac{d}{b}} \left (c + d x\right )^{\frac{5}{2}} \cos{\left (2 a - \frac{2 b c}{d} \right )} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle |{- \frac{b^{2} \left (c + d x\right )^{2}}{d^{2}}} \right )}}{4 d^{\frac{5}{2}} \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{9}{4}\right )} - \frac{\sqrt{b} \sqrt{\frac{d}{b}} \left (c + d x\right )^{\frac{3}{2}} \sin{\left (2 a - \frac{2 b c}{d} \right )} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{b^{2} \left (c + d x\right )^{2}}{d^{2}}} \right )}}{8 d^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{\pi } c \sqrt{\frac{d}{b}} \sin{\left (2 a - \frac{2 b c}{d} \right )} C\left (\frac{2 b \sqrt{c + d x}}{\sqrt{\pi } d \sqrt{\frac{b}{d}}}\right )}{2 d} + \frac{\sqrt{\pi } c \sqrt{\frac{d}{b}} \cos{\left (2 a - \frac{2 b c}{d} \right )} S\left (\frac{2 b \sqrt{c + d x}}{\sqrt{\pi } d \sqrt{\frac{b}{d}}}\right )}{2 d} + \frac{\sqrt{\pi } x \sqrt{\frac{d}{b}} \sin{\left (2 a - \frac{2 b c}{d} \right )} C\left (\frac{2 b \sqrt{c + d x}}{\sqrt{\pi } d \sqrt{\frac{b}{d}}}\right )}{2} + \frac{\sqrt{\pi } x \sqrt{\frac{d}{b}} \cos{\left (2 a - \frac{2 b c}{d} \right )} S\left (\frac{2 b \sqrt{c + d x}}{\sqrt{\pi } d \sqrt{\frac{b}{d}}}\right )}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15843, size = 316, normalized size = 2.23 \begin{align*} -\frac{\frac{\sqrt{\pi } d^{2} \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 )} b} + \frac{\sqrt{\pi } d^{2} \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 )} b} + \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{2 i \,{\left (d x + c\right )} b - 2 i \, b c + 2 i \, a d}{d}\right )}}{b} + \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{-2 i \,{\left (d x + c\right )} b + 2 i \, b c - 2 i \, a d}{d}\right )}}{b}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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