Optimal. Leaf size=304 \[ \frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{12 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b} \]
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Rubi [A] time = 0.469995, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{12 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b} \]
Antiderivative was successfully verified.
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Rule 4406
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 ^2(a+b x) \, dx &=\int \left (\frac{1}{4} \sqrt{c+d x} \cos (a+b x)-\frac{1}{4} \sqrt{c+d x} \cos (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int \sqrt{c+d x} \cos (a+b x) \, dx-\frac{1}{4} \int \sqrt{c+d x} \cos (3 a+3 b x) \, dx\\ &=\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}+\frac{d \int \frac{\sin (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{24 b}-\frac{d \int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx}{8 b}\\ &=\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}+\frac{\left (d \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{24 b}-\frac{\left (d \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{8 b}+\frac{\left (d \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{24 b}-\frac{\left (d \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{8 b}\\ &=\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{12 b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{4 b}+\frac{\sin \left (3 a-\frac{3 b c}{d}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{12 b}-\frac{\sin \left (a-\frac{b c}{d}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{4 b}\\ &=-\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{d} \sqrt{\frac{\pi }{6}} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{12 b^{3/2}}+\frac{\sqrt{d} \sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{12 b^{3/2}}-\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}\\ \end{align*}
Mathematica [C] time = 6.56647, size = 280, normalized size = 0.92 \[ -\frac{-\sqrt{2 \pi } \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-\sqrt{2 \pi } \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{i \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left (\frac{e^{2 i a} \text{Gamma}\left (\frac{3}{2},-\frac{i b (c+d x)}{d}\right )}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \text{Gamma}\left (\frac{3}{2},\frac{i b (c+d x)}{d}\right )}{\sqrt{\frac{i b (c+d x)}{d}}}\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 294, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( 1/8\,{\frac{d\sqrt{dx+c}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }-1/16\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ({\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ({\frac{ad-bc}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}}-1/24\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }+{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{3}}{144\,b} \left ( \cos \left ( 3\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 3\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\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 = 2.31223, size = 1648, normalized size = 5.42 \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 = 0.583368, size = 639, normalized size = 2.1 \begin{align*} \frac{\sqrt{6} \pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 9 \, \sqrt{2} \pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{S}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 9 \, \sqrt{2} \pi d \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{b c - a d}{d}\right ) + \sqrt{6} \pi d \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) - 24 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sqrt{d x + c} \sin \left (b x + a\right )}{72 \, b^{2}} \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 \sqrt{c + d x} \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.24356, size = 662, normalized size = 2.18 \begin{align*} -\frac{\frac{i \, \sqrt{6} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{3 i \, b c - 3 i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - \frac{9 i \, \sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{i \, b c - i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} + \frac{9 i \, \sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{-i \, b c + i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - \frac{i \, \sqrt{6} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{-3 i \, b c + 3 i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - \frac{6 i \, \sqrt{d x + c} d e^{\left (\frac{3 i \,{\left (d x + c\right )} b - 3 i \, b c + 3 i \, a d}{d}\right )}}{b} + \frac{18 i \, \sqrt{d x + c} d e^{\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} - \frac{18 i \, \sqrt{d x + c} d e^{\left (\frac{-i \,{\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b} + \frac{6 i \, \sqrt{d x + c} d e^{\left (\frac{-3 i \,{\left (d x + c\right )} b + 3 i \, b c - 3 i \, a d}{d}\right )}}{b}}{144 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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