Optimal. Leaf size=158 \[ -\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{8 b^{3/2}}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}+\frac{(c+d x)^{3/2}}{3 d} \]
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Rubi [A] time = 0.278329, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3312, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{8 b^{3/2}}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}+\frac{(c+d x)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \sqrt{c+d x} \cos ^2(a+b x) \, dx &=\int \left (\frac{1}{2} \sqrt{c+d x}+\frac{1}{2} \sqrt{c+d x} \cos (2 a+2 b x)\right ) \, dx\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{1}{2} \int \sqrt{c+d x} \cos (2 a+2 b x) \, dx\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}-\frac{d \int \frac{\sin (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{8 b}\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}-\frac{\left (d \cos \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{\left (d \sin \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{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}-\frac{\cos \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{\sin \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{(c+d x)^{3/2}}{3 d}-\frac{\sqrt{d} \sqrt{\pi } \cos \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{d} \sqrt{\pi } C\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}}+\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}\\ \end{align*}
Mathematica [C] time = 0.544999, size = 146, normalized size = 0.92 \[ \frac{1}{48} \sqrt{c+d x} \left (-\frac{3 i \sqrt{2} e^{2 i \left (a-\frac{b c}{d}\right )} \text{Gamma}\left (\frac{3}{2},-\frac{2 i b (c+d x)}{d}\right )}{b \sqrt{-\frac{i b (c+d x)}{d}}}+\frac{3 i \sqrt{2} e^{-2 i \left (a-\frac{b c}{d}\right )} \text{Gamma}\left (\frac{3}{2},\frac{2 i b (c+d x)}{d}\right )}{b \sqrt{\frac{i b (c+d x)}{d}}}+\frac{16 (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 150, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( 1/6\, \left ( dx+c \right ) ^{3/2}+1/8\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }-1/16\,{\frac{d\sqrt{\pi }}{b} \left ( \cos \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{b\sqrt{dx+c}}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 2\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{b\sqrt{dx+c}}{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 = 2.28497, size = 824, normalized size = 5.22 \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 = 1.88309, size = 369, normalized size = 2.34 \begin{align*} -\frac{3 \, \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 3 \, \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 4 \,{\left (2 \, b^{2} d x + 3 \, b d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, b^{2} c\right )} \sqrt{d x + c}}{24 \, b^{2} 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 \sqrt{c + d x} \cos ^{2}{\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 = 1.20211, size = 331, normalized size = 2.09 \begin{align*} -\frac{-\frac{3 i \, \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{3 i \, \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} - 16 \,{\left (d x + c\right )}^{\frac{3}{2}} + \frac{6 i \, \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{6 i \, \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}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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