Optimal. Leaf size=174 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{3/2}}{12 d} \]
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Rubi [A] time = 0.270369, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{3/2}}{12 d} \]
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 ^2(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{8} \sqrt{c+d x}-\frac{1}{8} \sqrt{c+d x} \cos (4 a+4 b x)\right ) \, dx\\ &=\frac{(c+d x)^{3/2}}{12 d}-\frac{1}{8} \int \sqrt{c+d x} \cos (4 a+4 b x) \, dx\\ &=\frac{(c+d x)^{3/2}}{12 d}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{d \int \frac{\sin (4 a+4 b x)}{\sqrt{c+d x}} \, dx}{64 b}\\ &=\frac{(c+d x)^{3/2}}{12 d}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{\left (d \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{64 b}+\frac{\left (d \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{64 b}\\ &=\frac{(c+d x)^{3/2}}{12 d}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{\cos \left (4 a-\frac{4 b c}{d}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{32 b}+\frac{\sin \left (4 a-\frac{4 b c}{d}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{32 b}\\ &=\frac{(c+d x)^{3/2}}{12 d}+\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} C\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (4 a-\frac{4 b c}{d}\right )}{64 b^{3/2}}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 0.885184, size = 161, normalized size = 0.93 \[ \frac{3 \sqrt{2 \pi } d \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+3 \sqrt{2 \pi } d \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} (8 b (c+d x)-3 d \sin (4 (a+b x)))}{384 d^2 \left (\frac{b}{d}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 159, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( 1/24\, \left ( dx+c \right ) ^{3/2}-{\frac{d\sqrt{dx+c}}{64\,b}\sin \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }+{\frac{d\sqrt{2}\sqrt{\pi }}{256\,b} \left ( \cos \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\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 = 1.93622, size = 817, normalized size = 4.7 \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 = 0.569092, size = 447, normalized size = 2.57 \begin{align*} \frac{3 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 3 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) + 16 \,{\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \,{\left (2 \, b d \cos \left (b x + a\right )^{3} - b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{384 \, b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.29019, size = 347, normalized size = 1.99 \begin{align*} -\frac{\frac{3 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 )}}{d}\right ) e^{\left (\frac{4 i \, b c - 4 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{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 )}}{d}\right ) e^{\left (\frac{-4 i \, b c + 4 i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - 64 \,{\left (d x + c\right )}^{\frac{3}{2}} - \frac{12 i \, \sqrt{d x + c} d e^{\left (\frac{4 i \,{\left (d x + c\right )} b - 4 i \, b c + 4 i \, a d}{d}\right )}}{b} + \frac{12 i \, \sqrt{d x + c} d e^{\left (\frac{-4 i \,{\left (d x + c\right )} b + 4 i \, b c - 4 i \, a d}{d}\right )}}{b}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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