Optimal. Leaf size=299 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.446811, antiderivative size = 299, 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} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4406
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \sqrt{c+d x} \cos ^3(a+b x) \sin (a+b x) \, dx &=\int \left (\frac{1}{4} \sqrt{c+d x} \sin (2 a+2 b x)+\frac{1}{8} \sqrt{c+d x} \sin (4 a+4 b x)\right ) \, dx\\ &=\frac{1}{8} \int \sqrt{c+d x} \sin (4 a+4 b x) \, dx+\frac{1}{4} \int \sqrt{c+d x} \sin (2 a+2 b x) \, dx\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}+\frac{d \int \frac{\cos (4 a+4 b x)}{\sqrt{c+d x}} \, dx}{64 b}+\frac{d \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{16 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}+\frac{\left (d \cos \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{\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}{16 b}-\frac{\left (d \sin \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 (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}{16 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}+\frac{\cos \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{\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 )}{8 b}-\frac{\sin \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 (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 )}{8 b}\\ &=-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}+\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} \cos \left (4 a-\frac{4 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{3/2}}+\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 )}{16 b^{3/2}}-\frac{\sqrt{d} \sqrt{\frac{\pi }{2}} S\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{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 )}{16 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.83486, size = 264, normalized size = 0.88 \[ \frac{\sqrt{2 \pi } \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+8 \sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-\sqrt{2 \pi } \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )-8 \sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-16 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{128 b \sqrt{\frac{b}{d}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 286, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( -1/16\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+1/32\,{\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}}}}}}-{\frac{d\sqrt{dx+c}}{64\,b}\cos \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 FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 2.1839, size = 1661, normalized size = 5.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.61023, size = 599, normalized size = 2. \begin{align*} \frac{\sqrt{2} \pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - \sqrt{2} \pi d \sqrt{\frac{b}{\pi d}} \operatorname{S}\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 ) + 8 \, \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 ) - 8 \, \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 ) - 4 \,{\left (8 \, b \cos \left (b x + a\right )^{4} - 3 \, b\right )} \sqrt{d x + c}}{128 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.3384, size = 643, normalized size = 2.15 \begin{align*} -\frac{\frac{\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{\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{8 \, \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{8 \, \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{4 \, \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{16 \, \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{16 \, \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{4 \, \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}}{256 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]