Optimal. Leaf size=353 \[ -\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.57135, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b} \]
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 (c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{4} (c+d x)^{3/2} \cos (a+b x)-\frac{1}{4} (c+d x)^{3/2} \cos (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int (c+d x)^{3/2} \cos (a+b x) \, dx-\frac{1}{4} \int (c+d x)^{3/2} \cos (3 a+3 b x) \, dx\\ &=\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac{d \int \sqrt{c+d x} \sin (3 a+3 b x) \, dx}{8 b}-\frac{(3 d) \int \sqrt{c+d x} \sin (a+b x) \, dx}{8 b}\\ &=\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac{d^2 \int \frac{\cos (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{48 b^2}-\frac{\left (3 d^2\right ) \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{16 b^2}\\ &=\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac{\left (d^2 \cos \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}{48 b^2}-\frac{\left (3 d^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{16 b^2}-\frac{\left (d^2 \sin \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}{48 b^2}+\frac{\left (3 d^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{16 b^2}\\ &=\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac{\left (d \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{24 b^2}-\frac{\left (3 d \cos \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{8 b^2}-\frac{\left (d \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{24 b^2}+\frac{\left (3 d \sin \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{8 b^2}\\ &=\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}-\frac{3 d^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{d^{3/2} \sqrt{\frac{\pi }{6}} \cos \left (3 a-\frac{3 b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}-\frac{d^{3/2} \sqrt{\frac{\pi }{6}} S\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 )}{24 b^{5/2}}+\frac{3 d^{3/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{8 b^{5/2}}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}\\ \end{align*}
Mathematica [C] time = 9.30628, size = 677, normalized size = 1.92 \[ -\frac{i c \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left (\frac{e^{2 i a} \text{Gamma}\left (\frac{3}{2},-\frac{i b (c+d x)}{d}\right )}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \text{Gamma}\left (\frac{3}{2},\frac{i b (c+d x)}{d}\right )}{\sqrt{\frac{i b (c+d x)}{d}}}\right )}{8 b}+\frac{d \left (\sqrt{2 \pi } \sqrt{\frac{b}{d}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right ) \left (2 b c \sin \left (a-\frac{b c}{d}\right )-3 d \cos \left (a-\frac{b c}{d}\right )\right )+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right ) \left (3 d \sin \left (a-\frac{b c}{d}\right )+2 b c \cos \left (a-\frac{b c}{d}\right )\right )+2 b \sqrt{c+d x} (2 b x \sin (a+b x)+3 \cos (a+b x))\right )}{16 b^3}-\frac{d \left (\sqrt{2 \pi } \sqrt{\frac{b}{d}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right ) \left (2 b c \sin \left (3 a-\frac{3 b c}{d}\right )-d \cos \left (3 a-\frac{3 b c}{d}\right )\right )+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right ) \left (d \sin \left (3 a-\frac{3 b c}{d}\right )+2 b c \cos \left (3 a-\frac{3 b c}{d}\right )\right )+2 \sqrt{3} b \sqrt{c+d x} (2 b x \sin (3 (a+b x))+\cos (3 (a+b x)))\right )}{48 \sqrt{3} b^3}-\frac{c \left (-\sqrt{2 \pi } \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-\sqrt{2 \pi } \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right )}{24 \sqrt{3} b \sqrt{\frac{b}{d}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 386, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( 1/8\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }-3/8\,{\frac{d}{b} \left ( -1/2\,{\frac{d\sqrt{dx+c}}{b}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }+1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \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 ) -\sin \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 ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) }-1/24\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }+1/8\,{\frac{d}{b} \left ( -1/6\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }+1/36\,{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{3}}{b} \left ( \cos \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 ) -\sin \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 ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 2.37528, size = 1787, normalized size = 5.06 \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 = 0.607138, size = 756, normalized size = 2.14 \begin{align*} \frac{\sqrt{6} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 27 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{C}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 27 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{S}\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^{2} \sqrt{\frac{b}{\pi d}} \operatorname{S}\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 d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right ) - 2 \,{\left (b^{2} d x + b^{2} c -{\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{144 \, b^{3}} \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 = 1.37434, size = 1517, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]