Optimal. Leaf size=203 \[ \frac{3 \sqrt{\pi } d^{3/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{32 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{32 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}-\frac{(c+d x)^{3/2} \sin (a+b x) \cos (a+b x)}{2 b}-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.359744, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3311, 32, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{3 \sqrt{\pi } d^{3/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{32 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{32 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}-\frac{(c+d x)^{3/2} \sin (a+b x) \cos (a+b x)}{2 b}-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3311
Rule 32
Rule 3312
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int (c+d x)^{3/2} \sin ^2(a+b x) \, dx &=-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}+\frac{1}{2} \int (c+d x)^{3/2} \, dx-\frac{\left (3 d^2\right ) \int \frac{\sin ^2(a+b x)}{\sqrt{c+d x}} \, dx}{16 b^2}\\ &=\frac{(c+d x)^{5/2}}{5 d}-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}-\frac{\left (3 d^2\right ) \int \left (\frac{1}{2 \sqrt{c+d x}}-\frac{\cos (2 a+2 b x)}{2 \sqrt{c+d x}}\right ) \, dx}{16 b^2}\\ &=-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}+\frac{\left (3 d^2\right ) \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{32 b^2}\\ &=-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}+\frac{\left (3 d^2 \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}{32 b^2}-\frac{\left (3 d^2 \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}{32 b^2}\\ &=-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}+\frac{\left (3 d \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{16 b^2}-\frac{\left (3 d \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{16 b^2}\\ &=-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}+\frac{3 d^{3/2} \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 )}{32 b^{5/2}}-\frac{3 d^{3/2} \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 )}{32 b^{5/2}}-\frac{(c+d x)^{3/2} \cos (a+b x) \sin (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}\\ \end{align*}
Mathematica [A] time = 1.69637, size = 175, normalized size = 0.86 \[ \frac{\sqrt{\frac{b}{d}} \left (15 \sqrt{\pi } d^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-15 \sqrt{\pi } d^2 \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left (4 b (c+d x) (4 b (c+d x)-5 d \sin (2 (a+b x)))-15 d^2 \cos (2 (a+b x))\right )\right )}{160 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 197, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( 1/10\, \left ( dx+c \right ) ^{5/2}-1/8\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }+3/8\,{\frac{d}{b} \left ( -1/4\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{da-cb}{d}} \right ) }+1/8\,{\frac{d\sqrt{\pi }}{b} \left ( \cos \left ( 2\,{\frac{da-cb}{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{da-cb}{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}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.82384, size = 899, normalized size = 4.43 \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 = 2.15745, size = 478, normalized size = 2.35 \begin{align*} \frac{15 \, \pi d^{3} \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 ) - 15 \, \pi d^{3} \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 ) + 2 \,{\left (16 \, b^{3} d^{2} x^{2} + 32 \, b^{3} c d x + 16 \, b^{3} c^{2} - 30 \, b d^{2} \cos \left (b x + a\right )^{2} + 15 \, b d^{2} - 40 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{160 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{\frac{3}{2}} \sin ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.26014, size = 772, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]