Optimal. Leaf size=250 \[ \frac{256 \sqrt{2 \pi } b^{15/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{675675}-\frac{256 \sqrt{2 \pi } b^{15/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{675675}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}} \]
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Rubi [A] time = 0.336934, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3416, 3297, 3306, 3305, 3351, 3304, 3352} \[ \frac{256 \sqrt{2 \pi } b^{15/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{675675}-\frac{256 \sqrt{2 \pi } b^{15/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{675675}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3416
Rule 3297
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^{17/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}-\frac{1}{5} (2 b) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{15/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{1}{65} \left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^{13/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}+\frac{1}{715} \left (8 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{\left (16 b^4\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )}{6435}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac{\left (32 b^5\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )}{45045}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{\left (64 b^6\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )}{225225}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}+\frac{\left (128 b^7\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (256 b^8\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (256 b^8 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}-\frac{\left (256 b^8 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (512 b^8 \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}-\frac{\left (512 b^8 \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{256 b^{15/2} \sqrt{2 \pi } \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac{256 b^{15/2} \sqrt{2 \pi } S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{675675}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}\\ \end{align*}
Mathematica [A] time = 0.316166, size = 238, normalized size = 0.95 \[ \frac{2 \left (128 \sqrt{2 \pi } b^{15/2} x^{5/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )-128 \sqrt{2 \pi } b^{15/2} x^{5/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )-128 b^7 x^{7/3} \sin \left (a+b \sqrt [3]{x}\right )+96 b^5 x^{5/3} \sin \left (a+b \sqrt [3]{x}\right )+64 b^6 x^2 \cos \left (a+b \sqrt [3]{x}\right )-240 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )+3780 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )-840 b^3 x \sin \left (a+b \sqrt [3]{x}\right )+20790 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )-135135 \cos \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 180, normalized size = 0.7 \begin{align*} -{\frac{2}{5}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{5}{2}}}}-{\frac{4\,b}{5} \left ( -{\frac{1}{13}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{13}{6}}}}+{\frac{2\,b}{13} \left ( -{\frac{1}{11}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{11}{6}}}}-{\frac{2\,b}{11} \left ( -{\frac{1}{9}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{3}{2}}}}+{\frac{2\,b}{9} \left ( -{\frac{1}{7}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{7}{6}}}}-{\frac{2\,b}{7} \left ( -{\frac{1}{5}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{5}{6}}}}+{\frac{2\,b}{5} \left ( -{\frac{1}{3}\cos \left ( a+b\sqrt [3]{x} \right ){\frac{1}{\sqrt{x}}}}-{\frac{2\,b}{3} \left ( -{\sin \left ( a+b\sqrt [3]{x} \right ){\frac{1}{\sqrt [6]{x}}}}+\sqrt{b}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) \right ) \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.43609, size = 362, normalized size = 1.45 \begin{align*} -\frac{3 \,{\left ({\left ({\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) - i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x^{\frac{1}{3}}{\left | b \right |}} b^{6}{\left | b \right |}}{4 \, x^{\frac{1}{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13131, size = 490, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (128 \, \sqrt{2} \pi b^{7} x^{3} \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 128 \, \sqrt{2} \pi b^{7} x^{3} \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) -{\left (240 \, b^{4} x^{\frac{11}{6}} - 3780 \, b^{2} x^{\frac{7}{6}} -{\left (64 \, b^{6} x^{2} - 135135\right )} \sqrt{x}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) + 2 \,{\left (48 \, b^{5} x^{\frac{13}{6}} - 420 \, b^{3} x^{\frac{3}{2}} -{\left (64 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac{5}{6}}\right )} \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{675675 \, x^{3}} \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 \frac{\cos{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{\frac{1}{3}} + a\right )}{x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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