Optimal. Leaf size=250 \[ \frac {256 \sqrt {2 \pi } b^{15/2} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 \sqrt {2 \pi } b^{15/2} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 250, normalized size of antiderivative = 1.00, 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 3297
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 3416
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*}
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Mathematica [A] time = 0.34, size = 238, normalized size = 0.95 \[ \frac {2 \left (128 \sqrt {2 \pi } b^{15/2} x^{5/2} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-128 \sqrt {2 \pi } b^{15/2} x^{5/2} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-128 b^7 x^{7/3} \sin \left (a+b \sqrt [3]{x}\right )+64 b^6 x^2 \cos \left (a+b \sqrt [3]{x}\right )+96 b^5 x^{5/3} \sin \left (a+b \sqrt [3]{x}\right )-240 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )-840 b^3 x \sin \left (a+b \sqrt [3]{x}\right )+3780 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )+20790 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )-135135 \cos \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 164, normalized size = 0.66 \[ \frac {2 \, {\left (128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) - {\left (240 \, b^{4} x^{\frac {11}{6}} - 3780 \, b^{2} x^{\frac {7}{6}} - {\left (64 \, b^{6} x^{2} - 135135\right )} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 2 \, {\left (48 \, b^{5} x^{\frac {13}{6}} - 420 \, b^{3} x^{\frac {3}{2}} - {\left (64 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac {5}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{675675 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 180, normalized size = 0.72 \[ -\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 76, normalized size = 0.30 \[ \frac {3 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \relax (a) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \relax (a)\right )} \sqrt {b x^{\frac {1}{3}}} b^{7}}{4 \, x^{\frac {1}{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\cos \left (a+b\,x^{1/3}\right )}{x^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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