Optimal. Leaf size=121 \[ \frac {a x^5}{5}-\frac {3 \sqrt {\frac {\pi }{2}} b \sin (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{4 d^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} b \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{4 d^{5/2}}+\frac {3 b x \sin \left (c+d x^2\right )}{4 d^2}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 3385, 3386, 3353, 3352, 3351} \[ \frac {a x^5}{5}-\frac {3 \sqrt {\frac {\pi }{2}} b \sin (c) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {d} x\right )}{4 d^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} b \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{4 d^{5/2}}+\frac {3 b x \sin \left (c+d x^2\right )}{4 d^2}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 3351
Rule 3352
Rule 3353
Rule 3385
Rule 3386
Rubi steps
\begin {align*} \int x^4 \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^4+b x^4 \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^5}{5}+b \int x^4 \sin \left (c+d x^2\right ) \, dx\\ &=\frac {a x^5}{5}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d}+\frac {(3 b) \int x^2 \cos \left (c+d x^2\right ) \, dx}{2 d}\\ &=\frac {a x^5}{5}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d}+\frac {3 b x \sin \left (c+d x^2\right )}{4 d^2}-\frac {(3 b) \int \sin \left (c+d x^2\right ) \, dx}{4 d^2}\\ &=\frac {a x^5}{5}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d}+\frac {3 b x \sin \left (c+d x^2\right )}{4 d^2}-\frac {(3 b \cos (c)) \int \sin \left (d x^2\right ) \, dx}{4 d^2}-\frac {(3 b \sin (c)) \int \cos \left (d x^2\right ) \, dx}{4 d^2}\\ &=\frac {a x^5}{5}-\frac {b x^3 \cos \left (c+d x^2\right )}{2 d}-\frac {3 b \sqrt {\frac {\pi }{2}} \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{4 d^{5/2}}-\frac {3 b \sqrt {\frac {\pi }{2}} C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)}{4 d^{5/2}}+\frac {3 b x \sin \left (c+d x^2\right )}{4 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 125, normalized size = 1.03 \[ \frac {a x^5}{5}-\frac {3 \sqrt {\frac {\pi }{2}} b \left (\sin (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+\cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )\right )}{4 d^{5/2}}-\frac {b x \cos \left (d x^2\right ) \left (2 d x^2 \cos (c)-3 \sin (c)\right )}{4 d^2}+\frac {b x \sin \left (d x^2\right ) \left (2 d x^2 \sin (c)+3 \cos (c)\right )}{4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 103, normalized size = 0.85 \[ \frac {8 \, a d^{3} x^{5} - 20 \, b d^{2} x^{3} \cos \left (d x^{2} + c\right ) - 15 \, \sqrt {2} \pi b \sqrt {\frac {d}{\pi }} \cos \relax (c) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) - 15 \, \sqrt {2} \pi b \sqrt {\frac {d}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) \sin \relax (c) + 30 \, b d x \sin \left (d x^{2} + c\right )}{40 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.50, size = 165, normalized size = 1.36 \[ \frac {1}{5} \, a x^{5} - \frac {3 i \, \sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}\right ) e^{\left (i \, c\right )}}{16 \, d^{2} {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}\right ) e^{\left (-i \, c\right )}}{16 \, d^{2} {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}} + \frac {i \, {\left (2 i \, b d x^{3} - 3 \, b x\right )} e^{\left (i \, d x^{2} + i \, c\right )}}{8 \, d^{2}} + \frac {i \, {\left (2 i \, b d x^{3} + 3 \, b x\right )} e^{\left (-i \, d x^{2} - i \, c\right )}}{8 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 89, normalized size = 0.74 \[ \frac {a \,x^{5}}{5}+b \left (-\frac {x^{3} \cos \left (d \,x^{2}+c \right )}{2 d}+\frac {\frac {3 x \sin \left (d \,x^{2}+c \right )}{4 d}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (c ) \mathrm {S}\left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \relax (c ) \FresnelC \left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 d^{\frac {3}{2}}}}{d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.64, size = 92, normalized size = 0.76 \[ \frac {1}{5} \, a x^{5} - \frac {{\left (16 \, d^{3} x^{3} \cos \left (d x^{2} + c\right ) - 24 \, d^{2} x \sin \left (d x^{2} + c\right ) - \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (3 i + 3\right ) \, \cos \relax (c) + \left (3 i - 3\right ) \, \sin \relax (c)\right )} \operatorname {erf}\left (\sqrt {i \, d} x\right ) + {\left (\left (3 i - 3\right ) \, \cos \relax (c) - \left (3 i + 3\right ) \, \sin \relax (c)\right )} \operatorname {erf}\left (\sqrt {-i \, d} x\right )\right )} d^{\frac {3}{2}}\right )} b}{32 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\left (a+b\,\sin \left (d\,x^2+c\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 4.83, size = 488, normalized size = 4.03 \[ \frac {a x^{5}}{5} - \frac {5 \sqrt {2} \sqrt {\pi } b x^{4} \sqrt {\frac {1}{d}} \sin {\relax (c )} C\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{32 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt {2} \sqrt {\pi } b x^{4} \sqrt {\frac {1}{d}} \sin {\relax (c )} C\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right )}{2} - \frac {21 \sqrt {2} \sqrt {\pi } b x^{4} \sqrt {\frac {1}{d}} \cos {\relax (c )} S\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{32 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {2} \sqrt {\pi } b x^{4} \sqrt {\frac {1}{d}} \cos {\relax (c )} S\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right )}{2} - \frac {15 \sqrt {2} \sqrt {\pi } b \sqrt {\frac {1}{d}} \sin {\relax (c )} C\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{128 d^{2} \Gamma \left (\frac {9}{4}\right )} - \frac {63 \sqrt {2} \sqrt {\pi } b \sqrt {\frac {1}{d}} \cos {\relax (c )} S\left (\frac {\sqrt {2} \sqrt {d} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{128 d^{2} \Gamma \left (\frac {11}{4}\right )} + \frac {5 b x^{3} \sqrt {\frac {1}{d}} \sin {\relax (c )} \sin {\left (d x^{2} \right )} \Gamma \left (\frac {1}{4}\right )}{32 \sqrt {d} \Gamma \left (\frac {9}{4}\right )} - \frac {21 b x^{3} \sqrt {\frac {1}{d}} \cos {\relax (c )} \cos {\left (d x^{2} \right )} \Gamma \left (\frac {3}{4}\right )}{32 \sqrt {d} \Gamma \left (\frac {11}{4}\right )} + \frac {15 b x \sqrt {\frac {1}{d}} \sin {\relax (c )} \cos {\left (d x^{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 d^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {63 b x \sqrt {\frac {1}{d}} \sin {\left (d x^{2} \right )} \cos {\relax (c )} \Gamma \left (\frac {3}{4}\right )}{64 d^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________