Optimal. Leaf size=114 \[ -\frac {a}{3 x^3}-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)-\frac {b \sin \left (c+d x^2\right )}{3 x^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 114, 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, 3468, 3469,
3434, 3433, 3432} \begin {gather*} -\frac {a}{3 x^3}-\frac {2}{3} \sqrt {2 \pi } b d^{3/2} \sin (c) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {d} x\right )-\frac {2}{3} \sqrt {2 \pi } b d^{3/2} \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {b \sin \left (c+d x^2\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 3432
Rule 3433
Rule 3434
Rule 3468
Rule 3469
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx &=\int \left (\frac {a}{x^4}+\frac {b \sin \left (c+d x^2\right )}{x^4}\right ) \, dx\\ &=-\frac {a}{3 x^3}+b \int \frac {\sin \left (c+d x^2\right )}{x^4} \, dx\\ &=-\frac {a}{3 x^3}-\frac {b \sin \left (c+d x^2\right )}{3 x^3}+\frac {1}{3} (2 b d) \int \frac {\cos \left (c+d x^2\right )}{x^2} \, dx\\ &=-\frac {a}{3 x^3}-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {b \sin \left (c+d x^2\right )}{3 x^3}-\frac {1}{3} \left (4 b d^2\right ) \int \sin \left (c+d x^2\right ) \, dx\\ &=-\frac {a}{3 x^3}-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {b \sin \left (c+d x^2\right )}{3 x^3}-\frac {1}{3} \left (4 b d^2 \cos (c)\right ) \int \sin \left (d x^2\right ) \, dx-\frac {1}{3} \left (4 b d^2 \sin (c)\right ) \int \cos \left (d x^2\right ) \, dx\\ &=-\frac {a}{3 x^3}-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)-\frac {b \sin \left (c+d x^2\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 119, normalized size = 1.04 \begin {gather*} -\frac {a}{3 x^3}-\frac {b \cos \left (d x^2\right ) \left (2 d x^2 \cos (c)+\sin (c)\right )}{3 x^3}-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \left (\cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)\right )+\frac {b \left (-\cos (c)+2 d x^2 \sin (c)\right ) \sin \left (d x^2\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 83, normalized size = 0.73
method | result | size |
default | \(-\frac {a}{3 x^{3}}+b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{3 x^{3}}+\frac {2 d \left (-\frac {\cos \left (d \,x^{2}+c \right )}{x}-\sqrt {d}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \mathrm {S}\left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (c \right ) \FresnelC \left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )\) | \(83\) |
risch | \(-\frac {a}{3 x^{3}}-\frac {i b \,d^{2} \sqrt {\pi }\, \erf \left (\sqrt {i d}\, x \right ) {\mathrm e}^{-i c}}{3 \sqrt {i d}}+\frac {i b \,d^{2} \sqrt {\pi }\, \erf \left (\sqrt {-i d}\, x \right ) {\mathrm e}^{i c}}{3 \sqrt {-i d}}-\frac {2 b d \cos \left (d \,x^{2}+c \right )}{3 x}-\frac {b \sin \left (d \,x^{2}+c \right )}{3 x^{3}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.61, size = 82, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {d x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, d x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, d x^{2}\right )\right )} \cos \left (c\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, d x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} b d}{8 \, x} - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 98, normalized size = 0.86 \begin {gather*} -\frac {2 \, \sqrt {2} \pi b d x^{3} \sqrt {\frac {d}{\pi }} \cos \left (c\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) + 2 \, \sqrt {2} \pi b d x^{3} \sqrt {\frac {d}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) \sin \left (c\right ) + 2 \, b d x^{2} \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) + a}{3 \, x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {a + b \sin {\left (c + d x^{2} \right )}}{x^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\sin \left (d\,x^2+c\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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