Optimal. Leaf size=44 \[ -\frac {3}{4 x^2}+\frac {\cos \left (2 x^2\right )}{4 x^2}+\text {Ci}\left (x^2\right )-\frac {\sin \left (x^2\right )}{x^2}+\frac {\text {Si}\left (2 x^2\right )}{2} \]
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Rubi [A]
time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3484, 3461,
3378, 3380, 3460, 3383} \begin {gather*} \text {CosIntegral}\left (x^2\right )+\frac {\text {Si}\left (2 x^2\right )}{2}-\frac {3}{4 x^2}-\frac {\sin \left (x^2\right )}{x^2}+\frac {\cos \left (2 x^2\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3460
Rule 3461
Rule 3484
Rubi steps
\begin {align*} \int \frac {\left (1+\sin \left (x^2\right )\right )^2}{x^3} \, dx &=\int \left (\frac {3}{2 x^3}-\frac {\cos \left (2 x^2\right )}{2 x^3}+\frac {2 \sin \left (x^2\right )}{x^3}\right ) \, dx\\ &=-\frac {3}{4 x^2}-\frac {1}{2} \int \frac {\cos \left (2 x^2\right )}{x^3} \, dx+2 \int \frac {\sin \left (x^2\right )}{x^3} \, dx\\ &=-\frac {3}{4 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {\cos (2 x)}{x^2} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {\sin (x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {\cos \left (2 x^2\right )}{4 x^2}-\frac {\sin \left (x^2\right )}{x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {\cos \left (2 x^2\right )}{4 x^2}+\text {Ci}\left (x^2\right )-\frac {\sin \left (x^2\right )}{x^2}+\frac {\text {Si}\left (2 x^2\right )}{2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 41, normalized size = 0.93 \begin {gather*} \frac {-3+\cos \left (2 x^2\right )+4 x^2 \text {Ci}\left (x^2\right )-4 \sin \left (x^2\right )+2 x^2 \text {Si}\left (2 x^2\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 39, normalized size = 0.89
method | result | size |
default | \(-\frac {3}{4 x^{2}}+\cosineIntegral \left (x^{2}\right )+\frac {\cos \left (2 x^{2}\right )}{4 x^{2}}+\frac {\sinIntegral \left (2 x^{2}\right )}{2}-\frac {\sin \left (x^{2}\right )}{x^{2}}\) | \(39\) |
risch | \(\cosineIntegral \left (x^{2}\right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (x^{2}\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right )}{2}-\frac {3}{4 x^{2}}-\frac {\pi \,\mathrm {csgn}\left (x^{2}\right )}{4}+\frac {\sinIntegral \left (2 x^{2}\right )}{2}-\frac {\sin \left (x^{2}\right )}{x^{2}}+\frac {\cos \left (2 x^{2}\right )}{4 x^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.34, size = 54, normalized size = 1.23 \begin {gather*} \frac {x^{2} {\left (i \, \Gamma \left (-1, 2 i \, x^{2}\right ) - i \, \Gamma \left (-1, -2 i \, x^{2}\right )\right )} - 1}{4 \, x^{2}} - \frac {1}{2 \, x^{2}} + \frac {1}{2} \, \Gamma \left (-1, i \, x^{2}\right ) + \frac {1}{2} \, \Gamma \left (-1, -i \, x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 47, normalized size = 1.07 \begin {gather*} \frac {x^{2} \operatorname {Ci}\left (-x^{2}\right ) + x^{2} \operatorname {Ci}\left (x^{2}\right ) + x^{2} \operatorname {Si}\left (2 \, x^{2}\right ) + \cos \left (x^{2}\right )^{2} - 2 \, \sin \left (x^{2}\right ) - 2}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.52, size = 51, normalized size = 1.16 \begin {gather*} - \log {\left (x^{2} \right )} + \frac {\log {\left (x^{4} \right )}}{2} + \operatorname {Ci}{\left (x^{2} \right )} + \frac {\operatorname {Si}{\left (2 x^{2} \right )}}{2} - \frac {\sin {\left (x^{2} \right )}}{x^{2}} + \frac {\cos {\left (2 x^{2} \right )}}{4 x^{2}} - \frac {3}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.05, size = 39, normalized size = 0.89 \begin {gather*} \frac {4 \, x^{2} \operatorname {Ci}\left (x^{2}\right ) + 2 \, x^{2} \operatorname {Si}\left (2 \, x^{2}\right ) + \cos \left (2 \, x^{2}\right ) - 4 \, \sin \left (x^{2}\right ) - 3}{4 \, x^{2}} \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 = -0.02 \begin {gather*} \mathrm {cosint}\left (x^2\right )+\frac {\mathrm {sinint}\left (2\,x^2\right )}{2}-\frac {\sin \left (x^2\right )}{x^2}+\frac {{\cos \left (x^2\right )}^2}{2\,x^2}-\frac {1}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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