Optimal. Leaf size=25 \[ b \text {CosIntegral}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6638, 12, 3378,
3383} \begin {gather*} b \text {CosIntegral}(b x)-\frac {\text {Si}(b x)}{x}-\frac {\sin (b x)}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3378
Rule 3383
Rule 6638
Rubi steps
\begin {align*} \int \frac {\text {Si}(b x)}{x^2} \, dx &=-\frac {\text {Si}(b x)}{x}+b \int \frac {\sin (b x)}{b x^2} \, dx\\ &=-\frac {\text {Si}(b x)}{x}+\int \frac {\sin (b x)}{x^2} \, dx\\ &=-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x}+b \int \frac {\cos (b x)}{x} \, dx\\ &=b \text {Ci}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} b \text {CosIntegral}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.30, size = 32, normalized size = 1.28
method | result | size |
derivativedivides | \(b \left (-\frac {\sinIntegral \left (b x \right )}{b x}-\frac {\sin \left (b x \right )}{b x}+\cosineIntegral \left (b x \right )\right )\) | \(32\) |
default | \(b \left (-\frac {\sinIntegral \left (b x \right )}{b x}-\frac {\sin \left (b x \right )}{b x}+\cosineIntegral \left (b x \right )\right )\) | \(32\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {2 b^{2} x^{2} \hypergeom \left (\left [1, 1, \frac {3}{2}\right ], \left [2, 2, \frac {5}{2}, \frac {5}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{9 \sqrt {\pi }}+\frac {8 \gamma -16+8 \ln \left (x \right )+8 \ln \left (b \right )}{\sqrt {\pi }}\right )}{8}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.30, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, b {\left (\Gamma \left (-1, i \, b x\right ) + \Gamma \left (-1, -i \, b x\right )\right )} - \frac {\operatorname {Si}\left (b x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 33, normalized size = 1.32 \begin {gather*} \frac {b x \operatorname {Ci}\left (b x\right ) + b x \operatorname {Ci}\left (-b x\right ) - 2 \, \sin \left (b x\right ) - 2 \, \operatorname {Si}\left (b x\right )}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.52, size = 36, normalized size = 1.44 \begin {gather*} - \frac {b^{3} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {3}{2} \\ 2, 2, \frac {5}{2}, \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{36} + \frac {b \log {\left (b^{2} x^{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 37, normalized size = 1.48 \begin {gather*} \frac {b x \operatorname {Ci}\left (b x\right ) + b x \operatorname {Ci}\left (-b x\right ) - 2 \, \sin \left (b x\right )}{2 \, x} - \frac {\operatorname {Si}\left (b x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} b\,\mathrm {cosint}\left (b\,x\right )-\frac {\mathrm {sinint}\left (b\,x\right )}{x}-\frac {\sin \left (b\,x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________