Optimal. Leaf size=74 \[ \frac {1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \sin (c) \text {Ci}\left (d x^2\right )+a b \cos (c) \text {Si}\left (d x^2\right )-\frac {1}{4} b^2 \cos (2 c) \text {Ci}\left (2 d x^2\right )+\frac {1}{4} b^2 \sin (2 c) \text {Si}\left (2 d x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3403, 6, 3378, 3376, 3375, 3377} \[ \frac {1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \sin (c) \text {CosIntegral}\left (d x^2\right )+a b \cos (c) \text {Si}\left (d x^2\right )-\frac {1}{4} b^2 \cos (2 c) \text {CosIntegral}\left (2 d x^2\right )+\frac {1}{4} b^2 \sin (2 c) \text {Si}\left (2 d x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 3375
Rule 3376
Rule 3377
Rule 3378
Rule 3403
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin \left (c+d x^2\right )\right )^2}{x} \, dx &=\int \left (\frac {a^2}{x}+\frac {b^2}{2 x}-\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+\frac {2 a b \sin \left (c+d x^2\right )}{x}\right ) \, dx\\ &=\int \left (\frac {a^2+\frac {b^2}{2}}{x}-\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+\frac {2 a b \sin \left (c+d x^2\right )}{x}\right ) \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) \log (x)+(2 a b) \int \frac {\sin \left (c+d x^2\right )}{x} \, dx-\frac {1}{2} b^2 \int \frac {\cos \left (2 c+2 d x^2\right )}{x} \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) \log (x)+(2 a b \cos (c)) \int \frac {\sin \left (d x^2\right )}{x} \, dx-\frac {1}{2} \left (b^2 \cos (2 c)\right ) \int \frac {\cos \left (2 d x^2\right )}{x} \, dx+(2 a b \sin (c)) \int \frac {\cos \left (d x^2\right )}{x} \, dx+\frac {1}{2} \left (b^2 \sin (2 c)\right ) \int \frac {\sin \left (2 d x^2\right )}{x} \, dx\\ &=-\frac {1}{4} b^2 \cos (2 c) \text {Ci}\left (2 d x^2\right )+\frac {1}{2} \left (2 a^2+b^2\right ) \log (x)+a b \text {Ci}\left (d x^2\right ) \sin (c)+a b \cos (c) \text {Si}\left (d x^2\right )+\frac {1}{4} b^2 \sin (2 c) \text {Si}\left (2 d x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 71, normalized size = 0.96 \[ \frac {1}{2} \left (2 a^2+b^2\right ) \log (x)-\frac {1}{4} b \left (-4 a \sin (c) \text {Ci}\left (d x^2\right )-4 a \cos (c) \text {Si}\left (d x^2\right )+b \cos (2 c) \text {Ci}\left (2 d x^2\right )-b \sin (2 c) \text {Si}\left (2 d x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 94, normalized size = 1.27 \[ \frac {1}{4} \, b^{2} \sin \left (2 \, c\right ) \operatorname {Si}\left (2 \, d x^{2}\right ) + a b \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) - \frac {1}{8} \, {\left (b^{2} \operatorname {Ci}\left (2 \, d x^{2}\right ) + b^{2} \operatorname {Ci}\left (-2 \, d x^{2}\right )\right )} \cos \left (2 \, c\right ) + \frac {1}{2} \, {\left (2 \, a^{2} + b^{2}\right )} \log \relax (x) + \frac {1}{2} \, {\left (a b \operatorname {Ci}\left (d x^{2}\right ) + a b \operatorname {Ci}\left (-d x^{2}\right )\right )} \sin \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 77, normalized size = 1.04 \[ -\frac {1}{4} \, b^{2} \cos \left (2 \, c\right ) \operatorname {Ci}\left (2 \, d x^{2}\right ) + a b \operatorname {Ci}\left (d x^{2}\right ) \sin \relax (c) + a b \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) - \frac {1}{4} \, b^{2} \sin \left (2 \, c\right ) \operatorname {Si}\left (-2 \, d x^{2}\right ) + \frac {1}{2} \, a^{2} \log \left (d x^{2}\right ) + \frac {1}{4} \, b^{2} \log \left (d x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.59, size = 157, normalized size = 2.12 \[ -\frac {\pi \,{\mathrm e}^{-i c} \mathrm {csgn}\left (d \,x^{2}\right ) a b}{2}+{\mathrm e}^{-i c} \Si \left (d \,x^{2}\right ) a b -\frac {i {\mathrm e}^{-i c} \Ei \left (1, -i d \,x^{2}\right ) a b}{2}+\ln \relax (x ) a^{2}+\frac {\ln \relax (x ) b^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (d \,x^{2}\right ) {\mathrm e}^{-2 i c} b^{2}}{8}+\frac {i {\mathrm e}^{-2 i c} \Si \left (2 d \,x^{2}\right ) b^{2}}{4}+\frac {{\mathrm e}^{-2 i c} \Ei \left (1, -2 i d \,x^{2}\right ) b^{2}}{8}+\frac {b^{2} {\mathrm e}^{2 i c} \Ei \left (1, -2 i d \,x^{2}\right )}{8}+\frac {i a b \,{\mathrm e}^{i c} \Ei \left (1, -i d \,x^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.50, size = 108, normalized size = 1.46 \[ -\frac {1}{2} \, {\left ({\left (i \, {\rm Ei}\left (i \, d x^{2}\right ) - i \, {\rm Ei}\left (-i \, d x^{2}\right )\right )} \cos \relax (c) - {\left ({\rm Ei}\left (i \, d x^{2}\right ) + {\rm Ei}\left (-i \, d x^{2}\right )\right )} \sin \relax (c)\right )} a b - \frac {1}{8} \, {\left ({\left ({\rm Ei}\left (2 i \, d x^{2}\right ) + {\rm Ei}\left (-2 i \, d x^{2}\right )\right )} \cos \left (2 \, c\right ) - {\left (-i \, {\rm Ei}\left (2 i \, d x^{2}\right ) + i \, {\rm Ei}\left (-2 i \, d x^{2}\right )\right )} \sin \left (2 \, c\right ) - 4 \, \log \relax (x)\right )} b^{2} + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sin {\left (c + d x^{2} \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________