Optimal. Leaf size=91 \[ \frac {3}{8} b \cos (a) \text {Ci}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {Ci}\left (3 b x^2\right )-\frac {3 \sin \left (a+b x^2\right )}{8 x^2}+\frac {\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sin (a) \text {Si}\left (b x^2\right )+\frac {3}{8} b \sin (3 a) \text {Si}\left (3 b x^2\right ) \]
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Rubi [A]
time = 0.15, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3484, 3460,
3378, 3384, 3380, 3383} \begin {gather*} \frac {3}{8} b \cos (a) \text {CosIntegral}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {CosIntegral}\left (3 b x^2\right )-\frac {3}{8} b \sin (a) \text {Si}\left (b x^2\right )+\frac {3}{8} b \sin (3 a) \text {Si}\left (3 b x^2\right )-\frac {3 \sin \left (a+b x^2\right )}{8 x^2}+\frac {\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3460
Rule 3484
Rubi steps
\begin {align*} \int \frac {\sin ^3\left (a+b x^2\right )}{x^3} \, dx &=\int \left (\frac {3 \sin \left (a+b x^2\right )}{4 x^3}-\frac {\sin \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sin \left (3 a+3 b x^2\right )}{x^3} \, dx\right )+\frac {3}{4} \int \frac {\sin \left (a+b x^2\right )}{x^3} \, dx\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x^2} \, dx,x,x^2\right )\right )+\frac {3}{8} \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 \sin \left (a+b x^2\right )}{8 x^2}+\frac {\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}+\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {3 \sin \left (a+b x^2\right )}{8 x^2}+\frac {\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}+\frac {1}{8} (3 b \cos (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \cos (3 a)) \text {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \sin (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \sin (3 a)) \text {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^2\right )\\ &=\frac {3}{8} b \cos (a) \text {Ci}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {Ci}\left (3 b x^2\right )-\frac {3 \sin \left (a+b x^2\right )}{8 x^2}+\frac {\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sin (a) \text {Si}\left (b x^2\right )+\frac {3}{8} b \sin (3 a) \text {Si}\left (3 b x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 90, normalized size = 0.99 \begin {gather*} \frac {3 b x^2 \cos (a) \text {Ci}\left (b x^2\right )-3 b x^2 \cos (3 a) \text {Ci}\left (3 b x^2\right )-3 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )-3 b x^2 \sin (a) \text {Si}\left (b x^2\right )+3 b x^2 \sin (3 a) \text {Si}\left (3 b x^2\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.39, size = 162, normalized size = 1.78
method | result | size |
risch | \(-\frac {3 i {\mathrm e}^{-3 i a} \mathrm {csgn}\left (b \,x^{2}\right ) \pi b}{16}+\frac {3 i {\mathrm e}^{-3 i a} \sinIntegral \left (3 b \,x^{2}\right ) b}{8}+\frac {3 \,{\mathrm e}^{-3 i a} \expIntegral \left (1, -3 i x^{2} b \right ) b}{16}+\frac {3 \,{\mathrm e}^{3 i a} b \expIntegral \left (1, -3 i x^{2} b \right )}{16}-\frac {3 \,{\mathrm e}^{i a} b \expIntegral \left (1, -i x^{2} b \right )}{16}+\frac {3 i \mathrm {csgn}\left (b \,x^{2}\right ) {\mathrm e}^{-i a} \pi b}{16}-\frac {3 i {\mathrm e}^{-i a} \sinIntegral \left (b \,x^{2}\right ) b}{8}-\frac {3 \expIntegral \left (1, -i x^{2} b \right ) {\mathrm e}^{-i a} b}{16}-\frac {3 \sin \left (b \,x^{2}+a \right )}{8 x^{2}}+\frac {\sin \left (3 b \,x^{2}+3 a \right )}{8 x^{2}}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.40, size = 97, normalized size = 1.07 \begin {gather*} -\frac {3}{16} \, {\left ({\left (\Gamma \left (-1, 3 i \, b x^{2}\right ) + \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) - {\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, 3 i \, b x^{2}\right ) + i \, \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) + {\left (i \, \Gamma \left (-1, i \, b x^{2}\right ) - i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 118, normalized size = 1.30 \begin {gather*} \frac {6 \, b x^{2} \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{2}\right ) - 6 \, b x^{2} \sin \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) - 3 \, {\left (b x^{2} \operatorname {Ci}\left (3 \, b x^{2}\right ) + b x^{2} \operatorname {Ci}\left (-3 \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + 3 \, {\left (b x^{2} \operatorname {Ci}\left (b x^{2}\right ) + b x^{2} \operatorname {Ci}\left (-b x^{2}\right )\right )} \cos \left (a\right ) + 8 \, {\left (\cos \left (b x^{2} + a\right )^{2} - 1\right )} \sin \left (b x^{2} + a\right )}{16 \, x^{2}} \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 {\sin ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs.
\(2 (80) = 160\).
time = 5.72, size = 186, normalized size = 2.04 \begin {gather*} -\frac {3 \, {\left (b x^{2} + a\right )} b^{2} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{2}\right ) - 3 \, a b^{2} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{2}\right ) - 3 \, {\left (b x^{2} + a\right )} b^{2} \cos \left (a\right ) \operatorname {Ci}\left (b x^{2}\right ) + 3 \, a b^{2} \cos \left (a\right ) \operatorname {Ci}\left (b x^{2}\right ) + 3 \, {\left (b x^{2} + a\right )} b^{2} \sin \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) - 3 \, a b^{2} \sin \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) + 3 \, {\left (b x^{2} + a\right )} b^{2} \sin \left (3 \, a\right ) \operatorname {Si}\left (-3 \, b x^{2}\right ) - 3 \, a b^{2} \sin \left (3 \, a\right ) \operatorname {Si}\left (-3 \, b x^{2}\right ) - b^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 3 \, b^{2} \sin \left (b x^{2} + a\right )}{8 \, b^{2} 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 = -0.01 \begin {gather*} \int \frac {{\sin \left (b\,x^2+a\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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