Optimal. Leaf size=165 \[ -\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \text {Ci}\left (b x^n\right ) \sin (a)}{8 n}+\frac {9 b^2 \text {Ci}\left (3 b x^n\right ) \sin (3 a)}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n} \]
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Rubi [A]
time = 0.19, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3506, 3460,
3378, 3384, 3380, 3383} \begin {gather*} -\frac {3 b^2 \sin (a) \text {CosIntegral}\left (b x^n\right )}{8 n}+\frac {9 b^2 \sin (3 a) \text {CosIntegral}\left (3 b x^n\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3460
Rule 3506
Rubi steps
\begin {align*} \int x^{-1-2 n} \sin ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^{-1-2 n} \sin \left (a+b x^n\right )-\frac {1}{4} x^{-1-2 n} \sin \left (3 a+3 b x^n\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^{-1-2 n} \sin \left (3 a+3 b x^n\right ) \, dx\right )+\frac {3}{4} \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x^3} \, dx,x,x^n\right )}{4 n}+\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}+\frac {(3 b) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^n\right )}{8 n}-\frac {(3 b) \text {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2\right ) \text {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {\left (3 b^2 \cos (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2 \cos (3 a)\right ) \text {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^n\right )}{8 n}-\frac {\left (3 b^2 \sin (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2 \sin (3 a)\right ) \text {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \text {Ci}\left (b x^n\right ) \sin (a)}{8 n}+\frac {9 b^2 \text {Ci}\left (3 b x^n\right ) \sin (3 a)}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 141, normalized size = 0.85 \begin {gather*} \frac {x^{-2 n} \left (-3 b x^n \cos \left (a+b x^n\right )+3 b x^n \cos \left (3 \left (a+b x^n\right )\right )-3 b^2 x^{2 n} \text {Ci}\left (b x^n\right ) \sin (a)+9 b^2 x^{2 n} \text {Ci}\left (3 b x^n\right ) \sin (3 a)-3 \sin \left (a+b x^n\right )+\sin \left (3 \left (a+b x^n\right )\right )-3 b^2 x^{2 n} \cos (a) \text {Si}\left (b x^n\right )+9 b^2 x^{2 n} \cos (3 a) \text {Si}\left (3 b x^n\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 144, normalized size = 0.87
method | result | size |
default | \(\frac {3 b^{2} \left (-\frac {x^{-2 n} \sin \left (a +b \,x^{n}\right )}{2 b^{2}}-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{2 b}-\frac {\sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )}{2}\right )}{4 n}-\frac {9 b^{2} \left (-\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-2 n}}{18 b^{2}}-\frac {\cos \left (3 a +3 b \,x^{n}\right ) x^{-n}}{6 b}-\frac {\sinIntegral \left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{2}-\frac {\cosineIntegral \left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{2}\right )}{4 n}\) | \(144\) |
risch | \(\frac {9 i b^{2} {\mathrm e}^{3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{16 n}-\frac {9 b^{2} {\mathrm e}^{-3 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{16 n}+\frac {9 b^{2} {\mathrm e}^{-3 i a} \sinIntegral \left (3 b \,x^{n}\right )}{8 n}-\frac {9 i b^{2} {\mathrm e}^{-3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{16 n}+\frac {3 b^{2} {\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{16 n}-\frac {3 b^{2} {\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{8 n}+\frac {3 i b^{2} {\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{16 n}-\frac {3 i b^{2} {\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{16 n}-\frac {3 b \cos \left (a +b \,x^{n}\right ) x^{-n}}{8 n}-\frac {3 x^{-2 n} \sin \left (a +b \,x^{n}\right )}{8 n}+\frac {3 b \cos \left (3 a +3 b \,x^{n}\right ) x^{-n}}{8 n}+\frac {x^{-2 n} \sin \left (3 a +3 b \,x^{n}\right )}{8 n}\) | \(253\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 183, normalized size = 1.11 \begin {gather*} \frac {24 \, b x^{n} \cos \left (b x^{n} + a\right )^{3} + 9 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + 9 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (-3 \, b x^{n}\right ) \sin \left (3 \, a\right ) - 3 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) - 3 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 18 \, b^{2} x^{2 \, n} \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) - 6 \, b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) - 24 \, b x^{n} \cos \left (b x^{n} + a\right ) + 8 \, {\left (\cos \left (b x^{n} + a\right )^{2} - 1\right )} \sin \left (b x^{n} + a\right )}{16 \, n x^{2 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\sin \left (a+b\,x^n\right )}^3}{x^{2\,n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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