Optimal. Leaf size=113 \[ \frac {3 b \cos (a) \text {Ci}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Ci}\left (3 b x^n\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n} \]
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Rubi [A] time = 0.21, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3425, 3379, 3297, 3303, 3299, 3302} \[ \frac {3 b \cos (a) \text {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3379
Rule 3425
Rubi steps
\begin {align*} \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^{-1-n} \sin \left (a+b x^n\right )-\frac {1}{4} x^{-1-n} \sin \left (3 a+3 b x^n\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^{-1-n} \sin \left (3 a+3 b x^n\right ) \, dx\right )+\frac {3}{4} \int x^{-1-n} \sin \left (a+b x^n\right ) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {(3 b \cos (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \cos (3 a)) \operatorname {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sin (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \sin (3 a)) \operatorname {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=\frac {3 b \cos (a) \text {Ci}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Ci}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 95, normalized size = 0.84 \[ \frac {x^{-n} \left (3 b \cos (a) x^n \text {Ci}\left (b x^n\right )-3 b \cos (3 a) x^n \text {Ci}\left (3 b x^n\right )-3 b \sin (a) x^n \text {Si}\left (b x^n\right )+3 b \sin (3 a) x^n \text {Si}\left (3 b x^n\right )-3 \sin \left (a+b x^n\right )+\sin \left (3 \left (a+b x^n\right )\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 127, normalized size = 1.12 \[ -\frac {3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{n}\right ) - 3 \, b x^{n} \cos \relax (a) \operatorname {Ci}\left (b x^{n}\right ) - 3 \, b x^{n} \cos \relax (a) \operatorname {Ci}\left (-b x^{n}\right ) + 3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (-3 \, b x^{n}\right ) - 6 \, b x^{n} \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) + 6 \, b x^{n} \sin \relax (a) \operatorname {Si}\left (b x^{n}\right ) - 8 \, {\left (\cos \left (b x^{n} + a\right )^{2} - 1\right )} \sin \left (b x^{n} + a\right )}{8 \, n x^{n}} \]
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 \[ \int x^{-n - 1} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 99, normalized size = 0.88 \[ \frac {3 b \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{b}-\Si \left (b \,x^{n}\right ) \sin \relax (a )+\Ci \left (b \,x^{n}\right ) \cos \relax (a )\right )}{4 n}-\frac {3 b \left (-\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-n}}{3 b}-\Si \left (3 b \,x^{n}\right ) \sin \left (3 a \right )+\Ci \left (3 b \,x^{n}\right ) \cos \left (3 a \right )\right )}{4 n} \]
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 \[ \int x^{-n - 1} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]
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 \[ \int \frac {{\sin \left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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