Optimal. Leaf size=46 \[ \frac {b \cos (a) \text {Ci}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n} \]
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Rubi [A]
time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3460, 3378,
3384, 3380, 3383} \begin {gather*} \frac {b \cos (a) \text {CosIntegral}\left (b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3460
Rubi steps
\begin {align*} \int x^{-1-n} \sin \left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}+\frac {b \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}+\frac {(b \cos (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{n}-\frac {(b \sin (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {b \cos (a) \text {Ci}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 47, normalized size = 1.02 \begin {gather*} \frac {x^{-n} \left (b x^n \cos (a) \text {Ci}\left (b x^n\right )-\sin \left (a+b x^n\right )-b x^n \sin (a) \text {Si}\left (b x^n\right )\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 44, normalized size = 0.96
method | result | size |
default | \(\frac {b \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{b}-\sinIntegral \left (b \,x^{n}\right ) \sin \left (a \right )+\cosineIntegral \left (b \,x^{n}\right ) \cos \left (a \right )\right )}{n}\) | \(44\) |
risch | \(-\frac {b \,{\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}+\frac {i b \,{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{2 n}-\frac {i b \,{\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{n}-\frac {b \,{\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{n}\) | \(97\) |
meijerg | error in int/gproduct: numeric exception: division by zero\ | N/A |
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 = 0.37, size = 62, normalized size = 1.35 \begin {gather*} \frac {b x^{n} \cos \left (a\right ) \operatorname {Ci}\left (b x^{n}\right ) + b x^{n} \cos \left (a\right ) \operatorname {Ci}\left (-b x^{n}\right ) - 2 \, b x^{n} \sin \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) - 2 \, \sin \left (b x^{n} + a\right )}{2 \, n x^{n}} \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 x^{- n - 1} \sin {\left (a + b x^{n} \right )}\, dx \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 = -0.02 \begin {gather*} \int \frac {\sin \left (a+b\,x^n\right )}{x^{n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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