Optimal. Leaf size=503 \[ -\frac {i c^3 e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {i c^3 e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i c^2 e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i c^2 e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i c e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i c e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {i e^{i a} (c+d x)^4 \left (-i b (c+d x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {i e^{-i a} (c+d x)^4 \left (i b (c+d x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i b (c+d x)^n\right )}{2 d^4 n} \]
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Rubi [A]
time = 0.27, antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3514, 3446,
2239, 3504, 2250} \begin {gather*} -\frac {i e^{i a} c^3 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {i e^{-i a} c^3 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i e^{i a} c^2 (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i e^{-i a} c^2 (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {i e^{i a} (c+d x)^4 \left (-i b (c+d x)^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i e^{i a} c (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i e^{-i a} c (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac {i e^{-i a} (c+d x)^4 \left (i b (c+d x)^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},i b (c+d x)^n\right )}{2 d^4 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2250
Rule 3446
Rule 3504
Rule 3514
Rubi steps
\begin {align*} \int x^3 \sin \left (a+b (c+d x)^n\right ) \, dx &=\frac {\text {Subst}\left (\int \left (-c^3 \sin \left (a+b x^n\right )+3 c^2 x \sin \left (a+b x^n\right )-3 c x^2 \sin \left (a+b x^n\right )+x^3 \sin \left (a+b x^n\right )\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {\text {Subst}\left (\int x^3 \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}-\frac {(3 c) \text {Subst}\left (\int x^2 \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int x \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}-\frac {c^3 \text {Subst}\left (\int \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {i \text {Subst}\left (\int e^{-i a-i b x^n} x^3 \, dx,x,c+d x\right )}{2 d^4}-\frac {i \text {Subst}\left (\int e^{i a+i b x^n} x^3 \, dx,x,c+d x\right )}{2 d^4}-\frac {(3 i c) \text {Subst}\left (\int e^{-i a-i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^4}+\frac {(3 i c) \text {Subst}\left (\int e^{i a+i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^4}+\frac {\left (3 i c^2\right ) \text {Subst}\left (\int e^{-i a-i b x^n} x \, dx,x,c+d x\right )}{2 d^4}-\frac {\left (3 i c^2\right ) \text {Subst}\left (\int e^{i a+i b x^n} x \, dx,x,c+d x\right )}{2 d^4}-\frac {\left (i c^3\right ) \text {Subst}\left (\int e^{-i a-i b x^n} \, dx,x,c+d x\right )}{2 d^4}+\frac {\left (i c^3\right ) \text {Subst}\left (\int e^{i a+i b x^n} \, dx,x,c+d x\right )}{2 d^4}\\ &=-\frac {i c^3 e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {i c^3 e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i c^2 e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i c^2 e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac {3 i c e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac {3 i c e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac {i e^{i a} (c+d x)^4 \left (-i b (c+d x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac {i e^{-i a} (c+d x)^4 \left (i b (c+d x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i b (c+d x)^n\right )}{2 d^4 n}\\ \end {align*}
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Mathematica [F]
time = 9.18, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \sin \left (a+b (c+d x)^n\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{3} \sin \left (a +b \left (d x +c \right )^{n}\right )\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sin {\left (a + b \left (c + d x\right )^{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 = -0.00 \begin {gather*} \int x^3\,\sin \left (a+b\,{\left (c+d\,x\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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