Optimal. Leaf size=369 \[ \frac {i e^{i a} c^2 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{-i a} c^2 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^3 n}+\frac {i e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{i a} c (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i b (c+d x)^n\right )}{d^3 n}+\frac {i e^{-i a} c (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i b (c+d x)^n\right )}{d^3 n}-\frac {i e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^3 n} \]
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Rubi [A] time = 0.25, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3433, 3365, 2208, 3423, 2218} \[ \frac {i e^{i a} c^2 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{-i a} c^2 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^3 n}+\frac {i e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{i a} c (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i b (c+d x)^n\right )}{d^3 n}+\frac {i e^{-i a} c (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i b (c+d x)^n\right )}{d^3 n}-\frac {i e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^3 n} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 3365
Rule 3423
Rule 3433
Rubi steps
\begin {align*} \int x^2 \sin \left (a+b (c+d x)^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (c^2 \sin \left (a+b x^n\right )-2 c x \sin \left (a+b x^n\right )+x^2 \sin \left (a+b x^n\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}-\frac {(2 c) \operatorname {Subst}\left (\int x \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}+\frac {c^2 \operatorname {Subst}\left (\int \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {i \operatorname {Subst}\left (\int e^{-i a-i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^3}-\frac {i \operatorname {Subst}\left (\int e^{i a+i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^3}-\frac {(i c) \operatorname {Subst}\left (\int e^{-i a-i b x^n} x \, dx,x,c+d x\right )}{d^3}+\frac {(i c) \operatorname {Subst}\left (\int e^{i a+i b x^n} x \, dx,x,c+d x\right )}{d^3}+\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int e^{-i a-i b x^n} \, dx,x,c+d x\right )}{2 d^3}-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int e^{i a+i b x^n} \, dx,x,c+d x\right )}{2 d^3}\\ &=\frac {i c^2 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^3 n}-\frac {i c^2 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^3 n}-\frac {i c 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 )}{d^3 n}+\frac {i c 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 )}{d^3 n}+\frac {i 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^3 n}-\frac {i 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^3 n}\\ \end {align*}
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Mathematica [F] time = 8.77, size = 0, normalized size = 0.00 \[ \int x^2 \sin \left (a+b (c+d x)^n\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \sin \left ({\left (d x + c\right )}^{n} b + a\right ), x\right ) \]
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 \[ \int x^{2} \sin \left ({\left (d x + c\right )}^{n} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{2} \sin \left (a +b \left (d x +c \right )^{n}\right )\, dx \]
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 \[ \int x^{2} \sin \left ({\left (d x + c\right )}^{n} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sin \left (a+b\,{\left (c+d\,x\right )}^n\right ) \,d x \]
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 \[ \int x^{2} \sin {\left (a + b \left (c + d x\right )^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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