Optimal. Leaf size=243 \[ \frac{i e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-i b (c+d x)^n\right )}{2 d^2 n}-\frac{i e^{i a} c (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d^2 n}+\frac{i e^{-i a} c (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d^2 n}-\frac{i e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},i b (c+d x)^n\right )}{2 d^2 n} \]
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Rubi [A] time = 0.131454, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3433, 3365, 2208, 3423, 2218} \[ \frac{i e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-i b (c+d x)^n\right )}{2 d^2 n}-\frac{i e^{i a} c (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d^2 n}+\frac{i e^{-i a} c (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d^2 n}-\frac{i e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},i b (c+d x)^n\right )}{2 d^2 n} \]
Antiderivative was successfully verified.
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Rule 3433
Rule 3365
Rule 2208
Rule 3423
Rule 2218
Rubi steps
\begin{align*} \int x \sin \left (a+b (c+d x)^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-c \sin \left (a+b x^n\right )+x \sin \left (a+b x^n\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int x \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{i \operatorname{Subst}\left (\int e^{-i a-i b x^n} x \, dx,x,c+d x\right )}{2 d^2}-\frac{i \operatorname{Subst}\left (\int e^{i a+i b x^n} x \, dx,x,c+d x\right )}{2 d^2}-\frac{(i c) \operatorname{Subst}\left (\int e^{-i a-i b x^n} \, dx,x,c+d x\right )}{2 d^2}+\frac{(i c) \operatorname{Subst}\left (\int e^{i a+i b x^n} \, dx,x,c+d x\right )}{2 d^2}\\ &=-\frac{i c 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^2 n}+\frac{i c 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^2 n}+\frac{i 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^2 n}-\frac{i 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^2 n}\\ \end{align*}
Mathematica [A] time = 0.828501, size = 192, normalized size = 0.79 \[ \frac{(c+d x) \left ((\sin (a)-i \cos (a)) \left (-i b (c+d x)^n\right )^{-2/n} \left (c \left (-i b (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )-(c+d x) \text{Gamma}\left (\frac{2}{n},-i b (c+d x)^n\right )\right )+(\sin (a)+i \cos (a)) \left (i b (c+d x)^n\right )^{-2/n} \left (c \left (i b (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )-(c+d x) \text{Gamma}\left (\frac{2}{n},i b (c+d x)^n\right )\right )\right )}{2 d^2 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int x\sin \left ( a+b \left ( dx+c \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin \left ({\left (d x + c\right )}^{n} b + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \sin \left ({\left (d x + c\right )}^{n} b + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin{\left (a + b \left (c + d x\right )^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin \left ({\left (d x + c\right )}^{n} b + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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