Optimal. Leaf size=503 \[ \frac{3 i e^{i a} c^2 (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac{i e^{i a} c^3 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac{i e^{-i a} c^3 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac{3 i e^{-i a} c^2 (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac{i e^{i a} (c+d x)^4 \left (-i b (c+d x)^n\right )^{-4/n} \text{Gamma}\left (\frac{4}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac{3 i e^{i a} c (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac{3 i e^{-i a} c (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac{i e^{-i a} (c+d x)^4 \left (i b (c+d x)^n\right )^{-4/n} \text{Gamma}\left (\frac{4}{n},i b (c+d x)^n\right )}{2 d^4 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.391409, antiderivative size = 503, normalized size of antiderivative = 1., 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 = {3433, 3365, 2208, 3423, 2218} \[ \frac{3 i e^{i a} c^2 (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac{i e^{i a} c^3 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac{i e^{-i a} c^3 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac{3 i e^{-i a} c^2 (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},i b (c+d x)^n\right )}{2 d^4 n}+\frac{i e^{i a} (c+d x)^4 \left (-i b (c+d x)^n\right )^{-4/n} \text{Gamma}\left (\frac{4}{n},-i b (c+d x)^n\right )}{2 d^4 n}-\frac{3 i e^{i a} c (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-i b (c+d x)^n\right )}{2 d^4 n}+\frac{3 i e^{-i a} c (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},i b (c+d x)^n\right )}{2 d^4 n}-\frac{i e^{-i a} (c+d x)^4 \left (i b (c+d x)^n\right )^{-4/n} \text{Gamma}\left (\frac{4}{n},i b (c+d x)^n\right )}{2 d^4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3433
Rule 3365
Rule 2208
Rule 3423
Rule 2218
Rubi steps
\begin{align*} \int x^3 \sin \left (a+b (c+d x)^n\right ) \, dx &=\frac{\operatorname{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{\operatorname{Subst}\left (\int x^3 \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}-\frac{(3 c) \operatorname{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 ) \operatorname{Subst}\left (\int x \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}-\frac{c^3 \operatorname{Subst}\left (\int \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac{i \operatorname{Subst}\left (\int e^{-i a-i b x^n} x^3 \, dx,x,c+d x\right )}{2 d^4}-\frac{i \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [F] time = 11.5296, size = 0, normalized size = 0. \[ \int x^3 \sin \left (a+b (c+d x)^n\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\sin \left ( a+b \left ( dx+c \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sin{\left (a + b \left (c + d x\right )^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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.
[In]
[Out]