Integrand size = 14, antiderivative size = 260 \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^4 (2 i-n)} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5203, 102, 152, 71} \[ \int e^{n \arctan (a+b x)} x^3 \, dx=-\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right ) (i a+i b x+1)^{1-\frac {i n}{2}}}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{3 b^4 (-n+2 i)}+\frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2} \]
[In]
[Out]
Rule 71
Rule 102
Rule 152
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int x^3 (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx \\ & = \frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}+\frac {\int x (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (-2 \left (1+a^2\right )-b (6 a+n) x\right ) \, dx}{4 b^2} \\ & = \frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}-\frac {\left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{24 b^3} \\ & = \frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^4 (2 i-n)} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05 \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\frac {(-i (i+a+b x))^{1+\frac {i n}{2}} \left (b^2 (2 i-n) x^2 (1+i a+i b x)^{1-\frac {i n}{2}}-2^{3-\frac {i n}{2}} (6 a+n) \operatorname {Hypergeometric2F1}\left (-2+\frac {i n}{2},1+\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )+2^{3-\frac {i n}{2}} (1+i a) (-i+5 a+n) \operatorname {Hypergeometric2F1}\left (-1+\frac {i n}{2},1+\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )+2^{1-\frac {i n}{2}} (-i+a)^2 (-2 i+4 a+n) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )\right )}{4 b^4 (2 i-n)} \]
[In]
[Out]
\[\int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{3}d x\]
[In]
[Out]
\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int { x^{3} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
[In]
[Out]
\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int x^{3} e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]
[In]
[Out]
\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int { x^{3} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
[In]
[Out]