Integrand size = 16, antiderivative size = 249 \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=\frac {3 \left (17-44 i a-36 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {3 \left (17 i+44 a-36 i a^2-8 a^3\right ) \text {arcsinh}(a+b x)}{8 b^4} \]
-3/8*(17*I+44*a-36*I*a^2-8*a^3)*arcsinh(b*x+a)/b^4-2*I*x^3*(1+I*a+I*b*x)^( 3/2)/b/(1-I*a-I*b*x)^(1/2)-9/4*x^2*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2) /b^2-1/8*I*(1+I*a+I*b*x)^(3/2)*(29*I+54*a-22*I*a^2-2*(11-10*I*a)*b*x)*(1-I *a-I*b*x)^(1/2)/b^4+3/8*(17-44*I*a-36*a^2+8*I*a^3)*(1-I*a-I*b*x)^(1/2)*(1+ I*a+I*b*x)^(1/2)/b^4
Time = 0.23 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.81 \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=\frac {\sqrt {1+i a+i b x} \left (80+78 i a^3+2 a^4-29 i b x+11 b^2 x^2+6 i b^3 x^3-2 b^4 x^4+a^2 (-233+22 i b x)-i a \left (237-54 i b x+10 b^2 x^2\right )\right )}{8 b^4 \sqrt {-i (i+a+b x)}}+\frac {3 \sqrt [4]{-1} \left (-17 i-44 a+36 i a^2+8 a^3\right ) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{4 b^{9/2}} \]
(Sqrt[1 + I*a + I*b*x]*(80 + (78*I)*a^3 + 2*a^4 - (29*I)*b*x + 11*b^2*x^2 + (6*I)*b^3*x^3 - 2*b^4*x^4 + a^2*(-233 + (22*I)*b*x) - I*a*(237 - (54*I)* b*x + 10*b^2*x^2)))/(8*b^4*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(-17* I - 44*a + (36*I)*a^2 + 8*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*S qrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))
Time = 0.41 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5618, 108, 27, 170, 25, 27, 164, 60, 62, 1090, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 e^{3 i \arctan (a+b x)} \, dx\) |
| \(\Big \downarrow \) 5618 |
| \(\displaystyle \int \frac {x^3 (i a+i b x+1)^{3/2}}{(-i a-i b x+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 i \int \frac {3 x^2 \sqrt {i a+i b x+1} (2 (i a+1)+3 i b x)}{2 \sqrt {-i a-i b x+1}}dx}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 i \int \frac {x^2 \sqrt {i a+i b x+1} (2 (i a+1)+3 i b x)}{\sqrt {-i a-i b x+1}}dx}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3 i \left (\frac {\int -\frac {b x \sqrt {i a+i b x+1} \left (6 i \left (a^2+1\right )-(11-10 i a) b x\right )}{\sqrt {-i a-i b x+1}}dx}{4 b^2}+\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\int \frac {b x \sqrt {i a+i b x+1} \left (6 i \left (a^2+1\right )-(11-10 i a) b x\right )}{\sqrt {-i a-i b x+1}}dx}{4 b^2}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\int \frac {x \sqrt {i a+i b x+1} \left (6 i \left (a^2+1\right )-(11-10 i a) b x\right )}{\sqrt {-i a-i b x+1}}dx}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\frac {\left (8 i a^3-36 a^2-44 i a+17\right ) \int \frac {\sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}dx}{2 b}+\frac {\sqrt {-i a-i b x+1} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right ) (i a+i b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\frac {\left (8 i a^3-36 a^2-44 i a+17\right ) \left (\int \frac {1}{\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx+\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b}\right )}{2 b}+\frac {\sqrt {-i a-i b x+1} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right ) (i a+i b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 62 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\frac {\left (8 i a^3-36 a^2-44 i a+17\right ) \left (\int \frac {1}{\sqrt {b^2 x^2+2 a b x+(1-i a) (i a+1)}}dx+\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b}\right )}{2 b}+\frac {\sqrt {-i a-i b x+1} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right ) (i a+i b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\frac {\left (8 i a^3-36 a^2-44 i a+17\right ) \left (\frac {\int \frac {1}{\sqrt {\frac {\left (2 x b^2+2 a b\right )^2}{4 b^2}+1}}d\left (2 x b^2+2 a b\right )}{2 b^2}+\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b}\right )}{2 b}+\frac {\sqrt {-i a-i b x+1} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right ) (i a+i b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {3 i \left (\frac {3 i x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b}-\frac {\frac {\sqrt {-i a-i b x+1} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right ) (i a+i b x+1)^{3/2}}{6 b^2}+\frac {\left (8 i a^3-36 a^2-44 i a+17\right ) \left (\frac {\text {arcsinh}\left (\frac {2 a b+2 b^2 x}{2 b}\right )}{b}+\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b}\right )}{2 b}}{4 b}\right )}{b}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}\) |
((-2*I)*x^3*(1 + I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) + ((3*I)*(( ((3*I)/4)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/b - ((Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(29*I + 54*a - (22*I)*a^2 - 2*(11 - (10*I)*a)*b*x))/(6*b^2) + ((17 - (44*I)*a - 36*a^2 + (8*I)*a^3)*((I*Sqrt[ 1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/b + ArcSinh[(2*a*b + 2*b^2*x)/(2*b )]/b))/(2*b))/(4*b)))/b
3.2.81.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Time = 1.38 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\frac {i \left (-2 b^{3} x^{3}+2 a \,b^{2} x^{2}+8 i x^{2} b^{2}-2 a^{2} b x -20 i a b x +2 a^{3}+44 i a^{2}+19 b x -93 a -48 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}+\frac {-\frac {51 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {132 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {24 a^{3} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {108 i a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {i \left (-32 i a^{3}+96 a^{2}+96 i a -32\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{2} \left (x +\frac {i+a}{b}\right )}}{8 b^{3}}\) | \(342\) |
| default | \(\text {Expression too large to display}\) | \(2972\) |
1/8*I*(-2*b^3*x^3+8*I*b^2*x^2+2*a*b^2*x^2-20*I*a*b*x-2*a^2*b*x+44*I*a^2+2* a^3+19*b*x-48*I-93*a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/b^4+1/8/b^3*(-51*I*ln( (b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-132*a*l n((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)+24*a^ 3*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)+10 8*I*a^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1 /2)-I*(96*a^2-32*I*a^3-32+96*I*a)/b^2/(x+(I+a)/b)*((x+(I+a)/b)^2*b^2-2*I*b *(x+(I+a)/b))^(1/2))
Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.87 \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=\frac {15 i \, a^{5} - 495 \, a^{4} - 1664 i \, a^{3} + {\left (15 i \, a^{4} - 480 \, a^{3} - 1184 i \, a^{2} + 968 \, a + 256 i\right )} b x + 2152 \, a^{2} - 24 \, {\left (8 \, a^{4} + 44 i \, a^{3} + {\left (8 \, a^{3} + 36 i \, a^{2} - 44 \, a - 17 i\right )} b x - 80 \, a^{2} - 61 i \, a + 17\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (2 i \, b^{4} x^{4} + 6 \, b^{3} x^{3} - {\left (10 \, a + 11 i\right )} b^{2} x^{2} - 2 i \, a^{4} + 78 \, a^{3} + {\left (22 \, a^{2} + 54 i \, a - 29\right )} b x + 233 i \, a^{2} - 237 \, a - 80 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1224 i \, a - 256}{64 \, {\left (b^{5} x + {\left (a + i\right )} b^{4}\right )}} \]
1/64*(15*I*a^5 - 495*a^4 - 1664*I*a^3 + (15*I*a^4 - 480*a^3 - 1184*I*a^2 + 968*a + 256*I)*b*x + 2152*a^2 - 24*(8*a^4 + 44*I*a^3 + (8*a^3 + 36*I*a^2 - 44*a - 17*I)*b*x - 80*a^2 - 61*I*a + 17)*log(-b*x - a + sqrt(b^2*x^2 + 2 *a*b*x + a^2 + 1)) - 8*(2*I*b^4*x^4 + 6*b^3*x^3 - (10*a + 11*I)*b^2*x^2 - 2*I*a^4 + 78*a^3 + (22*a^2 + 54*I*a - 29)*b*x + 233*I*a^2 - 237*a - 80*I)* sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1224*I*a - 256)/(b^5*x + (a + I)*b^4)
\[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=- i \left (\int \frac {i x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{6}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 i b^{2} x^{5}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{5}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \]
-I*(Integral(I*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*s qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2 *x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a*x** 3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt( a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b **2*x**2 + 1)), x) + Integral(-3*b*x**4/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x **2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a **2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**6/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x *sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b* *2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a **2*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**5/(a** 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b** 2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**5/(a**2*sqrt(a**2...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2295 vs. \(2 (175) = 350\).
Time = 0.21 (sec) , antiderivative size = 2295, normalized size of antiderivative = 9.22 \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=\text {Too large to display} \]
-1/4*I*b*x^5/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 315/4*I*a^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 3/4*I*a*x^4/sqrt(b^ 2*x^2 + 2*a*b*x + a^2 + 1) - 945/8*I*(a^2 + 1)*a^4*x/((a^2*b^2 - (a^2 + 1) *b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 21/8*I*a^2*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 105/8*I*(a^2 + 1)*a^5/((a^2*b^2 - (a^2 + 1)*b^2) *sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 105*(I*a*b^2 + b^2)*a^5*x/((a^2* b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) + 45*(I*a^2*b + 2*a*b - I*b)*a^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a ^2 + 1)*b^2) + 169/4*I*(a^2 + 1)^2*a^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b ^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^3*x/((a^ 2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 105/8*I*a^3* x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 5/8*(-I*a^2 - I)*x^3/(sqrt(b ^2*x^2 + 2*a*b*x + a^2 + 1)*b) - (I*a*b^2 + b^2)*x^4/(sqrt(b^2*x^2 + 2*a*b *x + a^2 + 1)*b^2) - 14*I*(a^2 + 1)^2*a^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt( b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 265/2*(I*a*b^2 + b^2)*(a^2 + 1)*a^3*x/ ((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 93/2*( I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2 *x^2 + 2*a*b*x + a^2 + 1)*b^2) - 15/8*I*(a^2 + 1)^3*x/((a^2*b^2 - (a^2 + 1 )*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 5*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*a*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^...
Time = 0.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.14 \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=-\frac {1}{8} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, x {\left (\frac {i \, x}{b} - \frac {i \, a b^{11} - 4 \, b^{11}}{b^{13}}\right )} - \frac {-2 i \, a^{2} b^{10} + 20 \, a b^{10} + 19 i \, b^{10}}{b^{13}}\right )} x - \frac {2 i \, a^{3} b^{9} - 44 \, a^{2} b^{9} - 93 i \, a b^{9} + 48 \, b^{9}}{b^{13}}\right )} - \frac {{\left (8 \, a^{3} + 36 i \, a^{2} - 44 \, a - 17 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
-1/8*sqrt((b*x + a)^2 + 1)*((2*x*(I*x/b - (I*a*b^11 - 4*b^11)/b^13) - (-2* I*a^2*b^10 + 20*a*b^10 + 19*I*b^10)/b^13)*x - (2*I*a^3*b^9 - 44*a^2*b^9 - 93*I*a*b^9 + 48*b^9)/b^13) - 1/8*(8*a^3 + 36*I*a^2 - 44*a - 17*I)*log(3*(x *abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b + a^3*b + (x*abs(b) - sqrt((b*x + a )^2 + 1))^3*abs(b) + 3*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*abs(b) + 2*I *(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b + 2*I*a^2*b + 4*(I*x*abs(b) - I*sq rt((b*x + a)^2 + 1))*a*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*a bs(b))/(b^3*abs(b))
Timed out. \[ \int e^{3 i \arctan (a+b x)} x^3 \, dx=\int \frac {x^3\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]