\(\int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx\) [94]

Optimal result

Integrand size = 16, antiderivative size = 325 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {973 (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {113 (1-i a x)^{7/4} (1+i a x)^{5/4}}{24 a^4}-\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x} \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}\right )}\right )}{64 \sqrt {2} a^4} \] Output:

475/64*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/a^4-4*I*x^3*(1+I*a*x)^(5/4)/a/(1-I* 
a*x)^(1/4)+973/96*(1-I*a*x)^(3/4)*(1+I*a*x)^(5/4)/a^4-17/4*x^2*(1-I*a*x)^( 
3/4)*(1+I*a*x)^(5/4)/a^2-113/24*(1-I*a*x)^(7/4)*(1+I*a*x)^(5/4)/a^4-475/12 
8*arctan(1-2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4))*2^(1/2)/a^4+475/128*ar 
ctan(1+2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4))*2^(1/2)/a^4-475/128*arctan 
h(2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)/(1+(1-I*a*x)^(1/2)/(1+I*a*x)^(1/ 
2)))*2^(1/2)/a^4
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.30 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\frac {-\sqrt [4]{1+i a x} (-i+a x)^2 \left (59-5 i a x+6 a^2 x^2\right )+380 \sqrt [4]{2} (1-i a x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-i a x)\right )}{24 a^4 \sqrt [4]{1-i a x}} \] Input:

Integrate[E^(((5*I)/2)*ArcTan[a*x])*x^3,x]
 

Output:

(-((1 + I*a*x)^(1/4)*(-I + a*x)^2*(59 - (5*I)*a*x + 6*a^2*x^2)) + 380*2^(1 
/4)*(1 - I*a*x)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - I*a*x)/2])/(24*a^4* 
(1 - I*a*x)^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.87 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {5585, 108, 27, 170, 27, 164, 60, 73, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int x^3 e^{\frac {5}{2} i \arctan (a x)} \, dx\)

\(\Big \downarrow \) 5585

\(\displaystyle \int \frac {x^3 (1+i a x)^{5/4}}{(1-i a x)^{5/4}}dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {4 i \int \frac {x^2 \sqrt [4]{i a x+1} (17 i a x+12)}{4 \sqrt [4]{1-i a x}}dx}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i \int \frac {x^2 \sqrt [4]{i a x+1} (17 i a x+12)}{\sqrt [4]{1-i a x}}dx}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {i \left (\frac {\int -\frac {a x (68 i-113 a x) \sqrt [4]{i a x+1}}{2 \sqrt [4]{1-i a x}}dx}{4 a^2}+\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\int \frac {x (68 i-113 a x) \sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}dx}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \int \frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}dx}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \int \frac {\sqrt {1-i a x}}{(i a x+1)^{3/4}}d\sqrt [4]{1-i a x}}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \int \frac {\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \int \frac {\sqrt {1-i a x}+1}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}+\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {i \left (\frac {17 i x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a}-\frac {\frac {(-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^2}+\frac {475 \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}\right )}{8 a}}{8 a}\right )}{a}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}\)

Input:

Int[E^(((5*I)/2)*ArcTan[a*x])*x^3,x]
 

Output:

((-4*I)*x^3*(1 + I*a*x)^(5/4))/(a*(1 - I*a*x)^(1/4)) + (I*((((17*I)/4)*x^2 
*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/a - (((521*I - 452*a*x)*(1 - I*a*x)^ 
(3/4)*(1 + I*a*x)^(5/4))/(12*a^2) + (475*((I*(1 - I*a*x)^(3/4)*(1 + I*a*x) 
^(1/4))/a + ((2*I)*((-(ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^ 
(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4) 
]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + 
 I*a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - I*a*x] + (Sqrt[2]*(1 - I*a*x 
)^(1/4))/(1 + I*a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/(8*a))/(8*a)))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 108

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])
 

rule 164

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]
 

rule 170

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]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 5585

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a 
*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))), x] /; FreeQ[{a, m, n}, x] &&  !Intege 
rQ[(I*n - 1)/2]
 

Maple [F]

\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{3}d x\]

Input:

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
 

Output:

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.77 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=-\frac {96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (48 \, a^{4} x^{4} - 136 i \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 521 i \, a x - 2467\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{4}} \] Input:

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="fricas")
 

Output:

-1/192*(96*a^4*sqrt(225625/4096*I/a^8)*log(64/475*a^4*sqrt(225625/4096*I/a 
^8) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) - 96*a^4*sqrt(225625/4096*I/a^8 
)*log(-64/475*a^4*sqrt(225625/4096*I/a^8) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x 
+ I))) + 96*a^4*sqrt(-225625/4096*I/a^8)*log(64/475*a^4*sqrt(-225625/4096* 
I/a^8) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) - 96*a^4*sqrt(-225625/4096*I 
/a^8)*log(-64/475*a^4*sqrt(-225625/4096*I/a^8) + sqrt(I*sqrt(a^2*x^2 + 1)/ 
(a*x + I))) + (48*a^4*x^4 - 136*I*a^3*x^3 - 226*a^2*x^2 + 521*I*a*x - 2467 
)*sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)))/a^4
 

Sympy [F(-1)]

Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\text {Timed out} \] Input:

integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(5/2)*x**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\int { x^{3} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \] Input:

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="maxima")
 

Output:

integrate(x^3*((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\int x^3\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \] Input:

int(x^3*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2),x)
 

Output:

int(x^3*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2), x)
 

Reduce [F]

\[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{3}d x \] Input:

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
                                                                                    
                                                                                    
 

Output:

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)*x^3,x)