\(\int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\) [230]

Optimal result

Integrand size = 25, antiderivative size = 164 \[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\frac {i \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+i a x}}{\sqrt [4]{a-i a x}}\right )}{a}-\frac {i \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a+i a x}}{\sqrt [4]{a-i a x}}\right )}{a}+\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a+i a x}}{\sqrt [4]{a-i a x} \left (1+\frac {\sqrt {a+i a x}}{\sqrt {a-i a x}}\right )}\right )}{a} \] Output:

I*2^(1/2)*arctan(1-2^(1/2)*(a+I*a*x)^(1/4)/(a-I*a*x)^(1/4))/a-I*2^(1/2)*ar 
ctan(1+2^(1/2)*(a+I*a*x)^(1/4)/(a-I*a*x)^(1/4))/a+I*2^(1/2)*arctanh(2^(1/2 
)*(a+I*a*x)^(1/4)/(a-I*a*x)^(1/4)/(1+(a+I*a*x)^(1/2)/(a-I*a*x)^(1/2)))/a
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 1.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 i 2^{3/4} \sqrt [4]{1+i x} \sqrt [4]{a-i a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2}-\frac {i x}{2}\right )}{a \sqrt [4]{a+i a x}} \] Input:

Integrate[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)),x]
 

Output:

((2*I)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(1/4)*Hypergeometric2F1[1/4, 1/ 
4, 5/4, 1/2 - (I/2)*x])/(a*(a + I*a*x)^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {73, 770, 755, 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.

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 i \int \frac {1}{\sqrt [4]{i x a+a}}d\sqrt [4]{a-i a x}}{a}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {4 i \int \frac {1}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{a}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \int \frac {\sqrt {a-i a x}+1}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{a}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \int \frac {1}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )\right )}{a}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {a-i a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {a-i a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{a}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 i \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\)

Input:

Int[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)),x]
 

Output:

((4*I)*((-(ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/Sqrt[ 
2]) + ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/Sqrt[2])/2 
 + (-1/2*Log[1 + Sqrt[a - I*a*x] - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x) 
^(1/4)]/Sqrt[2] + Log[1 + Sqrt[a - I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a 
 + I*a*x)^(1/4)]/(2*Sqrt[2]))/2))/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 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 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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   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 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 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]
 

Maple [F]

\[\int \frac {1}{\left (-i a x +a \right )^{\frac {3}{4}} \left (i a x +a \right )^{\frac {1}{4}}}d x\]

Input:

int(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x)
 

Output:

int(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\frac {1}{2} \, \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) - \frac {1}{2} \, \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) \] Input:

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")
 

Output:

1/2*sqrt(4*I/a^2)*log(1/2*((a^2*x - I*a^2)*sqrt(4*I/a^2) + 2*(I*a*x + a)^( 
3/4)*(-I*a*x + a)^(1/4))/(x - I)) - 1/2*sqrt(4*I/a^2)*log(-1/2*((a^2*x - I 
*a^2)*sqrt(4*I/a^2) - 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(x - I)) + 1 
/2*sqrt(-4*I/a^2)*log(1/2*((a^2*x - I*a^2)*sqrt(-4*I/a^2) + 2*(I*a*x + a)^ 
(3/4)*(-I*a*x + a)^(1/4))/(x - I)) - 1/2*sqrt(-4*I/a^2)*log(-1/2*((a^2*x - 
 I*a^2)*sqrt(-4*I/a^2) - 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(x - I))
 

Sympy [F]

\[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/(a-I*a*x)**(3/4)/(a+I*a*x)**(1/4),x)
 

Output:

Integral(1/((I*a*(x - I))**(1/4)*(-I*a*(x + I))**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")
 

Output:

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),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. \[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \] Input:

int(1/((a - a*x*1i)^(3/4)*(a + a*x*1i)^(1/4)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((a - a*x*1i)^(3/4)*(a + a*x*1i)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx=\frac {\int \frac {1}{\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {3}{4}}}d x}{a} \] Input:

int(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x)
 

Output:

int(1/((i*x + 1)**(1/4)*( - i*x + 1)**(3/4)),x)/a