\(\int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx\) [1178]

Optimal result

Integrand size = 25, antiderivative size = 256 \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=-\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \]

[Out]

-I*(a-I*a*x)^(1/4)*(a+I*a*x)^(3/4)/a-1/2*I*arctan(1-(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4))*2^(1/2)+1/2*I*arc
tan(1+(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4))*2^(1/2)-1/4*I*ln(1-(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4)+(a-I
*a*x)^(1/2)/(a+I*a*x)^(1/2))*2^(1/2)+1/4*I*ln(1+(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4)+(a-I*a*x)^(1/2)/(a+I*a
*x)^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {52, 65, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=-\frac {i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

((-I)*(a - I*a*x)^(1/4)*(a + I*a*x)^(3/4))/a - (I*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/S
qrt[2] + (I*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] - ((I/2)*Log[1 + Sqrt[a - I*a*x
]/Sqrt[a + I*a*x] - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + ((I/2)*Log[1 + Sqrt[a - I*a*x]/S
qrt[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

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

Rubi steps \begin{align*} \text {integral}& = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac {1}{2} a \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+2 i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2 a-x^4}} \, dx,x,\sqrt [4]{a-i a x}\right ) \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+2 i \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right ) \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+i \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+i \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right ) \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac {1}{2} i \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )-\frac {i \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {i \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {i \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {i \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}} \\ & = -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=\frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2}-\frac {i x}{2}\right )}{5 a \sqrt [4]{a+i a x}} \]

[In]

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

[Out]

(((2*I)/5)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(5/4)*Hypergeometric2F1[1/4, 5/4, 9/4, 1/2 - (I/2)*x])/(a*(a +
I*a*x)^(1/4))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.14 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.86

method	result	size
risch	\(\frac {i \left (x +i\right ) \left (x -i\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (\frac {-\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}-x^{3}-i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}-i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x -2 i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x -2 i x^{2}+\sqrt {-x^{4}-2 i x^{3}-2 i x +1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}+x}{\left (i x -1\right )^{2}}\right )}{2}-\frac {i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (-\frac {-i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x +x^{3}-i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}+i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}+2 i x^{2}+\sqrt {-x^{4}-2 i x^{3}-2 i x +1}-x}{\left (i x -1\right )^{2}}\right )}{2}\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (-\left (i x -1\right )^{3} \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(477\)

[In]

int((a-I*a*x)^(1/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

I*(x+I)*(x-I)*(-a*(I*x-1))^(1/4)/(I*x-1)/(a*(I*x+1))^(1/4)-(1/2*RootOf(_Z^2-I)*ln((-(1-x^4-2*I*x^3-2*I*x)^(1/4
)*RootOf(_Z^2-I)*x^2-x^3-I*RootOf(_Z^2-I)*(1-x^4-2*I*x^3-2*I*x)^(3/4)-I*(1-x^4-2*I*x^3-2*I*x)^(1/2)*x-2*I*Root
Of(_Z^2-I)*(1-x^4-2*I*x^3-2*I*x)^(1/4)*x-2*I*x^2+(1-x^4-2*I*x^3-2*I*x)^(1/2)+RootOf(_Z^2-I)*(1-x^4-2*I*x^3-2*I
*x)^(1/4)+x)/(I*x-1)^2)-1/2*I*RootOf(_Z^2-I)*ln(-(-I*(1-x^4-2*I*x^3-2*I*x)^(1/4)*RootOf(_Z^2-I)*x^2+2*RootOf(_
Z^2-I)*(1-x^4-2*I*x^3-2*I*x)^(1/4)*x+x^3-I*(1-x^4-2*I*x^3-2*I*x)^(1/2)*x-RootOf(_Z^2-I)*(1-x^4-2*I*x^3-2*I*x)^
(3/4)+I*RootOf(_Z^2-I)*(1-x^4-2*I*x^3-2*I*x)^(1/4)+2*I*x^2+(1-x^4-2*I*x^3-2*I*x)^(1/2)-x)/(I*x-1)^2))*(-a*(I*x
-1))^(1/4)/(I*x-1)*(-(I*x-1)^3*(I*x+1))^(1/4)/(a*(I*x+1))^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=\frac {\sqrt {i} a \log \left (\frac {\sqrt {i} {\left (a x - i \, a\right )} + {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{x - i}\right ) - \sqrt {i} a \log \left (-\frac {\sqrt {i} {\left (a x - i \, a\right )} - {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{x - i}\right ) + \sqrt {-i} a \log \left (\frac {\sqrt {-i} {\left (a x - i \, a\right )} + {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{x - i}\right ) - \sqrt {-i} a \log \left (-\frac {\sqrt {-i} {\left (a x - i \, a\right )} - {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{x - i}\right ) - 2 i \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, a} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=\int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:The choice was do
ne assuming 0=[0,0]ext_reduce Error: Bad Argument Typeintegrate(4*i*((sageVARa+(-i)*sageVARa*sageVARx)^(1/4))^
4/(-((sageVARa

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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