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

Optimal. Leaf size=256 \[ -\frac {i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}-\frac {i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}} \]

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Rubi [A]  time = 0.17, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {50, 63, 240, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\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}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[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 50

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 63

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 204

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

Rule 211

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 240

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 617

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 628

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 1162

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 1165

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*} \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 {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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*}

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Mathematica [C]  time = 0.02, size = 70, normalized size = 0.27 \begin {gather*} \frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{5/4} \, _2F_1\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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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IntegrateAlgebraic [A]  time = 0.47, size = 126, normalized size = 0.49 \begin {gather*} \frac {\sqrt [4]{-1} \sqrt [4]{x-i} \sqrt [4]{a-i a x} \left (-(-1)^{3/4} (x-i)^{3/4} \sqrt [4]{x+i}+\sqrt [4]{-1} \tan ^{-1}\left (\frac {\sqrt [4]{x+i}}{\sqrt [4]{x-i}}\right )+\sqrt [4]{-1} \tanh ^{-1}\left (\frac {\sqrt [4]{x+i}}{\sqrt [4]{x-i}}\right )\right )}{\sqrt [4]{x+i} \sqrt [4]{a+i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/4)*(-I + x)^(1/4)*(a - I*a*x)^(1/4)*(-((-1)^(3/4)*(-I + x)^(3/4)*(I + x)^(1/4)) + (-1)^(1/4)*ArcTan[(
I + x)^(1/4)/(-I + x)^(1/4)] + (-1)^(1/4)*ArcTanh[(I + x)^(1/4)/(-I + x)^(1/4)]))/((I + x)^(1/4)*(a + I*a*x)^(
1/4))

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fricas [A]  time = 1.49, size = 194, normalized size = 0.76 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.28, size = 477, normalized size = 1.86 \begin {gather*} \frac {i \left (x -i\right ) \left (x +i\right ) \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}}-\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}-i\right ) \ln \left (\frac {-x^{3}+\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}-i\right )-2 i x^{2}+2 i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}-i\right )-i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +x +i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-i\right )-\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-i\right )+\sqrt {-x^{4}-2 i x^{3}-2 i x +1}}{\left (i x -1\right )^{2}}\right )}{2}-\frac {i \RootOf \left (\textit {\_Z}^{2}-i\right ) \ln \left (\frac {-x^{3}+i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}-i\right )-2 i x^{2}-2 \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}-i\right )+i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +x +\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-i\right )-i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-i\right )-\sqrt {-x^{4}-2 i x^{3}-2 i x +1}}{\left (i x -1\right )^{2}}\right )}{2}\right ) \left (-\left (i x -1\right ) a \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 (\left (i x +1\right ) a \right )^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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