3.478 \(\int \frac {d-c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\)

Optimal. Leaf size=299 \[ \frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {1}{4} i x \sqrt [3]{c-i d}+\frac {1}{4} i x \sqrt [3]{c+i d} \]

[Out]

-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*ln(cos(f*x+e))/f-1/4*(c+I*d)^(1/3)*ln(cos(f*x+e
))/f-3/4*(c-I*d)^(1/3)*ln((c-I*d)^(1/3)-(c+d*tan(f*x+e))^(1/3))/f-3/4*(c+I*d)^(1/3)*ln((c+I*d)^(1/3)-(c+d*tan(
f*x+e))^(1/3))/f+1/2*(c-I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c-I*d)^(1/3))*3^(1/2))*3^(1/2)/f+1/
2*(c+I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+I*d)^(1/3))*3^(1/2))*3^(1/2)/f

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Rubi [A]  time = 0.34, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3539, 3537, 57, 617, 204, 31} \[ \frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {1}{4} i x \sqrt [3]{c-i d}+\frac {1}{4} i x \sqrt [3]{c+i d} \]

Antiderivative was successfully verified.

[In]

Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x])^(2/3),x]

[Out]

(-I/4)*(c - I*d)^(1/3)*x + (I/4)*(c + I*d)^(1/3)*x + (Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*
x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/(2*f) + (Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/
3))/(c + I*d)^(1/3))/Sqrt[3]])/(2*f) - ((c - I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - ((c + I*d)^(1/3)*Log[Cos[e
+ f*x]])/(4*f) - (3*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) - (3*(c + I*d)^(1
/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

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 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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {d-c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx &=\frac {1}{2} (-i c+d) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx+\frac {1}{2} (i c+d) \int \frac {1+i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\\ &=-\frac {(c-i d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) (c-i d x)^{2/3}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {(c+i d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) (c+i d x)^{2/3}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {\left (3 \sqrt [3]{c-i d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c-i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {\left (3 (c-i d)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(c-i d)^{2/3}+\sqrt [3]{c-i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {\left (3 \sqrt [3]{c+i d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c+i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {\left (3 (c+i d)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(c+i d)^{2/3}+\sqrt [3]{c+i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {\left (3 \sqrt [3]{c-i d}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}\right )}{2 f}-\frac {\left (3 \sqrt [3]{c+i d}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}\right )}{2 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x+\frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 330, normalized size = 1.10 \[ \frac {2 \sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-2 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )-2 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )+\sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )+\sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x])^(2/3),x]

[Out]

(2*Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(c
+ I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*(c - I*d)^(1/3)*Log[(c
- I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 2*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3
)] + (c - I*d)^(1/3)*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(
2/3)] + (c + I*d)^(1/3)*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x]
)^(2/3)])/(4*f)

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fricas [B]  time = 1.20, size = 2721, normalized size = 9.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d-c*tan(f*x+e))/(c+d*tan(f*x+e))^(2/3),x, algorithm="fricas")

[Out]

1/2*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2))*log(2*f*((c
*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^
2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*
x + e))^(2/3)) - 2*((c^2 + d^2)/f^6)^(1/6)*arctan((sqrt(2*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(
1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^
2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3))*f^5*((c^2 + d^2)/f^6)^(5/6) - f^
5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6) - (c^2 + d^2)*cos(2/3*arctan(
(f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))/((c^2 + d^2)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6
) + c*f^3)*sqrt(d^2/f^6)/d^2))))*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - (sqr
t(3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - ((c^2 +
d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*arctan(-(2*sqrt(3)*f^5
*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2
+ d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + 2*(f^5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2
+ d^2)/f^6)^(5/6) - 2*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sin
(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - 2*(sqrt(3)*f^5*((c^2 + d^2)/f^6)^(5/6)*c
os(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^5*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*ar
ctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sqrt(-sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e
))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6
)/d^2)) - f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6
*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f
*x + e))/cos(f*x + e))^(2/3)) - sqrt(3)*(c^2 + d^2))/(4*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6)
+ c*f^3)*sqrt(d^2/f^6)/d^2))^2 - c^2 - d^2)) + (sqrt(3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2
+ d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + ((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) +
c*f^3)*sqrt(d^2/f^6)/d^2)))*arctan((2*sqrt(3)*f^5*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^
2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - 2*(f^5*((c*cos(f*
x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(5/6) - 2*(c^2 + d^2)*cos(2/3*arctan((f^6*sqrt(
(c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6
)/d^2)) - 2*(sqrt(3)*f^5*((c^2 + d^2)/f^6)^(5/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f
^6)/d^2)) - f^5*((c^2 + d^2)/f^6)^(5/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2))
)*sqrt(sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan
((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^
(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c
^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) - sqrt(3)*(c^2 + d^2))/(4*(c^2
+ d^2)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2))^2 - c^2 - d^2)) + 1/4*(sqrt(3)*(
(c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - ((c^2 + d^2)/f
^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*log(sqrt(3)*f*((c*cos(f*x +
e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c
*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*c
os(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(
f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3)) - 1/4*(sqrt(3)*((c^2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqr
t((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) + ((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)
/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)))*log(-sqrt(3)*f*((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^
2 + d^2)/f^6)^(1/6)*sin(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*f^3)*sqrt(d^2/f^6)/d^2)) - f*((c*cos(f*x + e
) + d*sin(f*x + e))/cos(f*x + e))^(1/3)*((c^2 + d^2)/f^6)^(1/6)*cos(2/3*arctan((f^6*sqrt((c^2 + d^2)/f^6) + c*
f^3)*sqrt(d^2/f^6)/d^2)) + f^2*((c^2 + d^2)/f^6)^(1/3) + ((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))^(2/3
))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d-c*tan(f*x+e))/(c+d*tan(f*x+e))^(2/3),x, algorithm="giac")

[Out]

undef

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maple [C]  time = 0.31, size = 72, normalized size = 0.24 \[ -\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3} c +c^{2}+d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} c -c^{2}-d^{2}\right ) \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}}{2 f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d-c*tan(f*x+e))/(c+d*tan(f*x+e))^(2/3),x)

[Out]

-1/2/f*sum((_R^3*c-c^2-d^2)/(_R^5-_R^2*c)*ln((c+d*tan(f*x+e))^(1/3)-_R),_R=RootOf(_Z^6-2*_Z^3*c+c^2+d^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {c \tan \left (f x + e\right ) - d}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d-c*tan(f*x+e))/(c+d*tan(f*x+e))^(2/3),x, algorithm="maxima")

[Out]

-integrate((c*tan(f*x + e) - d)/(d*tan(f*x + e) + c)^(2/3), x)

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mupad [B]  time = 22.17, size = 4308, normalized size = 14.41 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d - c*tan(e + f*x))/(c + d*tan(e + f*x))^(2/3),x)

[Out]

log(((((1944*c*d^4*(c^2 + d^2)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) +
(7776*c*d^6*(c + d*tan(e + f*x))^(1/3))/f)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2
)^2))^(2/3))/16 - (972*d^8)/f^3)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3)
)/4 - (486*d^8*(c + d*tan(e + f*x))^(1/3))/f^4)*(((256*c^2*d^8*f^6 - d^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^
2*f^6))^(1/2) + 16*c*d^4*f^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3) + log(((((1944*c*d^4*(c^2 + d^2)
*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) + (7776*c*d^6*(c + d*tan(e + f
*x))^(1/3))/f)*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/16 - (972*d^8)/
f^3)*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/4 - (486*d^8*(c + d*tan(e
+ f*x))^(1/3))/f^4)*(-((256*c^2*d^8*f^6 - d^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) - 16*c*d^4*f
^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3) + log(- (486*c^4*d^4*(c + d*tan(e + f*x))^(1/3))/f^4 - (((
16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3)*(1944*c*d^4*(-c^8*d^2*f^6)^(1/
2) + 1944*c^4*d^6*f^3 + 243*c^5*d^4*f^5*(c + d*tan(e + f*x))^(1/3)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c
^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2/3) - 243*c*d^8*f^5*(c + d*tan(e + f*x))^(1/3)*((16*(-c^8*d^2*f^6)^(1/2) -
8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2/3)))/(4*f^6*(c^2 + d^2)))*((((16*c^5*f^3 - 16*c^3*d^2*f^3)^
2/4 - c^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(64*(c^4*f^6 + d^4*f
^6 + 2*c^2*d^2*f^6)))^(1/3) + log(((-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2)
)^(1/3)*(1944*c*d^4*(-c^8*d^2*f^6)^(1/2) - 1944*c^4*d^6*f^3 - 243*c^5*d^4*f^5*(c + d*tan(e + f*x))^(1/3)*(-(16
*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2/3) + 243*c*d^8*f^5*(c + d*tan(e + f
*x))^(1/3)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2/3)))/(4*f^6*(c^2 +
d^2)) - (486*c^4*d^4*(c + d*tan(e + f*x))^(1/3))/f^4)*(-(((16*c^5*f^3 - 16*c^3*d^2*f^3)^2/4 - c^6*(64*c^4*f^6
+ 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^
(1/3) + log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i + 1)*(972*c*d^4*(3^(1/2)*1i - 1)*(c^2 + d^2)*((16*(-c^8*d^2*f^6)^(
1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) + (3888*c*d^4*(c^2 - d^2)*(c + d*tan(e + f*x))^(1
/3))/f)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/32 - (972*c^4*d^4)/
f^3)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/8 - (486*c^4*d^4*(c +
d*tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*(((-256*c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(64*c^4*f^6 + 64
*d^4*f^6 + 128*c^2*d^2*f^6))^(1/3)*1i)/2 - (((-256*c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(64*c^4*f^6
+ 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/3)/2) + log(((3^(1/2)*1i - 1)*(((3^(1/2)*1i + 1)*(972*c*d^4*(3^(1/2)*1i -
1)*(c^2 + d^2)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) + (3888*c*d
^4*(c^2 - d^2)*(c + d*tan(e + f*x))^(1/3))/f)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^
2 + d^2)^2))^(2/3))/32 - (972*c^4*d^4)/f^3)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2
+ d^2)^2))^(1/3))/8 - (486*c^4*d^4*(c + d*tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*(-((-256*c^8*d^2*f^6)^(1/2) + 8*
c^5*f^3 - 8*c^3*d^2*f^3)/(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/3)*1i)/2 - (-((-256*c^8*d^2*f^6)^(1/2
) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/3)/2) + log((((3^(1/2)*1i)/2 -
1/2)*((972*d^8)/f^3 + (((3^(1/2)*1i)/2 + 1/2)*((7776*c*d^6*(c + d*tan(e + f*x))^(1/3))/f + 1944*c*d^4*((3^(1/2
)*1i)/2 - 1/2)*(c^2 + d^2)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))*((8*
(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/16)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1
/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/4 + (486*d^8*(c + d*tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2
- 1/2)*(((256*c^2*d^8*f^6 - d^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) + 16*c*d^4*f^3)/(64*(c^4*f^
6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3) - log((486*d^8*(c + d*tan(e + f*x))^(1/3))/f^4 - (((3^(1/2)*1i)/2 + 1/2)*
((972*d^8)/f^3 - (((3^(1/2)*1i)/2 - 1/2)*((7776*c*d^6*(c + d*tan(e + f*x))^(1/3))/f - 1944*c*d^4*((3^(1/2)*1i)
/2 + 1/2)*(c^2 + d^2)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))*((8*(-d^6
*f^6*(c^2 - d^2)^2)^(1/2) + 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/16)*((8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) +
16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/4)*((3^(1/2)*1i)/2 + 1/2)*(((256*c^2*d^8*f^6 - d^6*(64*c^4*f^6 + 64
*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) + 16*c*d^4*f^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3) + log((((3^
(1/2)*1i)/2 - 1/2)*((972*d^8)/f^3 + (((3^(1/2)*1i)/2 + 1/2)*((7776*c*d^6*(c + d*tan(e + f*x))^(1/3))/f + 1944*
c*d^4*((3^(1/2)*1i)/2 - 1/2)*(c^2 + d^2)*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^
2))^(1/3))*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/16)*(-(8*(-d^6*f^6*
(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/4 + (486*d^8*(c + d*tan(e + f*x))^(1/3))/f^4)
*((3^(1/2)*1i)/2 - 1/2)*(-((256*c^2*d^8*f^6 - d^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) - 16*c*d^
4*f^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3) - log((486*d^8*(c + d*tan(e + f*x))^(1/3))/f^4 - (((3^(
1/2)*1i)/2 + 1/2)*((972*d^8)/f^3 - (((3^(1/2)*1i)/2 - 1/2)*((7776*c*d^6*(c + d*tan(e + f*x))^(1/3))/f - 1944*c
*d^4*((3^(1/2)*1i)/2 + 1/2)*(c^2 + d^2)*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2
))^(1/3))*(-(8*(-d^6*f^6*(c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(2/3))/16)*(-(8*(-d^6*f^6*(
c^2 - d^2)^2)^(1/2) - 16*c*d^4*f^3)/(f^6*(c^2 + d^2)^2))^(1/3))/4)*((3^(1/2)*1i)/2 + 1/2)*(-((256*c^2*d^8*f^6
- d^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) - 16*c*d^4*f^3)/(64*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^
6)))^(1/3) - log((((3^(1/2)*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*(1944*c*d^4*((3^(1/2)*1i)/2 + 1/2)*(c^2 + d^
2)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) - (3888*c*d^4*(c^2 - d^2)
*(c + d*tan(e + f*x))^(1/3))/f)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(2
/3))/16 - (972*c^4*d^4)/f^3)*((16*(-c^8*d^2*f^6)^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3)
)/4 + (486*c^4*d^4*(c + d*tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2 + 1/2)*((((16*c^5*f^3 - 16*c^3*d^2*f^3)^2/
4 - c^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) - 8*c^5*f^3 + 8*c^3*d^2*f^3)/(64*(c^4*f^6 + d^4*f^6
+ 2*c^2*d^2*f^6)))^(1/3) - log((((3^(1/2)*1i)/2 + 1/2)*((((3^(1/2)*1i)/2 - 1/2)*(1944*c*d^4*((3^(1/2)*1i)/2 +
1/2)*(c^2 + d^2)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^2 + d^2)^2))^(1/3) - (3888*c
*d^4*(c^2 - d^2)*(c + d*tan(e + f*x))^(1/3))/f)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(
c^2 + d^2)^2))^(2/3))/16 - (972*c^4*d^4)/f^3)*(-(16*(-c^8*d^2*f^6)^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(f^6*(c^
2 + d^2)^2))^(1/3))/4 + (486*c^4*d^4*(c + d*tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2 + 1/2)*(-(((16*c^5*f^3 -
16*c^3*d^2*f^3)^2/4 - c^6*(64*c^4*f^6 + 64*d^4*f^6 + 128*c^2*d^2*f^6))^(1/2) + 8*c^5*f^3 - 8*c^3*d^2*f^3)/(64
*(c^4*f^6 + d^4*f^6 + 2*c^2*d^2*f^6)))^(1/3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {d}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {2}{3}}}\right )\, dx - \int \frac {c \tan {\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d-c*tan(f*x+e))/(c+d*tan(f*x+e))**(2/3),x)

[Out]

-Integral(-d/(c + d*tan(e + f*x))**(2/3), x) - Integral(c*tan(e + f*x)/(c + d*tan(e + f*x))**(2/3), x)

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