3.18 \(\int (b \tan (c+d x))^{2/3} \, dx\)

Optimal. Leaf size=224 \[ \frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}+\frac {b^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt {3}\right )}{2 d}+\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}-\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d} \]

[Out]

b^(2/3)*arctan((b*tan(d*x+c))^(1/3)/b^(1/3))/d+1/2*b^(2/3)*arctan(-3^(1/2)+2*(b*tan(d*x+c))^(1/3)/b^(1/3))/d+1
/2*b^(2/3)*arctan(3^(1/2)+2*(b*tan(d*x+c))^(1/3)/b^(1/3))/d+1/4*b^(2/3)*ln(b^(2/3)-b^(1/3)*3^(1/2)*(b*tan(d*x+
c))^(1/3)+(b*tan(d*x+c))^(2/3))*3^(1/2)/d-1/4*b^(2/3)*ln(b^(2/3)+b^(1/3)*3^(1/2)*(b*tan(d*x+c))^(1/3)+(b*tan(d
*x+c))^(2/3))*3^(1/2)/d

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Rubi [A]  time = 0.39, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3476, 329, 295, 634, 618, 204, 628, 203} \[ \frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}+\frac {b^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt {3}\right )}{2 d}+\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}-\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x])^(2/3),x]

[Out]

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

Rule 203

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

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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int (b \tan (c+d x))^{2/3} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^{2/3}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^4}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}\\ &=\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}+\frac {\sqrt {3} x}{2}}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}+\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}-\frac {\sqrt {3} x}{2}}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b^{2/3}+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}\\ &=\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}+\frac {\left (\sqrt {3} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 d}-\frac {\left (\sqrt {3} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 d}\\ &=\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}+\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}-\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}+\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} d}-\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} d}\\ &=\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 d}+\frac {b^{2/3} \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 d}+\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}-\frac {\sqrt {3} b^{2/3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 40, normalized size = 0.18 \[ \frac {3 (b \tan (c+d x))^{5/3} \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\tan ^2(c+d x)\right )}{5 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x])^(2/3),x]

[Out]

(3*Hypergeometric2F1[5/6, 1, 11/6, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(5/3))/(5*b*d)

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fricas [B]  time = 0.57, size = 583, normalized size = 2.60 \[ -\frac {1}{4} \, \sqrt {3} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} \log \left (\sqrt {3} b^{3} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {5}{6}} + b^{4} d^{4} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {2}{3}} + b^{6} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + \frac {1}{4} \, \sqrt {3} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} \log \left (-\sqrt {3} b^{3} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {5}{6}} + b^{4} d^{4} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {2}{3}} + b^{6} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) - \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} b^{4} + 2 \, b^{3} d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} - 2 \, \sqrt {\sqrt {3} b^{3} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {5}{6}} + b^{4} d^{4} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {2}{3}} + b^{6} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} d \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}}}{b^{4}}\right ) - \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} b^{4} - 2 \, b^{3} d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} + 2 \, \sqrt {-\sqrt {3} b^{3} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {5}{6}} + b^{4} d^{4} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {2}{3}} + b^{6} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} d \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}}}{b^{4}}\right ) - 2 \, \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} \arctan \left (-\frac {b^{3} d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}} - \sqrt {b^{4} d^{4} \left (\frac {b^{4}}{d^{6}}\right )^{\frac {2}{3}} + b^{6} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} d \left (\frac {b^{4}}{d^{6}}\right )^{\frac {1}{6}}}{b^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3)*(b^4/d^6)^(1/6)*log(sqrt(3)*b^3*d^5*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(5/6) + b^4*d^4
*(b^4/d^6)^(2/3) + b^6*(b*sin(d*x + c)/cos(d*x + c))^(2/3)) + 1/4*sqrt(3)*(b^4/d^6)^(1/6)*log(-sqrt(3)*b^3*d^5
*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(5/6) + b^4*d^4*(b^4/d^6)^(2/3) + b^6*(b*sin(d*x + c)/cos(d*x +
c))^(2/3)) - (b^4/d^6)^(1/6)*arctan(-(sqrt(3)*b^4 + 2*b^3*d*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(1/
6) - 2*sqrt(sqrt(3)*b^3*d^5*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(5/6) + b^4*d^4*(b^4/d^6)^(2/3) + b^
6*(b*sin(d*x + c)/cos(d*x + c))^(2/3))*d*(b^4/d^6)^(1/6))/b^4) - (b^4/d^6)^(1/6)*arctan((sqrt(3)*b^4 - 2*b^3*d
*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(1/6) + 2*sqrt(-sqrt(3)*b^3*d^5*(b*sin(d*x + c)/cos(d*x + c))^(
1/3)*(b^4/d^6)^(5/6) + b^4*d^4*(b^4/d^6)^(2/3) + b^6*(b*sin(d*x + c)/cos(d*x + c))^(2/3))*d*(b^4/d^6)^(1/6))/b
^4) - 2*(b^4/d^6)^(1/6)*arctan(-(b^3*d*(b*sin(d*x + c)/cos(d*x + c))^(1/3)*(b^4/d^6)^(1/6) - sqrt(b^4*d^4*(b^4
/d^6)^(2/3) + b^6*(b*sin(d*x + c)/cos(d*x + c))^(2/3))*d*(b^4/d^6)^(1/6))/b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*tan(d*x + c))^(2/3), x)

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maple [A]  time = 0.12, size = 202, normalized size = 0.90 \[ \frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 d \left (b^{2}\right )^{\frac {1}{6}}}+\frac {b \arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{d \left (b^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 d \left (b^{2}\right )^{\frac {1}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c))^(2/3),x)

[Out]

1/4/d/b*3^(1/2)*(b^2)^(5/6)*ln((b*tan(d*x+c))^(2/3)-3^(1/2)*(b^2)^(1/6)*(b*tan(d*x+c))^(1/3)+(b^2)^(1/3))+1/2/
d*b/(b^2)^(1/6)*arctan(2*(b*tan(d*x+c))^(1/3)/(b^2)^(1/6)-3^(1/2))+1/d*b/(b^2)^(1/6)*arctan((b*tan(d*x+c))^(1/
3)/(b^2)^(1/6))-1/4/d/b*3^(1/2)*(b^2)^(5/6)*ln((b*tan(d*x+c))^(2/3)+3^(1/2)*(b^2)^(1/6)*(b*tan(d*x+c))^(1/3)+(
b^2)^(1/3))+1/2/d*b/(b^2)^(1/6)*arctan(2*(b*tan(d*x+c))^(1/3)/(b^2)^(1/6)+3^(1/2))

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maxima [A]  time = 0.48, size = 168, normalized size = 0.75 \[ -\frac {{\left (\frac {\sqrt {3} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} b^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} b^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}}\right )} b}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(3)*log(sqrt(3)*(b*tan(d*x + c))^(1/3)*b^(1/3) + (b*tan(d*x + c))^(2/3) + b^(2/3))/b^(1/3) - sqrt(3)
*log(-sqrt(3)*(b*tan(d*x + c))^(1/3)*b^(1/3) + (b*tan(d*x + c))^(2/3) + b^(2/3))/b^(1/3) - 2*arctan((sqrt(3)*b
^(1/3) + 2*(b*tan(d*x + c))^(1/3))/b^(1/3))/b^(1/3) - 2*arctan(-(sqrt(3)*b^(1/3) - 2*(b*tan(d*x + c))^(1/3))/b
^(1/3))/b^(1/3) - 4*arctan((b*tan(d*x + c))^(1/3)/b^(1/3))/b^(1/3))*b/d

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mupad [B]  time = 2.95, size = 259, normalized size = 1.16 \[ \frac {{\left (-1\right )}^{1/6}\,b^{2/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {{\left (-1\right )}^{1/6}\,b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {486\,{\left (-1\right )}^{1/6}\,b^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {{\left (-1\right )}^{1/6}\,b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {486\,{\left (-1\right )}^{1/6}\,b^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-1\right )}^{1/6}\,b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {486\,{\left (-1\right )}^{1/6}\,b^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {{\left (-1\right )}^{1/6}\,b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {486\,{\left (-1\right )}^{1/6}\,b^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(c + d*x))^(2/3),x)

[Out]

((-1)^(1/6)*b^(2/3)*atan(((-1)^(2/3)*(b*tan(c + d*x))^(1/3))/b^(1/3))*1i)/d - ((-1)^(1/6)*b^(2/3)*log((972*b^9
)/d^3 + (486*(-1)^(1/6)*b^(26/3)*(3^(1/2)*1i - 1)*(b*tan(c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/2 - 1/2))/(2*d) -
((-1)^(1/6)*b^(2/3)*log((972*b^9)/d^3 + (486*(-1)^(1/6)*b^(26/3)*(3^(1/2)*1i + 1)*(b*tan(c + d*x))^(1/3))/d^3
)*((3^(1/2)*1i)/2 + 1/2))/(2*d) + ((-1)^(1/6)*b^(2/3)*log((972*b^9)/d^3 - (486*(-1)^(1/6)*b^(26/3)*(3^(1/2)*1i
- 1)*(b*tan(c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/4 - 1/4))/d + ((-1)^(1/6)*b^(2/3)*log((972*b^9)/d^3 - (486*(-
1)^(1/6)*b^(26/3)*(3^(1/2)*1i + 1)*(b*tan(c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/4 + 1/4))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c))**(2/3),x)

[Out]

Integral((b*tan(c + d*x))**(2/3), x)

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