3.1.19 \(\int \sqrt [3]{b \tan (c+d x)} \, dx\) [19]

Optimal. Leaf size=131 \[ -\frac {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3557, 335, 281, 298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x])^(1/3),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 281

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

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 335

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 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 648

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 3557

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.05, size = 40, normalized size = 0.31 \begin {gather*} \frac {3 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{4/3}}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x])^(1/3),x]

[Out]

(3*Hypergeometric2F1[2/3, 1, 5/3, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(4/3))/(4*b*d)

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Maple [A]
time = 0.08, size = 108, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {3 b \left (-\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {4}{3}}-\left (b^{2}\right )^{\frac {1}{3}} \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\)	\(108\)
default	\(\frac {3 b \left (-\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {4}{3}}-\left (b^{2}\right )^{\frac {1}{3}} \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c))^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/d*b*(-1/6/(b^2)^(1/3)*ln((b*tan(d*x+c))^(2/3)+(b^2)^(1/3))+1/12/(b^2)^(1/3)*ln((b*tan(d*x+c))^(4/3)-(b^2)^(1
/3)*(b*tan(d*x+c))^(2/3)+(b^2)^(2/3))+1/6*3^(1/2)/(b^2)^(1/3)*arctan(1/3*3^(1/2)*(2/(b^2)^(1/3)*(b*tan(d*x+c))
^(2/3)-1)))

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Maxima [A]
time = 0.49, size = 98, normalized size = 0.75 \begin {gather*} \frac {2 \, \sqrt {3} b^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} - b^{\frac {2}{3}}\right )}}{3 \, b^{\frac {2}{3}}}\right ) + b^{\frac {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {4}{3}} - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} b^{\frac {2}{3}} + b^{\frac {4}{3}}\right ) - 2 \, b^{\frac {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{4 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c))^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 124, normalized size = 0.95 \begin {gather*} \frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} + \sqrt {3} b}{3 \, b}\right ) - \left (-b\right )^{\frac {1}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b \tan \left (d x + c\right ) - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - \left (-b\right )^{\frac {1}{3}} b\right ) + 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + \left (-b\right )^{\frac {2}{3}}\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c))^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{b \tan {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c))**(1/3),x)

[Out]

Integral((b*tan(c + d*x))**(1/3), x)

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Giac [A]
time = 0.46, size = 127, normalized size = 0.97 \begin {gather*} \frac {1}{4} \, b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2} d} + \frac {{\left | b \right |}^{\frac {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b \tan \left (d x + c\right ) - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}}\right )}{b^{2} d} - \frac {2 \, {\left | b \right |}^{\frac {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{2} d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*tan(d*x+c))^(1/3),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.63, size = 146, normalized size = 1.11 \begin {gather*} \frac {{\left (-b\right )}^{1/3}\,\ln \left (81\,{\left (-b\right )}^{16/3}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}+81\,b^6\right )}{2\,d}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {81\,b^6}{d^4}-\frac {81\,{\left (-b\right )}^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {81\,b^6}{d^4}+\frac {162\,{\left (-b\right )}^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(c + d*x))^(1/3),x)

[Out]

((-b)^(1/3)*log(81*(-b)^(16/3)*(b*tan(c + d*x))^(2/3) + 81*b^6))/(2*d) - ((-b)^(1/3)*log((81*b^6)/d^4 - (81*(-
b)^(16/3)*((3^(1/2)*1i)/2 + 1/2)*(b*tan(c + d*x))^(2/3))/d^4)*((3^(1/2)*1i)/2 + 1/2))/(2*d) + ((-b)^(1/3)*log(
(81*b^6)/d^4 + (162*(-b)^(16/3)*((3^(1/2)*1i)/4 - 1/4)*(b*tan(c + d*x))^(2/3))/d^4)*((3^(1/2)*1i)/4 - 1/4))/d

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