3.7.91 \(\int \sqrt [3]{a+b \tan (c+d x)} \, dx\) [691]

Optimal. Leaf size=415 \[ -\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3566, 726, 52, 59, 631, 210, 31} \begin {gather*} \frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {1}{4} x \sqrt [3]{a-\sqrt {-b^2}}-\frac {1}{4} x \sqrt [3]{a+\sqrt {-b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(1/3),x]

[Out]

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

Rule 31

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

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 59

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 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 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 726

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 294, normalized size = 0.71 \begin {gather*} \frac {-i \sqrt [3]{a-i b} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a-i b)^{2/3}+\sqrt [3]{a-i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )+i \sqrt [3]{a+i b} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a+i b)^{2/3}+\sqrt [3]{a+i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^(1/3),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.40, size = 60, normalized size = 0.14

method	result	size
derivativedivides	\(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\)	\(60\)
default	\(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3320 vs. \(2 (325) = 650\).
time = 1.55, size = 3320, normalized size = 8.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2
))*log(2*a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*si
n(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + a^2*d^2*((a^2 + b^2)/d^6)
^(1/3) + a^2*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(2/3)) + 2*((a^2 + b^2)/d^6)^(1/6)*arctan(-(a*d^
8*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6) - sqrt(2*a*d^4*
((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan((d
^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + a^2*d^2*((a^2 + b^2)/d^6)^(1/3) + a^2*((
a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(2/3))*d^8*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6) + (a^4 + a^2*b
^2)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))/((a^4 + a^2*b^2)*cos
(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2))))*sin(2/3*arctan((d^6*sqrt(a
^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + (sqrt(3)*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((
d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) - ((a^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan(
(d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*arctan(-(2*a*d^8*((a*cos(d*x + c) + b*s
in(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt(
(a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) - 2*(sqrt(3)*a*d^8*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x +
c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6) + 2*(a^4 + a^2*b^2)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^
2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sq
rt(a^2/d^6))/a^2)) + 2*(sqrt(3)*d^8*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sq
rt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) - d^8*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*cos(2/3*arctan((d
^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*sqrt(sqrt(3)*a*d^4*((a*cos(d*x + c) + b*s
in(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt(
(a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) - a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*s
qrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/
d^6))/a^2)) + a^2*d^2*((a^2 + b^2)/d^6)^(1/3) + a^2*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(2/3)) +
sqrt(3)*(a^4 + a^2*b^2))/(3*a^4 + 3*a^2*b^2 - 4*(a^4 + a^2*b^2)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 +
b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2))^2)) + (sqrt(3)*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6
)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + ((a^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^
6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*arctan((2*a*d^8*((a*cos(d*x + c) + b*sin(d*x + c))/cos(
d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6)
+ b*d^3*sqrt(a^2/d^6))/a^2)) + 2*(sqrt(3)*a*d^8*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^
2/d^6)*((a^2 + b^2)/d^6)^(5/6) - 2*(a^4 + a^2*b^2)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b
*d^3*sqrt(a^2/d^6))/a^2)))*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)
) - 2*(sqrt(3)*d^8*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^
6) + b*d^3*sqrt(a^2/d^6))/a^2)) + d^8*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(5/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*
sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*sqrt(-sqrt(3)*a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos
(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6)
+ b*d^3*sqrt(a^2/d^6))/a^2)) - a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a
^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + a^
2*d^2*((a^2 + b^2)/d^6)^(1/3) + a^2*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(2/3)) - sqrt(3)*(a^4 + a
^2*b^2))/(3*a^4 + 3*a^2*b^2 - 4*(a^4 + a^2*b^2)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^
3*sqrt(a^2/d^6))/a^2))^2)) - 1/4*(sqrt(3)*((a^2 + b^2)/d^6)^(1/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2
+ b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)) + ((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2
+ b^2)/d^6) + b*d^3*sqrt(a^2/d^6))/a^2)))*log(sqrt(3)*a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^
(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d^6)^(1/6)*cos(2/3*arctan((d^6*sqrt(a^2/d^6)*sqrt((a^2 + b^2)/d^6) + b*d^3*sq
rt(a^2/d^6))/a^2)) - a*d^4*((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))^(1/3)*sqrt(a^2/d^6)*((a^2 + b^2)/d
^6)^(1/6)*sin(2/3*arctan((d^6*sqrt(a^2/d^6)*sqr...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 7.01, size = 863, normalized size = 2.08 \begin {gather*} \ln \left ({\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+d\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (-a\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+b\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}+d^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{4/3}+2\,b\,d\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}-\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {\left (\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}-3888\,a\,b^4\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (a^2+b^2\right )\right )}{4}+\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {\left (\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}+3888\,a\,b^4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (a^2+b^2\right )\right )}{4}-\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}-\ln \left (-\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}+\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}-3888\,a\,b^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^2+b^2\right )\right )}{4}+\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}+3888\,a\,b^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^2+b^2\right )\right )}{4}-\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x))^(1/3),x)

[Out]

log((a + b*tan(c + d*x))^(1/3) + d*(-(a*1i + b)/d^3)^(1/3)*1i)*(-(a*1i + b)/(8*d^3))^(1/3) + log(b*(a + b*tan(
c + d*x))^(1/3)*1i - a*(a + b*tan(c + d*x))^(1/3) + d^4*((a*1i - b)/d^3)^(4/3) + 2*b*d*((a*1i - b)/d^3)^(1/3))
*((a*1i - b)/(8*d^3))^(1/3) - log((486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^(1/3))/d^4 - (((((3^(1/2)*1i)/2 -
1/2)*(-(a*1i + b)/d^3)^(2/3)*((3888*b^5*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/3))/d - 3888*a*b^4*((3^(1/2)*1i)/2
+ 1/2)*(-(a*1i + b)/d^3)^(1/3)*(a^2 + b^2)))/4 + (1944*a*b^5*(a^2 + b^2))/d^3)*((3^(1/2)*1i)/2 + 1/2)*(-(a*1i
+ b)/d^3)^(1/3))/2)*((3^(1/2)*1i)/2 + 1/2)*(-(a*1i + b)/(8*d^3))^(1/3) + log((486*(b^8 - a^4*b^4)*(a + b*tan(
c + d*x))^(1/3))/d^4 - (((((3^(1/2)*1i)/2 + 1/2)*(-(a*1i + b)/d^3)^(2/3)*((3888*b^5*(a^2 + b^2)*(a + b*tan(c +
d*x))^(1/3))/d + 3888*a*b^4*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/d^3)^(1/3)*(a^2 + b^2)))/4 - (1944*a*b^5*(a^2
+ b^2))/d^3)*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/d^3)^(1/3))/2)*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/(8*d^3))^
(1/3) - log((((a*1i - b)/d^3)^(1/3)*((3^(1/2)*1i)/2 + 1/2)*((((a*1i - b)/d^3)^(2/3)*((3^(1/2)*1i)/2 - 1/2)*((3
888*b^5*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/3))/d - 3888*a*b^4*((a*1i - b)/d^3)^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(
a^2 + b^2)))/4 + (1944*a*b^5*(a^2 + b^2))/d^3))/2 - (486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^(1/3))/d^4)*((3^
(1/2)*1i)/2 + 1/2)*((a*1i - b)/(8*d^3))^(1/3) + log((486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^(1/3))/d^4 - (((
a*1i - b)/d^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*((((a*1i - b)/d^3)^(2/3)*((3^(1/2)*1i)/2 + 1/2)*((3888*b^5*(a^2 +
b^2)*(a + b*tan(c + d*x))^(1/3))/d + 3888*a*b^4*((a*1i - b)/d^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(a^2 + b^2)))/4
- (1944*a*b^5*(a^2 + b^2))/d^3))/2)*((3^(1/2)*1i)/2 - 1/2)*((a*1i - b)/(8*d^3))^(1/3)

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