3.1.64 \(\int \frac {1}{(a+b \tan (c+d \sqrt [3]{x}))^2} \, dx\) [64]

Optimal. Leaf size=610 \[ -\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}+\frac {6 b^2 x^{2/3}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {3 i b^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b \sqrt [3]{x} \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {3 i b^2 \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3} \]

[Out]

-6*I*b^2*x^(2/3)/(a^2+b^2)^2/d+6*b^2*x^(2/3)/(a+I*b)/(I*a+b)^2/d/(I*a-b+(I*a+b)*exp(2*I*(c+d*x^(1/3))))+x/(a-I
*b)^2+4*b*x/(I*a-b)/(a-I*b)^2-4*b^2*x/(a^2+b^2)^2+6*b^2*x^(1/3)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(
a^2+b^2)^2/d^2+6*b*x^(2/3)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d-6*I*b^2*x^(2/3)*ln
(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d-3*I*b^2*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+
I*b))/(a^2+b^2)^2/d^3+6*b*x^(1/3)*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^2-6*b
^2*x^(1/3)*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^2+3*b*polylog(3,-(a-I*b)*exp(2*I*(
c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^3-3*I*b^2*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b
^2)^2/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.00, antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3824, 3815, 2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438} \begin {gather*} -\frac {3 i b^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {3 i b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 b^2 \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac {6 i b^2 x^{2/3}}{d \left (a^2+b^2\right )^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x^{2/3}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt [3]{x}\right )}+i a-b\right )}+\frac {3 b \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}+\frac {6 b \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac {4 b x}{(-b+i a) (a-i b)^2}+\frac {x}{(a-i b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x^(1/3)])^(-2),x]

[Out]

((-6*I)*b^2*x^(2/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(2/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x/(a - I*b)^2 + (4*b*x)/((I*a - b)*(a - I*b)^2) - (4*b^2*x)/(a^2 + b^2)^2 + (6*b^2*x^(1
/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(2/3)*Log[1 + ((a -
I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - ((6*I)*b^2*x^(2/3)*Log[1 + ((a - I*b)
*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((3*I)*b^2*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d
*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (6*b*x^(1/3)*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))
/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (6*b^2*x^(1/3)*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/
(a + I*b))])/((a^2 + b^2)^2*d^2) + (3*b*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a -
I*b)^2*(a + I*b)*d^3) - ((3*I)*b^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^3)

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2222

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1
)*Log[F])), x] - Dist[d*(m/(b*f*g*n*(p + 1)*Log[F])), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3815

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

Rule 3824

Int[((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Ta
n[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx &=3 \text {Subst}\left (\int \frac {x^2}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {x^2}{(a-i b)^2}-\frac {4 b^2 x^2}{(i a+b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2}+\frac {4 b x^2}{(a-i b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {x}{(a-i b)^2}+\frac {(12 b) \text {Subst}\left (\int \frac {x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{(i a+b)^2}\\ &=\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2}-\frac {(12 b) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}\\ &=-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a+i b)^2 (i a+b)}-\frac {(12 b) \text {Subst}\left (\int x \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}\\ &=-\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {(6 b) \text {Subst}\left (\int \text {Li}_2\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b) (a+i b)^2 d}+\frac {\left (12 i b^2\right ) \text {Subst}\left (\int x \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}\\ &=-\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}+\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (1-\frac {i b}{a}\right ) x}{1+\frac {i b}{a}}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^3}\\ &=-\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {3 i b^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {3 i b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 4.71, size = 356, normalized size = 0.58 \begin {gather*} \frac {\frac {i b e^{2 i c} \left (\frac {6 b x^{2/3}}{-i a+b}+\frac {4 a d x}{-i a+b}-\frac {3 e^{-2 i c} \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (2 d \left (b+a d \sqrt [3]{x}\right ) \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )-i \left (b+2 a d \sqrt [3]{x}\right ) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )+a \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )\right )}{\left (a^2+b^2\right ) d^2}\right )}{d \left (b-b e^{2 i c}-i a \left (1+e^{2 i c}\right )\right )}+\frac {x (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {3 b^2 x^{2/3} \sin \left (d \sqrt [3]{x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt [3]{x}\right )+b \sin \left (c+d \sqrt [3]{x}\right )\right )}}{a^2+b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x^(1/3)])^(-2),x]

[Out]

((I*b*E^((2*I)*c)*((6*b*x^(2/3))/((-I)*a + b) + (4*a*d*x)/((-I)*a + b) - (3*((-I)*b*(-1 + E^((2*I)*c)) + a*(1
+ E^((2*I)*c)))*(2*d*(b + a*d*x^(1/3))*x^(1/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)] - I*(b
+ 2*a*d*x^(1/3))*PolyLog[2, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))] + a*PolyLog[3, -(((a - I*b)*E
^((2*I)*(c + d*x^(1/3))))/(a + I*b))]))/((a^2 + b^2)*d^2*E^((2*I)*c))))/(d*(b - b*E^((2*I)*c) - I*a*(1 + E^((2
*I)*c)))) + (x*(a*Cos[c] - b*Sin[c]))/(a*Cos[c] + b*Sin[c]) + (3*b^2*x^(2/3)*Sin[d*x^(1/3)])/(d*(a*Cos[c] + b*
Sin[c])*(a*Cos[c + d*x^(1/3)] + b*Sin[c + d*x^(1/3)])))/(a^2 + b^2)

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Maple [F]
time = 1.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(c+d*x^(1/3)))^2,x)

[Out]

int(1/(a+b*tan(c+d*x^(1/3)))^2,x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1732 vs. \(2 (491) = 982\).
time = 0.87, size = 1732, normalized size = 2.84 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="maxima")

[Out]

(3*(2*a*b*log(b*tan(d*x^(1/3) + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*x^(1/3) + c)^2 + 1)/(a^4 + 2*a
^2*b^2 + b^4) + (a^2 - b^2)*(d*x^(1/3) + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x^(
1/3) + c)))*c^2 + ((a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^3 - 3*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^
(1/3) + c)^2*c - 6*((I*a*b^2 + b^3)*c*cos(2*d*x^(1/3) + 2*c) - (a*b^2 - I*b^3)*c*sin(2*d*x^(1/3) + 2*c) + (I*a
*b^2 - b^3)*c)*arctan2(-b*cos(2*d*x^(1/3) + 2*c) + a*sin(2*d*x^(1/3) + 2*c) + b, a*cos(2*d*x^(1/3) + 2*c) + b*
sin(2*d*x^(1/3) + 2*c) + a) - 6*((I*a^2*b - a*b^2)*(d*x^(1/3) + c)^2 + (I*a*b^2 - b^3 + 2*(-I*a^2*b + a*b^2)*c
)*(d*x^(1/3) + c) + ((I*a^2*b + a*b^2)*(d*x^(1/3) + c)^2 + (I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*x^(1/3)
+ c))*cos(2*d*x^(1/3) + 2*c) - ((a^2*b - I*a*b^2)*(d*x^(1/3) + c)^2 + (a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)
*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*arctan2((2*a*b*cos(2*d*x^(1/3) + 2*c) - (a^2 - b^2)*sin(2*d*x^(1/3)
+ 2*c))/(a^2 + b^2), (2*a*b*sin(2*d*x^(1/3) + 2*c) + a^2 + b^2 + (a^2 - b^2)*cos(2*d*x^(1/3) + 2*c))/(a^2 + b^
2)) + ((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*(d*x^(1/3) + c)^3 - 3*(2*I*a*b^2 + 2*b^3 + (a^3 - 3*I*a^2*b - 3*a*b
^2 + I*b^3)*c)*(d*x^(1/3) + c)^2 - 12*(-I*a*b^2 - b^3)*(d*x^(1/3) + c)*c)*cos(2*d*x^(1/3) + 2*c) - 3*(I*a*b^2
- b^3 + 2*(I*a^2*b - a*b^2)*(d*x^(1/3) + c) + 2*(-I*a^2*b + a*b^2)*c + (I*a*b^2 + b^3 + 2*(I*a^2*b + a*b^2)*(d
*x^(1/3) + c) + 2*(-I*a^2*b - a*b^2)*c)*cos(2*d*x^(1/3) + 2*c) - (a*b^2 - I*b^3 + 2*(a^2*b - I*a*b^2)*(d*x^(1/
3) + c) - 2*(a^2*b - I*a*b^2)*c)*sin(2*d*x^(1/3) + 2*c))*dilog((I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b))
- 3*((a*b^2 - I*b^3)*c*cos(2*d*x^(1/3) + 2*c) + (I*a*b^2 + b^3)*c*sin(2*d*x^(1/3) + 2*c) + (a*b^2 + I*b^3)*c)
*log((a^2 + b^2)*cos(2*d*x^(1/3) + 2*c)^2 + 4*a*b*sin(2*d*x^(1/3) + 2*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^
2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^(1/3) + 2*c)) + 3*((a^2*b + I*a*b^2)*(d*x^(1/3) + c)^2 + (a*b^2 + I*b^
3 - 2*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c) + ((a^2*b - I*a*b^2)*(d*x^(1/3) + c)^2 + (a*b^2 - I*b^3 - 2*(a^2*b
- I*a*b^2)*c)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - ((-I*a^2*b - a*b^2)*(d*x^(1/3) + c)^2 + (-I*a*b^2 - b^
3 + 2*(I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*log(((a^2 + b^2)*cos(2*d*x^(1/3) + 2*c)^2
+ 4*a*b*sin(2*d*x^(1/3) + 2*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^(1
/3) + 2*c))/(a^2 + b^2)) + 3*(a^2*b + I*a*b^2 + (a^2*b - I*a*b^2)*cos(2*d*x^(1/3) + 2*c) - (-I*a^2*b - a*b^2)*
sin(2*d*x^(1/3) + 2*c))*polylog(3, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + ((I*a^3 + 3*a^2*b - 3*I*a
*b^2 - b^3)*(d*x^(1/3) + c)^3 + 3*(2*a*b^2 - 2*I*b^3 - (I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^
2 - 12*(a*b^2 - I*b^3)*(d*x^(1/3) + c)*c)*sin(2*d*x^(1/3) + 2*c))/(a^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a
*b^4 + I*b^5 + (a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 - I*b^5)*cos(2*d*x^(1/3) + 2*c) - (-I*a^5 - a^
4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*sin(2*d*x^(1/3) + 2*c)))/d^3

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (491) = 982\).
time = 0.43, size = 1187, normalized size = 1.95 \begin {gather*} -\frac {6 \, b^{3} d^{2} x^{\frac {2}{3}} - 2 \, {\left (a^{3} - a b^{2}\right )} d^{3} x + 2 \, {\left (a^{3} - a b^{2}\right )} d^{3} + 3 \, {\left (-2 i \, a^{2} b d x^{\frac {1}{3}} - i \, a b^{2} + {\left (-2 i \, a b^{2} d x^{\frac {1}{3}} - i \, b^{3}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) + 3 \, {\left (2 i \, a^{2} b d x^{\frac {1}{3}} + i \, a b^{2} + {\left (2 i \, a b^{2} d x^{\frac {1}{3}} + i \, b^{3}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - 6 \, {\left (a^{2} b d^{2} x^{\frac {2}{3}} - a^{2} b c^{2} + a b^{2} d x^{\frac {1}{3}} + a b^{2} c + {\left (a b^{2} d^{2} x^{\frac {2}{3}} - a b^{2} c^{2} + b^{3} d x^{\frac {1}{3}} + b^{3} c\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) - 6 \, {\left (a^{2} b d^{2} x^{\frac {2}{3}} - a^{2} b c^{2} + a b^{2} d x^{\frac {1}{3}} + a b^{2} c + {\left (a b^{2} d^{2} x^{\frac {2}{3}} - a b^{2} c^{2} + b^{3} d x^{\frac {1}{3}} + b^{3} c\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) - 6 \, {\left (a^{2} b c^{2} - a b^{2} c + {\left (a b^{2} c^{2} - b^{3} c\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{\tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 1}\right ) - 6 \, {\left (a^{2} b c^{2} - a b^{2} c + {\left (a b^{2} c^{2} - b^{3} c\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{\tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 1}\right ) - 3 \, {\left (a b^{2} \tan \left (d x^{\frac {1}{3}} + c\right ) + a^{2} b\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) - 3 \, {\left (a b^{2} \tan \left (d x^{\frac {1}{3}} + c\right ) + a^{2} b\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) - 2 \, {\left (3 \, a b^{2} d^{2} x^{\frac {2}{3}} + {\left (a^{2} b - b^{3}\right )} d^{3} x - {\left (a^{2} b - b^{3}\right )} d^{3}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{3} \tan \left (d x^{\frac {1}{3}} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="fricas")

[Out]

-1/2*(6*b^3*d^2*x^(2/3) - 2*(a^3 - a*b^2)*d^3*x + 2*(a^3 - a*b^2)*d^3 + 3*(-2*I*a^2*b*d*x^(1/3) - I*a*b^2 + (-
2*I*a*b^2*d*x^(1/3) - I*b^3)*tan(d*x^(1/3) + c))*dilog(2*((I*a*b - b^2)*tan(d*x^(1/3) + c)^2 - a^2 - I*a*b + (
I*a^2 - 2*a*b - I*b^2)*tan(d*x^(1/3) + c))/((a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 + b^2) + 1) + 3*(2*I*a^2*b*
d*x^(1/3) + I*a*b^2 + (2*I*a*b^2*d*x^(1/3) + I*b^3)*tan(d*x^(1/3) + c))*dilog(2*((-I*a*b - b^2)*tan(d*x^(1/3)
+ c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(d*x^(1/3) + c))/((a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 +
b^2) + 1) - 6*(a^2*b*d^2*x^(2/3) - a^2*b*c^2 + a*b^2*d*x^(1/3) + a*b^2*c + (a*b^2*d^2*x^(2/3) - a*b^2*c^2 + b^
3*d*x^(1/3) + b^3*c)*tan(d*x^(1/3) + c))*log(-2*((I*a*b - b^2)*tan(d*x^(1/3) + c)^2 - a^2 - I*a*b + (I*a^2 - 2
*a*b - I*b^2)*tan(d*x^(1/3) + c))/((a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 + b^2)) - 6*(a^2*b*d^2*x^(2/3) - a^2
*b*c^2 + a*b^2*d*x^(1/3) + a*b^2*c + (a*b^2*d^2*x^(2/3) - a*b^2*c^2 + b^3*d*x^(1/3) + b^3*c)*tan(d*x^(1/3) + c
))*log(-2*((-I*a*b - b^2)*tan(d*x^(1/3) + c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(d*x^(1/3) + c))/((
a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 + b^2)) - 6*(a^2*b*c^2 - a*b^2*c + (a*b^2*c^2 - b^3*c)*tan(d*x^(1/3) + c
))*log(((I*a*b + b^2)*tan(d*x^(1/3) + c)^2 - a^2 + I*a*b + (I*a^2 + I*b^2)*tan(d*x^(1/3) + c))/(tan(d*x^(1/3)
+ c)^2 + 1)) - 6*(a^2*b*c^2 - a*b^2*c + (a*b^2*c^2 - b^3*c)*tan(d*x^(1/3) + c))*log(((I*a*b - b^2)*tan(d*x^(1/
3) + c)^2 + a^2 + I*a*b + (I*a^2 + I*b^2)*tan(d*x^(1/3) + c))/(tan(d*x^(1/3) + c)^2 + 1)) - 3*(a*b^2*tan(d*x^(
1/3) + c) + a^2*b)*polylog(3, ((a^2 + 2*I*a*b - b^2)*tan(d*x^(1/3) + c)^2 - a^2 - 2*I*a*b + b^2 - 2*(-I*a^2 +
2*a*b + I*b^2)*tan(d*x^(1/3) + c))/((a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 + b^2)) - 3*(a*b^2*tan(d*x^(1/3) +
c) + a^2*b)*polylog(3, ((a^2 - 2*I*a*b - b^2)*tan(d*x^(1/3) + c)^2 - a^2 + 2*I*a*b + b^2 - 2*(I*a^2 + 2*a*b -
I*b^2)*tan(d*x^(1/3) + c))/((a^2 + b^2)*tan(d*x^(1/3) + c)^2 + a^2 + b^2)) - 2*(3*a*b^2*d^2*x^(2/3) + (a^2*b -
b^3)*d^3*x - (a^2*b - b^3)*d^3)*tan(d*x^(1/3) + c))/((a^4*b + 2*a^2*b^3 + b^5)*d^3*tan(d*x^(1/3) + c) + (a^5
+ 2*a^3*b^2 + a*b^4)*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (c + d \sqrt [3]{x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x**(1/3)))**2,x)

[Out]

Integral((a + b*tan(c + d*x**(1/3)))**(-2), 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(1/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="giac")

[Out]

integrate((b*tan(d*x^(1/3) + c) + a)^(-2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x^(1/3)))^2,x)

[Out]

int(1/(a + b*tan(c + d*x^(1/3)))^2, x)

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