3.4.78 \(\int \frac {1}{a+b \tan ^3(c+d x)} \, dx\) [378]

Optimal. Leaf size=256 \[ \frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3742, 6857, 649, 209, 266, 1885, 1874, 31, 648, 631, 210, 642} \begin {gather*} -\frac {b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d \left (a^2+b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} d \left (a^2+b^2\right )}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} d \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^3)^(-1),x]

[Out]

(a*x)/(a^2 + b^2) + (b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])
/(Sqrt[3]*a^(2/3)*(a^2 + b^2)*d) - (b*Log[a*Cos[c + d*x]^3 + b*Sin[c + d*x]^3])/(3*(a^2 + b^2)*d) + (b^(1/3)*(
a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(2/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^(4/3) + b^(4/3
))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(6*a^(2/3)*(a^2 + 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 209

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

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 266

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

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 649

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

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{a+b \tan ^3(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^3\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+b x}{\left (a^2+b^2\right ) \left (1+x^2\right )}-\frac {b \left (-b+a x+b x^2\right )}{\left (a^2+b^2\right ) \left (a+b x^3\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {-b+a x+b x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {-b+a x}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b^2 \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (a^{4/3}-2 b^{4/3}\right )+\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}+\frac {\left (b^{2/3} \left (a^{4/3}+b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}\\ &=\frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {\left (b^{2/3} \left (a^{4/3}-b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [3]{a} \left (a^2+b^2\right ) d}-\frac {\left (\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d}\\ &=\frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {\left (\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} \left (a^2+b^2\right ) d}\\ &=\frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) 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 = 0.41, size = 278, normalized size = 1.09 \begin {gather*} \frac {-2 \sqrt {3} b^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-3 i a^{5/3} \log (i-\tan (c+d x))+3 a^{2/3} b \log (i-\tan (c+d x))+3 i a^{5/3} \log (i+\tan (c+d x))+3 a^{2/3} b \log (i+\tan (c+d x))+2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )-b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )-2 a^{2/3} b \log \left (a+b \tan ^3(c+d x)\right )-3 a^{2/3} b \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b \tan ^3(c+d x)}{a}\right ) \tan ^2(c+d x)}{6 a^{2/3} \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] - (3*I)*a^(5/3)*Log[I - Tan[c
+ d*x]] + 3*a^(2/3)*b*Log[I - Tan[c + d*x]] + (3*I)*a^(5/3)*Log[I + Tan[c + d*x]] + 3*a^(2/3)*b*Log[I + Tan[c
+ d*x]] + 2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]] - b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x]
+ b^(2/3)*Tan[c + d*x]^2] - 2*a^(2/3)*b*Log[a + b*Tan[c + d*x]^3] - 3*a^(2/3)*b*Hypergeometric2F1[2/3, 1, 5/3,
-((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(6*a^(2/3)*(a^2 + b^2)*d)

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Maple [A]
time = 0.37, size = 293, normalized size = 1.14

method	result	size
risch	\(\frac {x}{i b +a}+\frac {2 i a^{2} b \,d^{3} x}{a^{4} d^{3}+a^{2} b^{2} d^{3}}+\frac {2 i a^{2} b \,d^{2} c}{a^{4} d^{3}+a^{2} b^{2} d^{3}}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (27 a^{4} d^{3}+27 a^{2} b^{2} d^{3}\right ) \textit {\_Z}^{3}+27 \textit {\_Z}^{2} a^{2} b \,d^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {18 d^{2} a^{4}}{a^{2}-b^{2}}-\frac {18 d^{2} b^{2} a^{2}}{a^{2}-b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {6 i d \,a^{3}}{a^{2}-b^{2}}-\frac {6 i d \,b^{2} a}{a^{2}-b^{2}}-\frac {6 d b \,a^{2}}{a^{2}-b^{2}}\right ) \textit {\_R} +\frac {a^{2}}{a^{2}-b^{2}}+\frac {b^{2}}{a^{2}-b^{2}}\right )\right )\)	\(265\)
derivativedivides	\(\frac {\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (-b \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\ln \left (a +b \left (\tan ^{3}\left (d x +c \right )\right )\right )}{3}\right ) b}{a^{2}+b^{2}}}{d}\)	\(293\)
default	\(\frac {\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (-b \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\ln \left (a +b \left (\tan ^{3}\left (d x +c \right )\right )\right )}{3}\right ) b}{a^{2}+b^{2}}}{d}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 291, normalized size = 1.14 \begin {gather*} -\frac {\frac {2 \, \sqrt {3} {\left (a {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2\right )} - b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, a}{b}\right )}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \tan \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {18 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {3 \, {\left (b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\tan \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {6 \, {\left (b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right )\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{18 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*sqrt(3)*(a*(3*(a/b)^(2/3) - 2) - b*(3*(a/b)^(1/3) - 2*a/b))*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*tan(
d*x + c))/(a/b)^(1/3))/((a^2*(a/b)^(2/3) + b^2*(a/b)^(2/3))*(a/b)^(1/3)) - 18*(d*x + c)*a/(a^2 + b^2) + 3*(b*(
2*(a/b)^(2/3) + 1) + a*(a/b)^(1/3))*log(tan(d*x + c)^2 - (a/b)^(1/3)*tan(d*x + c) + (a/b)^(2/3))/(a^2*(a/b)^(2
/3) + b^2*(a/b)^(2/3)) - 9*b*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 6*(b*((a/b)^(2/3) - 1) - a*(a/b)^(1/3))*log
((a/b)^(1/3) + tan(d*x + c))/(a^2*(a/b)^(2/3) + b^2*(a/b)^(2/3)))/d

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Fricas [C] Result contains complex when optimal does not.
time = 7.75, size = 4817, normalized size = 18.82 \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(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/24*(2*(a^2 + b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2
- b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 +
a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*
d*log(-1/4*(4*b^2*tan(d*x + c)^2 - ((a^4 + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + a^2*b^2)*d^2)*((1/2)^(1/3)*(I*
sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/
3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^
3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2 + 2*(a^2*b*d*tan(d*x + c)^2 - a^2*b
*d + 2*(a^3 - a*b^2)*d*tan(d*x + c))*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d +
b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^
2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(
a^2*d + b^2*d)) - 4*a^2)/(tan(d*x + c)^2 + 1)) - 24*a*d*x - ((a^2 + b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*
d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^
2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a
^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 3*sqrt(1/3)*(a^2 + b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^
4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b
^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b
^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*b +
b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2
+ b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) -
2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/((a
^4 + 2*a^2*b^2 + b^4)*d^2)) - 6*b)*log(1/4*(8*a^4 - 16*a^2*b^2 - ((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)
^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d +
b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)
^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/
(a^2*d + b^2*d))^2 + 8*(2*a^2*b^2 - b^4)*tan(d*x + c)^2 + 2*((a^4*b + a^2*b^3)*d*tan(d*x + c)^2 + 2*(a^5 - a*b
^4)*d*tan(d*x + c) - (a^4*b + a^2*b^3)*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2
*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^
2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) +
2*b/(a^2*d + b^2*d)) + 3*sqrt(1/3)*(4*(a^4*b + a^2*b^3)*d*tan(d*x + c)^2 - 4*(a^5 - a*b^4)*d*tan(d*x + c) - ((
a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*
(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^
(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^
2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) - 4*(a^4*b + a^2*b^3)*d)*sqrt(-((a^4 + 2*a^2*b^2 +
b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2
+ b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) -
2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*
b + b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a
^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3)
- 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/
((a^4 + 2*a^2*b^2 + b^4)*d^2)) - 24*(a^3*b - a*b^3)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) - ((a^2 + b^2)*((1/2)^
(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*
d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d
+ b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d + 3*sqrt(1/3)*(a^2 + b^2)*
d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*
d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b
/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.68, size = 334, normalized size = 1.30 \begin {gather*} \frac {\frac {2 \, {\left (a^{3} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} - b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right ) \right |}\right )}{a^{5} b + 2 \, a^{3} b^{3} + a b^{5}} + \frac {6 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \tan \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )}}{\sqrt {3} a^{3} b + \sqrt {3} a b^{3}} + \frac {6 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\tan \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{3} b + a b^{3}} + \frac {3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b \log \left ({\left | b \tan \left (d x + c\right )^{3} + a \right |}\right )}{a^{2} + b^{2}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*(a^3*b^2*(-a/b)^(1/3) + a*b^4*(-a/b)^(1/3) - a^2*b^3 - b^5)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + tan(d*
x + c)))/(a^5*b + 2*a^3*b^3 + a*b^5) + 6*(pi*floor((d*x + c)/pi + 1/2)*sgn((-a/b)^(1/3)) + arctan(1/3*sqrt(3)*
((-a/b)^(1/3) + 2*tan(d*x + c))/(-a/b)^(1/3)))*((-a*b^2)^(1/3)*b^2 + (-a*b^2)^(2/3)*a)/(sqrt(3)*a^3*b + sqrt(3
)*a*b^3) + 6*(d*x + c)*a/(a^2 + b^2) + ((-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*log(tan(d*x + c)^2 + (-a/b)^(1/
3)*tan(d*x + c) + (-a/b)^(2/3))/(a^3*b + a*b^3) + 3*b*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 2*b*log(abs(b*tan(
d*x + c)^3 + a))/(a^2 + b^2))/d

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Mupad [B]
time = 12.65, size = 342, normalized size = 1.34 \begin {gather*} \frac {\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (12\,b^6-69\,a^2\,b^4\right )+\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (36\,a\,b^6-180\,a^3\,b^4+\mathrm {tan}\left (c+d\,x\right )\,\left (162\,a^2\,b^5-54\,a^4\,b^3\right )\right )-36\,a\,b^5+27\,a^3\,b^3\right )+13\,a\,b^4-16\,b^5\,\mathrm {tan}\left (c+d\,x\right )\right )+5\,b^4\,\mathrm {tan}\left (c+d\,x\right )\right )\right )\,\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x)^3),x)

[Out]

symsum(log(root(27*a^2*b^2*z^3 + 27*a^4*z^3 + 27*a^2*b*z^2 - b, z, k)*(root(27*a^2*b^2*z^3 + 27*a^4*z^3 + 27*a
^2*b*z^2 - b, z, k)*(root(27*a^2*b^2*z^3 + 27*a^4*z^3 + 27*a^2*b*z^2 - b, z, k)*(tan(c + d*x)*(12*b^6 - 69*a^2
*b^4) + root(27*a^2*b^2*z^3 + 27*a^4*z^3 + 27*a^2*b*z^2 - b, z, k)*(36*a*b^6 - 180*a^3*b^4 + tan(c + d*x)*(162
*a^2*b^5 - 54*a^4*b^3)) - 36*a*b^5 + 27*a^3*b^3) + 13*a*b^4 - 16*b^5*tan(c + d*x)) + 5*b^4*tan(c + d*x)))*root
(27*a^2*b^2*z^3 + 27*a^4*z^3 + 27*a^2*b*z^2 - b, z, k), k, 1, 3)/d + log(tan(c + d*x) - 1i)/(2*d*(a*1i + b)) +
(log(tan(c + d*x) + 1i)*1i)/(2*d*(a + b*1i))

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