Integrand size = 14, antiderivative size = 558 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}-b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2+b^2\right ) d}-\frac {2 a b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} \left (a^2+b^2\right ) d}+\frac {b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )} \]
(a^2-b^2)*x/(a^2+b^2)^2-2/3*a*b*ln(a*cos(d*x+c)^3+b*sin(d*x+c)^3)/(a^2+b^2 )^2/d+1/3*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)-b^2)*ln(a^(1/3)+b^(1/3)*tan(d*x+c ))/a^(1/3)/(a^2+b^2)^2/d+1/9*b^(1/3)*(a^(4/3)+2*b^(4/3))*ln(a^(1/3)+b^(1/3 )*tan(d*x+c))/a^(5/3)/(a^2+b^2)/d-1/6*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)-b^2)* ln(a^(2/3)-a^(1/3)*b^(1/3)*tan(d*x+c)+b^(2/3)*tan(d*x+c)^2)/a^(1/3)/(a^2+b ^2)^2/d-1/18*b^(1/3)*(a^(4/3)+2*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^(5/3)/(a^2+b^2)/d+1/3*b^(1/3)*(a^2-2*a^(2/3)* b^(4/3)-b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*tan(d*x+c))/a^(1/3)*3^(1/2))/a^ (1/3)/(a^2+b^2)^2/d*3^(1/2)+1/9*b^(1/3)*(a^(4/3)-2*b^(4/3))*arctan(1/3*(a^ (1/3)-2*b^(1/3)*tan(d*x+c))/a^(1/3)*3^(1/2))/a^(5/3)/(a^2+b^2)/d*3^(1/2)+1 /3*b*(a+tan(d*x+c)*(b-a*tan(d*x+c)))/a/(a^2+b^2)/d/(a+b*tan(d*x+c)^3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.36 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=-\frac {2 \sqrt [3]{a} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \left (a^2+b^2\right )^2 d}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2+b^2\right ) d}-\frac {i \log (i-\tan (c+d x))}{2 (a-i b)^2 d}+\frac {i \log (i+\tan (c+d x))}{2 (a+i b)^2 d}+\frac {2 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac {2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{a} b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}-\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac {2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}-\frac {(a-b) b (a+b) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \tan ^3(c+d x)}{a}\right ) \tan ^2(c+d x)}{2 a \left (a^2+b^2\right )^2 d}-\frac {b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \tan ^3(c+d x)}{a}\right ) \tan ^2(c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b}{3 \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )} \]
(-2*a^(1/3)*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^( 1/3))])/(Sqrt[3]*(a^2 + b^2)^2*d) - (2*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3) *Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 + b^2)*d) - ((I /2)*Log[I - Tan[c + d*x]])/((a - I*b)^2*d) + ((I/2)*Log[I + Tan[c + d*x]]) /((a + I*b)^2*d) + (2*a^(1/3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]]) /(3*(a^2 + b^2)^2*d) + (2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9* a^(5/3)*(a^2 + b^2)*d) - (a^(1/3)*b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Ta n[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(3*(a^2 + b^2)^2*d) - (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])/(9*a^(5/3 )*(a^2 + b^2)*d) - (2*a*b*Log[a + b*Tan[c + d*x]^3])/(3*(a^2 + b^2)^2*d) - ((a - b)*b*(a + b)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x]^3)/a) ]*Tan[c + d*x]^2)/(2*a*(a^2 + b^2)^2*d) - (b*Hypergeometric2F1[2/3, 2, 5/3 , -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(2*a*(a^2 + b^2)*d) + b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3)) + (b^2*Tan[c + d*x])/(3*a*(a^2 + b^2)*d* (a + b*Tan[c + d*x]^3))
Time = 0.86 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4144, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \tan (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {1}{\left (\tan ^2(c+d x)+1\right ) \left (b \tan ^3(c+d x)+a\right )^2}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {a^2+2 b \tan (c+d x) a-b^2}{\left (a^2+b^2\right )^2 \left (\tan ^2(c+d x)+1\right )}+\frac {b \left (-2 a b \tan ^2(c+d x)-\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{\left (a^2+b^2\right )^2 \left (b \tan ^3(c+d x)+a\right )}-\frac {b \left (b \tan ^2(c+d x)+a \tan (c+d x)-b\right )}{\left (a^2+b^2\right ) \left (b \tan ^3(c+d x)+a\right )^2}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \arctan (\tan (c+d x))}{\left (a^2+b^2\right )^2}+\frac {b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a \left (a^2+b^2\right ) \left (a+b \tan ^3(c+d x)\right )}-\frac {2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2}+\frac {a b \log \left (\tan ^2(c+d x)+1\right )}{\left (a^2+b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2+b^2\right )}+\frac {\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2-b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2+b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} \left (a^2+b^2\right )}-\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2+b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right )}+\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2}}{d}\) |
(((a^2 - b^2)*ArcTan[Tan[c + d*x]])/(a^2 + b^2)^2 + (b^(1/3)*(a^2 - 2*a^(2 /3)*b^(4/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1 /3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^2)^2) + (b^(1/3)*(a^(4/3) - 2*b^(4/3))*Ar cTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^( 5/3)*(a^2 + b^2)) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(1/3)*(a^2 + b^2)^2) + (b^(1/3)*(a^(4/3) + 2* b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9*a^(5/3)*(a^2 + b^2)) + (a *b*Log[1 + Tan[c + d*x]^2])/(a^2 + b^2)^2 - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4 /3) - b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d* x]^2])/(6*a^(1/3)*(a^2 + b^2)^2) - (b^(1/3)*(a^(4/3) + 2*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])/(18*a^(5/3)* (a^2 + b^2)) - (2*a*b*Log[a + b*Tan[c + d*x]^3])/(3*(a^2 + b^2)^2) + (b*(a + Tan[c + d*x]*(b - a*Tan[c + d*x])))/(3*a*(a^2 + b^2)*(a + b*Tan[c + d*x ]^3)))/d
3.4.79.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.39 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {-\frac {b \left (\frac {\left (\frac {a^{2}}{3}+\frac {b^{2}}{3}\right ) \tan \left (d x +c \right )^{2}-\frac {b \left (a^{2}+b^{2}\right ) \tan \left (d x +c \right )}{3 a}-\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \tan \left (d x +c \right )^{3}}+\frac {\frac {2 \left (-4 a^{2} b -b^{3}\right ) \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}-a \,b^{2}\right ) \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 a^{2} \ln \left (a +b \tan \left (d x +c \right )^{3}\right )}{3}}{a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{d}\) | \(411\) |
| default | \(\frac {-\frac {b \left (\frac {\left (\frac {a^{2}}{3}+\frac {b^{2}}{3}\right ) \tan \left (d x +c \right )^{2}-\frac {b \left (a^{2}+b^{2}\right ) \tan \left (d x +c \right )}{3 a}-\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \tan \left (d x +c \right )^{3}}+\frac {\frac {2 \left (-4 a^{2} b -b^{3}\right ) \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}-a \,b^{2}\right ) \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 a^{2} \ln \left (a +b \tan \left (d x +c \right )^{3}\right )}{3}}{a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{d}\) | \(411\) |
| risch | \(\text {Expression too large to display}\) | \(1178\) |
1/d*(-b/(a^4+2*a^2*b^2+b^4)*(((1/3*a^2+1/3*b^2)*tan(d*x+c)^2-1/3*b*(a^2+b^ 2)/a*tan(d*x+c)-1/3*a^2-1/3*b^2)/(a+b*tan(d*x+c)^3)+2/3/a*((-4*a^2*b-b^3)* (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)))+(2*a^3-a*b^2)*(-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)))+a^2*ln(a+b*tan(d*x+c)^3)))+1/(a^4+2*a^2*b^2+b ^4)*(a*b*ln(1+tan(d*x+c)^2)+(a^2-b^2)*arctan(tan(d*x+c))))
Result contains complex when optimal does not.
Time = 1.27 (sec) , antiderivative size = 11554, normalized size of antiderivative = 20.71 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\frac {\frac {9 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, \sqrt {3} {\left (2 \, a^{3} {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - 2 \, a^{2} b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {a}{b}\right )} - a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{3} \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 )}{{\left (a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {9 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (2 \, a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} + 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + b^{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^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4\right )} - 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - b^{3}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right )\right )}{a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {3 \, {\left (a b \tan \left (d x + c\right )^{2} - b^{2} \tan \left (d x + c\right ) - a b\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )^{3}}}{9 \, d} \]
1/9*(9*a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2*sqrt(3)*(2* a^3*((a/b)^(2/3) - 1) - 2*a^2*b*(2*(a/b)^(1/3) - a/b) - a*b^2*(a/b)^(2/3) - b^3*(a/b)^(1/3))*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*tan(d*x + c))/(a/b )^(1/3))/((a^5*(a/b)^(2/3) + 2*a^3*b^2*(a/b)^(2/3) + a*b^4*(a/b)^(2/3))*(a /b)^(1/3)) + 9*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (2*a^2*b*(3 *(a/b)^(2/3) + 2) + 2*a^3*(a/b)^(1/3) - a*b^2*(a/b)^(1/3) + b^3)*log(tan(d *x + c)^2 - (a/b)^(1/3)*tan(d*x + c) + (a/b)^(2/3))/(a^5*(a/b)^(2/3) + 2*a ^3*b^2*(a/b)^(2/3) + a*b^4*(a/b)^(2/3)) - 2*(a^2*b*(3*(a/b)^(2/3) - 4) - 2 *a^3*(a/b)^(1/3) + a*b^2*(a/b)^(1/3) - b^3)*log((a/b)^(1/3) + tan(d*x + c) )/(a^5*(a/b)^(2/3) + 2*a^3*b^2*(a/b)^(2/3) + a*b^4*(a/b)^(2/3)) - 3*(a*b*t an(d*x + c)^2 - b^2*tan(d*x + c) - a*b)/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*t an(d*x + c)^3))/d
Time = 0.58 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\frac {\frac {9 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {6 \, a b \log \left ({\left | b \tan \left (d x + c\right )^{3} + a \right |}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, a^{8} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a^{6} b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{7} b^{3} - 9 \, a^{5} b^{5} - 6 \, a^{3} b^{7} - a b^{9}\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^{11} b + 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}} + \frac {9 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \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 (2 \, a^{3} - a b^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} + {\left (4 \, a^{2} b^{2} + b^{4}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}}{\sqrt {3} a^{6} b + 2 \, \sqrt {3} a^{4} b^{3} + \sqrt {3} a^{2} b^{5}} - \frac {{\left ({\left (2 \, a^{3} - a b^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} - {\left (4 \, a^{2} b^{2} + b^{4}\right )} \left (-a b^{2}\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^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} + \frac {3 \, {\left (2 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} - a^{3} b \tan \left (d x + c\right )^{2} - a b^{3} \tan \left (d x + c\right )^{2} + a^{2} b^{2} \tan \left (d x + c\right ) + b^{4} \tan \left (d x + c\right ) + 3 \, a^{3} b + a b^{3}\right )}}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (b \tan \left (d x + c\right )^{3} + a\right )}}}{9 \, d} \]
1/9*(9*a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 6*a*b*log(abs (b*tan(d*x + c)^3 + a))/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*a^8*b^2*(-a/b)^(1/3 ) + 3*a^6*b^4*(-a/b)^(1/3) - a^2*b^8*(-a/b)^(1/3) - 4*a^7*b^3 - 9*a^5*b^5 - 6*a^3*b^7 - a*b^9)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + tan(d*x + c)))/( a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9) + 9*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 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)) )*((2*a^3 - a*b^2)*(-a*b^2)^(2/3) + (4*a^2*b^2 + b^4)*(-a*b^2)^(1/3))/(sqr t(3)*a^6*b + 2*sqrt(3)*a^4*b^3 + sqrt(3)*a^2*b^5) - ((2*a^3 - a*b^2)*(-a*b ^2)^(2/3) - (4*a^2*b^2 + b^4)*(-a*b^2)^(1/3))*log(tan(d*x + c)^2 + (-a/b)^ (1/3)*tan(d*x + c) + (-a/b)^(2/3))/(a^6*b + 2*a^4*b^3 + a^2*b^5) + 3*(2*a^ 2*b^2*tan(d*x + c)^3 - a^3*b*tan(d*x + c)^2 - a*b^3*tan(d*x + c)^2 + a^2*b ^2*tan(d*x + c) + b^4*tan(d*x + c) + 3*a^3*b + a*b^3)/((a^5 + 2*a^3*b^2 + a*b^4)*(b*tan(d*x + c)^3 + a)))/d
Time = 12.90 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (-\frac {\frac {16\,a^2\,b^4}{27}+\frac {8\,b^6}{27}}{a^7+2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3+729\,a^5\,b^4\,z^3+729\,a^9\,z^3+1458\,a^6\,b\,z^2+108\,a^3\,b^2\,z-64\,a^2\,b-8\,b^3,z,k\right )\,\left (\frac {\frac {32\,a\,b^7}{27}-\frac {128\,a^3\,b^5}{27}}{a^7+2\,a^5\,b^2+a^3\,b^4}-\mathrm {root}\left (1458\,a^7\,b^2\,z^3+729\,a^5\,b^4\,z^3+729\,a^9\,z^3+1458\,a^6\,b\,z^2+108\,a^3\,b^2\,z-64\,a^2\,b-8\,b^3,z,k\right )\,\left (\mathrm {root}\left (1458\,a^7\,b^2\,z^3+729\,a^5\,b^4\,z^3+729\,a^9\,z^3+1458\,a^6\,b\,z^2+108\,a^3\,b^2\,z-64\,a^2\,b-8\,b^3,z,k\right )\,\left (\frac {-27\,a^9\,b^3+34\,a^7\,b^5+77\,a^5\,b^7+16\,a^3\,b^9}{a^7+2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3+729\,a^5\,b^4\,z^3+729\,a^9\,z^3+1458\,a^6\,b\,z^2+108\,a^3\,b^2\,z-64\,a^2\,b-8\,b^3,z,k\right )\,\left (\frac {180\,a^{10}\,b^4+324\,a^8\,b^6+108\,a^6\,b^8-36\,a^4\,b^{10}}{a^7+2\,a^5\,b^2+a^3\,b^4}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-1458\,a^{11}\,b^3+1458\,a^9\,b^5+7290\,a^7\,b^7+4374\,a^5\,b^9\right )}{27\,\left (a^7+2\,a^5\,b^2+a^3\,b^4\right )}\right )-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2484\,a^8\,b^4-1836\,a^6\,b^6+864\,a^4\,b^8+216\,a^2\,b^{10}\right )}{27\,\left (a^7+2\,a^5\,b^2+a^3\,b^4\right )}\right )-\frac {\frac {388\,a^6\,b^4}{9}+\frac {353\,a^4\,b^6}{9}+\frac {64\,a^2\,b^8}{9}}{a^7+2\,a^5\,b^2+a^3\,b^4}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (447\,a^5\,b^5+408\,a^3\,b^7+96\,a\,b^9\right )}{27\,\left (a^7+2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (236\,a^4\,b^4+134\,a^2\,b^6-16\,b^8\right )}{27\,\left (a^7+2\,a^5\,b^2+a^3\,b^4\right )}\right )-\frac {8\,a\,b^5\,\mathrm {tan}\left (c+d\,x\right )}{9\,\left (a^7+2\,a^5\,b^2+a^3\,b^4\right )}\right )\,\mathrm {root}\left (1458\,a^7\,b^2\,z^3+729\,a^5\,b^4\,z^3+729\,a^9\,z^3+1458\,a^6\,b\,z^2+108\,a^3\,b^2\,z-64\,a^2\,b-8\,b^3,z,k\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}+\frac {\frac {b}{3\,\left (a^2+b^2\right )}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3\,\left (a^2+b^2\right )}+\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{3\,a\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3+a\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]
symsum(log(root(1458*a^7*b^2*z^3 + 729*a^5*b^4*z^3 + 729*a^9*z^3 + 1458*a^ 6*b*z^2 + 108*a^3*b^2*z - 64*a^2*b - 8*b^3, z, k)*(((32*a*b^7)/27 - (128*a ^3*b^5)/27)/(a^7 + a^3*b^4 + 2*a^5*b^2) - root(1458*a^7*b^2*z^3 + 729*a^5* b^4*z^3 + 729*a^9*z^3 + 1458*a^6*b*z^2 + 108*a^3*b^2*z - 64*a^2*b - 8*b^3, z, k)*(root(1458*a^7*b^2*z^3 + 729*a^5*b^4*z^3 + 729*a^9*z^3 + 1458*a^6*b *z^2 + 108*a^3*b^2*z - 64*a^2*b - 8*b^3, z, k)*((16*a^3*b^9 + 77*a^5*b^7 + 34*a^7*b^5 - 27*a^9*b^3)/(a^7 + a^3*b^4 + 2*a^5*b^2) + root(1458*a^7*b^2* z^3 + 729*a^5*b^4*z^3 + 729*a^9*z^3 + 1458*a^6*b*z^2 + 108*a^3*b^2*z - 64* a^2*b - 8*b^3, z, k)*((108*a^6*b^8 - 36*a^4*b^10 + 324*a^8*b^6 + 180*a^10* b^4)/(a^7 + a^3*b^4 + 2*a^5*b^2) - (tan(c + d*x)*(4374*a^5*b^9 + 7290*a^7* b^7 + 1458*a^9*b^5 - 1458*a^11*b^3))/(27*(a^7 + a^3*b^4 + 2*a^5*b^2))) - ( tan(c + d*x)*(216*a^2*b^10 + 864*a^4*b^8 - 1836*a^6*b^6 - 2484*a^8*b^4))/( 27*(a^7 + a^3*b^4 + 2*a^5*b^2))) - ((64*a^2*b^8)/9 + (353*a^4*b^6)/9 + (38 8*a^6*b^4)/9)/(a^7 + a^3*b^4 + 2*a^5*b^2) + (tan(c + d*x)*(96*a*b^9 + 408* a^3*b^7 + 447*a^5*b^5))/(27*(a^7 + a^3*b^4 + 2*a^5*b^2))) + (tan(c + d*x)* (134*a^2*b^6 - 16*b^8 + 236*a^4*b^4))/(27*(a^7 + a^3*b^4 + 2*a^5*b^2))) - ((8*b^6)/27 + (16*a^2*b^4)/27)/(a^7 + a^3*b^4 + 2*a^5*b^2) - (8*a*b^5*tan( c + d*x))/(9*(a^7 + a^3*b^4 + 2*a^5*b^2)))*root(1458*a^7*b^2*z^3 + 729*a^5 *b^4*z^3 + 729*a^9*z^3 + 1458*a^6*b*z^2 + 108*a^3*b^2*z - 64*a^2*b - 8*b^3 , z, k), k, 1, 3)/d + (log(tan(c + d*x) - 1i)*1i)/(2*d*(a*b*2i - a^2 + ...