Integrand size = 26, antiderivative size = 282 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {x}{8 \sqrt [3]{2} a^{4/3}}+\frac {i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d} \] Output:
-1/16*x*2^(2/3)/a^(4/3)+1/8*I*3^(1/2)*arctan(1/3*(a^(1/3)+2^(2/3)*(a+I*a*t an(d*x+c))^(1/3))*3^(1/2)/a^(1/3))*2^(2/3)/a^(4/3)/d+1/16*I*ln(cos(d*x+c)) *2^(2/3)/a^(4/3)/d+3/16*I*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^( 2/3)/a^(4/3)/d-39/40*I*tan(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(4/3)+3/5*tan(d*x +c)^3/d/(a+I*a*tan(d*x+c))^(4/3)-51/10*I/a/d/(a+I*a*tan(d*x+c))^(1/3)-87/4 0*I*(a+I*a*tan(d*x+c))^(2/3)/a^2/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.40 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {15 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},\frac {1}{2} (1+i \tan (c+d x))\right ) (-i+\tan (c+d x))^2+6 \left (-97-126 i \tan (c+d x)+16 \tan ^2(c+d x)-8 i \tan ^3(c+d x)\right )}{80 a d (-i+\tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)}} \] Input:
Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(4/3),x]
Output:
(15*Hypergeometric2F1[2/3, 1, 5/3, (1 + I*Tan[c + d*x])/2]*(-I + Tan[c + d *x])^2 + 6*(-97 - (126*I)*Tan[c + d*x] + 16*Tan[c + d*x]^2 - (8*I)*Tan[c + d*x]^3))/(80*a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))
Time = 1.01 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.88, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 4043, 27, 3042, 4078, 27, 3042, 4075, 3042, 4009, 3042, 3962, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^4}{(a+i a \tan (c+d x))^{4/3}}dx\) |
| \(\Big \downarrow \) 4043 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {3 \int \frac {\tan ^2(c+d x) (9 a-4 i a \tan (c+d x))}{3 (i \tan (c+d x) a+a)^{4/3}}dx}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {\tan ^2(c+d x) (9 a-4 i a \tan (c+d x))}{(i \tan (c+d x) a+a)^{4/3}}dx}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {\tan (c+d x)^2 (9 a-4 i a \tan (c+d x))}{(i \tan (c+d x) a+a)^{4/3}}dx}{5 a}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {3 \int \frac {2 \tan (c+d x) \left (29 \tan (c+d x) a^2+39 i a^2\right )}{3 \sqrt [3]{i \tan (c+d x) a+a}}dx}{8 a^2}}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {\tan (c+d x) \left (29 \tan (c+d x) a^2+39 i a^2\right )}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {\tan (c+d x) \left (29 \tan (c+d x) a^2+39 i a^2\right )}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {39 i a^2 \tan (c+d x)-29 a^2}{\sqrt [3]{i \tan (c+d x) a+a}}dx-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\int \frac {39 i a^2 \tan (c+d x)-29 a^2}{\sqrt [3]{i \tan (c+d x) a+a}}dx-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 4009 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {5 a \int (i \tan (c+d x) a+a)^{2/3}dx-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {5 a \int (i \tan (c+d x) a+a)^{2/3}dx-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {-\frac {5 i a^2 \int \frac {1}{(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{d}-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {-\frac {5 i a^2 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {-\frac {5 i a^2 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {-\frac {5 i a^2 \left (\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {\frac {39 i a \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {-\frac {5 i a^2 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}-\frac {102 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i a (a+i a \tan (c+d x))^{2/3}}{2 d}}{4 a^2}}{5 a}\) |
Input:
Int[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(4/3),x]
Output:
(3*Tan[c + d*x]^3)/(5*d*(a + I*a*Tan[c + d*x])^(4/3)) - ((((39*I)/8)*a*Tan [c + d*x]^2)/(d*(a + I*a*Tan[c + d*x])^(4/3)) - (((-5*I)*a^2*(((-I)*Sqrt[3 ]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(1/3)*a^(1/3)) - (3*Log[2^(1/3)*a^ (1/3) - I*a*Tan[c + d*x]])/(2*2^(1/3)*a^(1/3)) + Log[a - I*a*Tan[c + d*x]] /(2*2^(1/3)*a^(1/3))))/d - ((102*I)*a^2)/(d*(a + I*a*Tan[c + d*x])^(1/3)) - (((87*I)/2)*a*(a + I*a*Tan[c + d*x])^(2/3))/d)/(4*a^2))/(5*a)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d S ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b , c, d, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a *f*m)), x] + Simp[(b*c + a*d)/(2*a*b) Int[(a + b*Tan[e + f*x])^(m + 1), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 , 0] && LtQ[m, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[1/(a*(m + n - 1)) Int[(a + b *Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)*Simp[d*(b*c*m + a*d*(-1 + n)) - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B *d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-(A*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f* x])^(n - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a *A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
Time = 0.97 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {3 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\frac {7 a^{2}}{4 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {a^{3}}{8 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {a^{2} \left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right )}{4}\right )}{d \,a^{3}}\) | \(212\) |
| default | \(\frac {3 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\frac {7 a^{2}}{4 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {a^{3}}{8 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {a^{2} \left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right )}{4}\right )}{d \,a^{3}}\) | \(212\) |
Input:
int(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(4/3),x,method=_RETURNVERBOSE)
Output:
3*I/d/a^3*(1/5*(a+I*a*tan(d*x+c))^(5/3)-a*(a+I*a*tan(d*x+c))^(2/3)-7/4*a^2 /(a+I*a*tan(d*x+c))^(1/3)+1/8*a^3/(a+I*a*tan(d*x+c))^(4/3)+1/4*a^2*(1/6*2^ (2/3)/a^(1/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/12*2^(2/3)/a^ (1/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d*x+c))^(1/3) +2^(2/3)*a^(2/3))+1/6*3^(1/2)*2^(2/3)/a^(1/3)*arctan(1/3*3^(1/2)*(2^(2/3)/ a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (201) = 402\).
Time = 0.09 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.57 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {3 \cdot 2^{\frac {2}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 425 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 125 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i\right )} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} - 160 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (32 \, a^{3} d^{2} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) + 80 \, {\left ({\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) + 80 \, {\left ({\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (-i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right )}{160 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \] Input:
integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")
Output:
-1/160*(3*2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(231*I*e^(6*I*d*x + 6*I*c) + 425*I*e^(4*I*d*x + 4*I*c) + 125*I*e^(2*I*d*x + 2*I*c) - 5*I)*e^(4 /3*I*d*x + 4/3*I*c) - 160*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*(-1/128*I/(a^4*d^3))^(1/3)*log(32*a^3*d^2*(-1/128*I/(a^4*d^3))^(2/ 3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 80*((-I*sqrt(3)*a^2*d + a^2*d)*e^(6*I*d*x + 6*I*c) + (-I*sqrt(3)*a^2*d + a^2*d)*e^(4*I*d*x + 4*I*c))*(-1/128*I/(a^4*d^3))^(1/3)*log(-16*(I*sqrt(3) *a^3*d^2 + a^3*d^2)*(-1/128*I/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 80*((I*sqrt(3)*a^2*d + a^2*d )*e^(6*I*d*x + 6*I*c) + (I*sqrt(3)*a^2*d + a^2*d)*e^(4*I*d*x + 4*I*c))*(-1 /128*I/(a^4*d^3))^(1/3)*log(-16*(-I*sqrt(3)*a^3*d^2 + a^3*d^2)*(-1/128*I/( a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))
\[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\tan ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \] Input:
integrate(tan(d*x+c)**4/(a+I*a*tan(d*x+c))**(4/3),x)
Output:
Integral(tan(c + d*x)**4/(I*a*(tan(c + d*x) - I))**(4/3), x)
Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.74 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {i \, {\left (10 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {11}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 5 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 10 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) + 48 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}} a^{2} - 240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} a^{3} - \frac {30 \, {\left (14 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} - a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\right )}}{80 \, a^{5} d} \] Input:
integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")
Output:
1/80*I*(10*sqrt(3)*2^(2/3)*a^(11/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^ (1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3)) - 5*2^(2/3)*a^(11/3)*log( 2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan( d*x + c) + a)^(2/3)) + 10*2^(2/3)*a^(11/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan (d*x + c) + a)^(1/3)) + 48*(I*a*tan(d*x + c) + a)^(5/3)*a^2 - 240*(I*a*tan (d*x + c) + a)^(2/3)*a^3 - 30*(14*(I*a*tan(d*x + c) + a)*a^4 - a^5)/(I*a*t an(d*x + c) + a)^(4/3))/(a^5*d)
Time = 0.48 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.73 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {i \, {\left (10 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {8}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 5 \cdot 2^{\frac {2}{3}} a^{\frac {8}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 10 \cdot 2^{\frac {2}{3}} a^{\frac {8}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) + 48 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}} a - 240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} a^{2} + \frac {30 \, {\left (14 \, {\left (-i \, a \tan \left (d x + c\right ) - a\right )} a^{3} + a^{4}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\right )}}{80 \, a^{4} d} \] Input:
integrate(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")
Output:
1/80*I*(10*sqrt(3)*2^(2/3)*a^(8/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^( 1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3)) - 5*2^(2/3)*a^(8/3)*log(2^ (2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d* x + c) + a)^(2/3)) + 10*2^(2/3)*a^(8/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d* x + c) + a)^(1/3)) + 48*(I*a*tan(d*x + c) + a)^(5/3)*a - 240*(I*a*tan(d*x + c) + a)^(2/3)*a^2 + 30*(14*(-I*a*tan(d*x + c) - a)*a^3 + a^4)/(I*a*tan(d *x + c) + a)^(4/3))/(a^4*d)
Time = 1.80 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.93 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\frac {3{}\mathrm {i}}{8\,d}-\frac {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,21{}\mathrm {i}}{4\,a\,d}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{2/3}\,3{}\mathrm {i}}{a^2\,d}+\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/3}\,3{}\mathrm {i}}{5\,a^3\,d}-\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (9\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}\right )}{a^{4/3}\,d}+\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (-\frac {9\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{16\,a^2\,d^2}-\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{32\,a^{5/3}\,d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}\,d}-\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (-\frac {9\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{16\,a^2\,d^2}+\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{32\,a^{5/3}\,d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}\,d} \] Input:
int(tan(c + d*x)^4/(a + a*tan(c + d*x)*1i)^(4/3),x)
Output:
(3i/(8*d) - ((a + a*tan(c + d*x)*1i)*21i)/(4*a*d))/(a + a*tan(c + d*x)*1i) ^(4/3) - ((a + a*tan(c + d*x)*1i)^(2/3)*3i)/(a^2*d) + ((a + a*tan(c + d*x) *1i)^(5/3)*3i)/(5*a^3*d) - ((1i/128)^(1/3)*log(9*(a*(tan(c + d*x)*1i + 1)) ^(1/3) + 9*(-1)^(1/3)*2^(1/3)*a^(1/3)))/(a^(4/3)*d) + ((1i/128)^(1/3)*log( - (9*(a + a*tan(c + d*x)*1i)^(1/3))/(16*a^2*d^2) - (9*(-1)^(1/3)*2^(1/3)*( 3^(1/2)*1i - 1))/(32*a^(5/3)*d^2))*((3^(1/2)*1i)/2 + 1/2))/(a^(4/3)*d) - ( (1i/128)^(1/3)*log((9*(-1)^(1/3)*2^(1/3)*(3^(1/2)*1i + 1))/(32*a^(5/3)*d^2 ) - (9*(a + a*tan(c + d*x)*1i)^(1/3))/(16*a^2*d^2))*((3^(1/2)*1i)/2 - 1/2) )/(a^(4/3)*d)
\[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\int \frac {\tan \left (d x +c \right )^{4}}{\left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}} \tan \left (d x +c \right ) i +\left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}}}d x}{a^{\frac {4}{3}}} \] Input:
int(tan(d*x+c)^4/(a+I*a*tan(d*x+c))^(4/3),x)
Output:
int(tan(c + d*x)**4/((tan(c + d*x)*i + 1)**(1/3)*tan(c + d*x)*i + (tan(c + d*x)*i + 1)**(1/3)),x)/(a**(1/3)*a)