Integrand size = 28, antiderivative size = 107 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {8 \sqrt [4]{-1} a^3 \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {16 a^3 \sqrt {d \tan (e+f x)}}{3 d f}-\frac {2 \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}{3 d f} \] Output:
-8*(-1)^(1/4)*a^3*arctan((-1)^(3/4)*(d*tan(f*x+e))^(1/2)/d^(1/2))/d^(1/2)/ f-16/3*a^3*(d*tan(f*x+e))^(1/2)/d/f-2/3*(d*tan(f*x+e))^(1/2)*(a^3+I*a^3*ta n(f*x+e))/d/f
Time = 0.77 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 i a^3 \left (-12 (-1)^{3/4} \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+\sqrt {d \tan (e+f x)} (-9 i+\tan (e+f x))\right )}{3 d f} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]
Output:
(((-2*I)/3)*a^3*(-12*(-1)^(3/4)*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] + Sqrt[d*Tan[e + f*x]]*(-9*I + Tan[e + f*x])))/(d*f)
Time = 0.54 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4039, 27, 3042, 4075, 3042, 4016, 218}
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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4039 |
| \(\displaystyle \frac {2 a \int \frac {2 (i \tan (e+f x) a+a) (a d+2 i a \tan (e+f x) d)}{\sqrt {d \tan (e+f x)}}dx}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 a \int \frac {(i \tan (e+f x) a+a) (a d+2 i a \tan (e+f x) d)}{\sqrt {d \tan (e+f x)}}dx}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \int \frac {(i \tan (e+f x) a+a) (a d+2 i a \tan (e+f x) d)}{\sqrt {d \tan (e+f x)}}dx}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \frac {4 a \left (-\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\int \frac {3 d a^2+3 i d \tan (e+f x) a^2}{\sqrt {d \tan (e+f x)}}dx\right )}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \left (-\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\int \frac {3 d a^2+3 i d \tan (e+f x) a^2}{\sqrt {d \tan (e+f x)}}dx\right )}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle \frac {4 a \left (-\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {18 a^4 d^2 \int \frac {1}{3 a^2 d^2-3 i a^2 d^2 \tan (e+f x)}d\sqrt {d \tan (e+f x)}}{f}\right )}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4 a \left (-\frac {6 \sqrt [4]{-1} a^2 \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}\right )}{3 d}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]
Output:
(-2*Sqrt[d*Tan[e + f*x]]*(a^3 + I*a^3*Tan[e + f*x]))/(3*d*f) + (4*a*((-6*( -1)^(1/4)*a^2*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (4*a^2*Sqrt[d*Tan[e + f*x]])/f))/(3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[a/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x ] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (88 ) = 176\).
Time = 1.76 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.91
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-3 d \sqrt {d \tan \left (f x +e \right )}+4 d^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(311\) |
| default | \(\frac {2 a^{3} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-3 d \sqrt {d \tan \left (f x +e \right )}+4 d^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(311\) |
| parts | \(\frac {a^{3} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f d}-\frac {2 i a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,d^{2}}+\frac {3 i a^{3} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f \left (d^{2}\right )^{\frac {1}{4}}}-\frac {6 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{f d}\) | \(592\) |
Input:
int((a+I*a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/f*a^3/d^2*(-1/3*I*(d*tan(f*x+e))^(3/2)-3*d*(d*tan(f*x+e))^(1/2)+4*d^2*(1 /8/d*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2 )*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1 /2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*a rctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))+1/8*I/(d^2)^(1/4)*2^(1 /2)*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2) )/(d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*a rctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^ (1/4)*(d*tan(f*x+e))^(1/2)+1))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (87) = 174\).
Time = 0.08 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.03 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {3 \, \sqrt {-\frac {64 i \, a^{6}}{d f^{2}}} {\left (d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \log \left (\frac {{\left (-8 i \, a^{3} d e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt {-\frac {64 i \, a^{6}}{d f^{2}}} {\left (d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) - 3 \, \sqrt {-\frac {64 i \, a^{6}}{d f^{2}}} {\left (d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \log \left (\frac {{\left (-8 i \, a^{3} d e^{\left (2 i \, f x + 2 i \, e\right )} - \sqrt {-\frac {64 i \, a^{6}}{d f^{2}}} {\left (d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) - 16 \, {\left (5 \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, a^{3}\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )}} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
1/12*(3*sqrt(-64*I*a^6/(d*f^2))*(d*f*e^(2*I*f*x + 2*I*e) + d*f)*log(1/4*(- 8*I*a^3*d*e^(2*I*f*x + 2*I*e) + sqrt(-64*I*a^6/(d*f^2))*(d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/a^3) - 3*sqrt(-64*I*a^6/(d*f^2))*(d*f*e^(2*I*f* x + 2*I*e) + d*f)*log(1/4*(-8*I*a^3*d*e^(2*I*f*x + 2*I*e) - sqrt(-64*I*a^6 /(d*f^2))*(d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/a^3) - 16*(5*a^3*e^ (2*I*f*x + 2*I*e) + 4*a^3)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f *x + 2*I*e) + 1)))/(d*f*e^(2*I*f*x + 2*I*e) + d*f)
\[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=- i a^{3} \left (\int \frac {i}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\right )\, dx\right ) \] Input:
integrate((a+I*a*tan(f*x+e))**3/(d*tan(f*x+e))**(1/2),x)
Output:
-I*a**3*(Integral(I/sqrt(d*tan(e + f*x)), x) + Integral(-3*tan(e + f*x)/sq rt(d*tan(e + f*x)), x) + Integral(tan(e + f*x)**3/sqrt(d*tan(e + f*x)), x) + Integral(-3*I*tan(e + f*x)**2/sqrt(d*tan(e + f*x)), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (87) = 174\).
Time = 0.12 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.87 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {3 \, a^{3} d {\left (\frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + \frac {2 \, {\left (-i \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} - 9 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d\right )}}{d}}{3 \, d f} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
1/3*(3*a^3*d*((2*I + 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sq rt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) + (2*I + 2)*sqrt(2)*arctan(-1/2*sqrt( 2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - (I - 1)*s qrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt (d) + (I - 1)*sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sq rt(d) + d)/sqrt(d)) + 2*(-I*(d*tan(f*x + e))^(3/2)*a^3 - 9*sqrt(d*tan(f*x + e))*a^3*d)/d)/(d*f)
Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \, a^{3} {\left (-\frac {12 i \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\sqrt {d} {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )}} + \frac {i \, \sqrt {d \tan \left (f x + e\right )} d^{5} \tan \left (f x + e\right ) + 9 \, \sqrt {d \tan \left (f x + e\right )} d^{5}}{d^{6}}\right )}}{3 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
-2/3*a^3*(-12*I*sqrt(2)*arctan(2*sqrt(d*tan(f*x + e))*abs(d)/(I*sqrt(2)*d^ (3/2) + sqrt(2)*sqrt(d)*abs(d)))/(sqrt(d)*(I*d/abs(d) + 1)) + (I*sqrt(d*ta n(f*x + e))*d^5*tan(f*x + e) + 9*sqrt(d*tan(f*x + e))*d^5)/d^6)/f
Time = 1.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {6\,a^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{d\,f}-\frac {a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}}{3\,d^2\,f}+\frac {\sqrt {16{}\mathrm {i}}\,a^3\,\mathrm {atan}\left (\frac {\sqrt {16{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{4\,\sqrt {-d}}\right )\,2{}\mathrm {i}}{\sqrt {-d}\,f} \] Input:
int((a + a*tan(e + f*x)*1i)^3/(d*tan(e + f*x))^(1/2),x)
Output:
(16i^(1/2)*a^3*atan((16i^(1/2)*(d*tan(e + f*x))^(1/2))/(4*(-d)^(1/2)))*2i) /((-d)^(1/2)*f) - (a^3*(d*tan(e + f*x))^(3/2)*2i)/(3*d^2*f) - (6*a^3*(d*ta n(e + f*x))^(1/2))/(d*f)
\[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {d}\, a^{3} \left (-6 \sqrt {\tan \left (f x +e \right )}+4 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) f +3 \left (\int \sqrt {\tan \left (f x +e \right )}d x \right ) f i -\left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) f i \right )}{d f} \] Input:
int((a+I*a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x)
Output:
(sqrt(d)*a**3*( - 6*sqrt(tan(e + f*x)) + 4*int(sqrt(tan(e + f*x))/tan(e + f*x),x)*f + 3*int(sqrt(tan(e + f*x)),x)*f*i - int(sqrt(tan(e + f*x))*tan(e + f*x)**2,x)*f*i))/(d*f)