Integrand size = 28, antiderivative size = 254 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {23 (-1)^{3/4} a^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{8 d}+\frac {(2+2 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {i a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{8 d}+\frac {7 a \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d} \]
23/8*(-1)^(3/4)*a^(3/2)*arctan((-1)^(3/4)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a* tan(d*x+c))^(1/2))/d+(2+2*I)*a^(3/2)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2 )/(a+I*a*tan(d*x+c))^(1/2))/d-9/8*I*a*tan(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^ (1/2)/d+7/12*a*(a+I*a*tan(d*x+c))^(1/2)*tan(d*x+c)^(3/2)/d+1/3*I*a^2*tan(d *x+c)^(5/2)/d/(a+I*a*tan(d*x+c))^(1/2)-1/3*a^2*tan(d*x+c)^(7/2)/d/(a+I*a*t an(d*x+c))^(1/2)
Time = 6.39 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.33 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {2 \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {2 a^{3/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt [4]{-1} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {a+i a \tan (c+d x)}}{8 d \sqrt {1+i \tan (c+d x)}}-\frac {9 i a \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{8 d}+\frac {7 a \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}+\frac {i a \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]
(2*Sqrt[2]*a*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (2*a^(3/2)*ArcSin h[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sqrt[1 + I*Tan[c + d*x]]*Sqrt[I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (7*(-1)^(1/4) *a*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d *Sqrt[1 + I*Tan[c + d*x]]) - (((9*I)/8)*a*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a* Tan[c + d*x]])/d + (7*a*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(12 *d) + ((I/3)*a*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/d
Time = 1.80 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {3042, 4039, 27, 3042, 4078, 3042, 4080, 27, 3042, 4080, 27, 3042, 4084, 3042, 4027, 218, 4082, 65, 216}
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.
| \(\displaystyle \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^{5/2} (a+i a \tan (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 4039 |
| \(\displaystyle \frac {1}{3} a \int \frac {\tan ^{\frac {5}{2}}(c+d x) (11 i \tan (c+d x) a+13 a)}{2 \sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \int \frac {\tan ^{\frac {5}{2}}(c+d x) (11 i \tan (c+d x) a+13 a)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} a \int \frac {\tan (c+d x)^{5/2} (11 i \tan (c+d x) a+13 a)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\int \tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a} \left (5 i a^2-7 a^2 \tan (c+d x)\right )dx}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\int \tan (c+d x)^{3/2} \sqrt {i \tan (c+d x) a+a} \left (5 i a^2-7 a^2 \tan (c+d x)\right )dx}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {\int \frac {3}{2} \sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a} \left (9 i \tan (c+d x) a^3+7 a^3\right )dx}{2 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \int \sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a} \left (9 i \tan (c+d x) a^3+7 a^3\right )dx}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \int \sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a} \left (9 i \tan (c+d x) a^3+7 a^3\right )dx}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {\int -\frac {\sqrt {i \tan (c+d x) a+a} \left (9 i a^4-23 a^4 \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)}}dx}{a}+\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (9 i a^4-23 a^4 \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (9 i a^4-23 a^4 \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4084 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {32 i a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-23 i a^3 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {32 i a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-23 i a^3 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {64 a^6 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-23 i a^3 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {(32+32 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-23 i a^3 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {(32+32 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {23 i a^5 \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {(32+32 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {46 i a^5 \int \frac {1}{1-\frac {i a \tan (c+d x)}{i \tan (c+d x) a+a}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{6} a \left (\frac {2 i a \tan ^{\frac {5}{2}}(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\frac {3 \left (\frac {9 i a^3 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {46 (-1)^{3/4} a^{9/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {(32+32 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a}\right )}{4 a}-\frac {7 a^2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}}{a^2}\right )-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}\) |
-1/3*(a^2*Tan[c + d*x]^(7/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (a*(((2*I)* a*Tan[c + d*x]^(5/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((-7*a^2*Tan[c + d* x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d) + (3*(-1/2*((46*(-1)^(3/4)*a^(9 /2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d* x]]])/d + ((32 + 32*I)*a^(9/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]] )/Sqrt[a + I*a*Tan[c + d*x]]])/d)/a + ((9*I)*a^3*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d))/(4*a))/a^2))/6
3.2.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[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_)])^(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]
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[B*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(f*(m + n))), x] + Simp[ 1/(a*(m + n)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Sim p[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*T an[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]
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[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
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)/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] - Simp[B/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[ e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (200 ) = 400\).
Time = 1.07 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.76
| method | result | size |
| derivativedivides | \(-\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a \left (-16 i \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+24 i \sqrt {2}\, \sqrt {i a}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +69 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a +24 \sqrt {2}\, \sqrt {i a}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +54 i \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-28 \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )+96 \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \sqrt {-i a}\right )}{48 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(447\) |
| default | \(-\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a \left (-16 i \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+24 i \sqrt {2}\, \sqrt {i a}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +69 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a +24 \sqrt {2}\, \sqrt {i a}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +54 i \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-28 \sqrt {-i a}\, \sqrt {i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )+96 \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \sqrt {-i a}\right )}{48 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(447\) |
-1/48/d*tan(d*x+c)^(1/2)*(a*(1+I*tan(d*x+c)))^(1/2)*a*(-16*I*(-I*a)^(1/2)* (I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^2+24*I*2^(1/2 )*(I*a)^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^( 1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a+69*I*(-I*a)^(1/2)*ln(1/2*(2*I*a *tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^( 1/2))*a+24*2^(1/2)*(I*a)^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1 +I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a+54*I*(-I*a)^(1 /2)*(I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-28*(-I*a)^(1/2)*(I*a )^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)+96*ln(1/2*(2*I*a* tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1 /2))*a*(-I*a)^(1/2))/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(I*a)^(1/2)/(-I *a)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (188) = 376\).
Time = 0.27 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.65 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (-49 i \, a e^{\left (5 i \, d x + 5 i \, c\right )} - 38 i \, a e^{\left (3 i \, d x + 3 i \, c\right )} - 21 i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {529 i \, a^{3}}{64 \, d^{2}}} \log \left (\frac {{\left (23 \, \sqrt {2} {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 16 i \, \sqrt {\frac {529 i \, a^{3}}{64 \, d^{2}}} d e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{23 \, a}\right ) - 12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {529 i \, a^{3}}{64 \, d^{2}}} \log \left (\frac {{\left (23 \, \sqrt {2} {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} - 16 i \, \sqrt {\frac {529 i \, a^{3}}{64 \, d^{2}}} d e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{23 \, a}\right ) - 12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a^{3}}{d^{2}}} \log \left (\frac {{\left (2 \, \sqrt {2} {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + i \, \sqrt {\frac {8 i \, a^{3}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{2 \, a}\right ) + 12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a^{3}}{d^{2}}} \log \left (\frac {{\left (2 \, \sqrt {2} {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} - i \, \sqrt {\frac {8 i \, a^{3}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{2 \, a}\right )}{24 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
1/24*(sqrt(2)*(-49*I*a*e^(5*I*d*x + 5*I*c) - 38*I*a*e^(3*I*d*x + 3*I*c) - 21*I*a*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I* d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)) + 12*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(529/64*I*a^3/d^2)*log(1/23*(23*sqrt(2)* (a*e^(2*I*d*x + 2*I*c) + a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^( 2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)) + 16*I*sqrt(529/64*I*a^3/ d^2)*d*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a) - 12*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(529/64*I*a^3/d^2)*log(1/23*(23*sqrt(2)*( a*e^(2*I*d*x + 2*I*c) + a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2 *I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)) - 16*I*sqrt(529/64*I*a^3/d ^2)*d*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a) - 12*(d*e^(4*I*d*x + 4*I*c) + 2 *d*e^(2*I*d*x + 2*I*c) + d)*sqrt(8*I*a^3/d^2)*log(1/2*(2*sqrt(2)*(a*e^(2*I *d*x + 2*I*c) + a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)) + I*sqrt(8*I*a^3/d^2)*d*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a) + 12*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(8*I*a^3/d^2)*log(1/2*(2*sqrt(2)*(a*e^(2*I*d*x + 2*I*c) + a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^ (2*I*d*x + 2*I*c) + 1)) - I*sqrt(8*I*a^3/d^2)*d*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0]W arning, replacing 0 by -35, a substitution variable should perhaps be purg ed.Warnin
Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]