Integrand size = 46, antiderivative size = 190 \[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 (-1)^{3/4} a^{5/2} B \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {(2+2 i) a^{3/2} (2 a+3 i b) B \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a (a+3 i b) B \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)} \]
2*(-1)^(3/4)*a^(5/2)*B*arctan((-1)^(3/4)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*t an(d*x+c))^(1/2))/d+(2+2*I)*a^(3/2)*(2*a+3*I*b)*B*arctanh((1+I)*a^(1/2)*ta n(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/d-2*a*(a+3*I*b)*B*(a+I*a*tan(d*x+ c))^(1/2)/d/tan(d*x+c)^(1/2)-b*B*(a+I*a*tan(d*x+c))^(3/2)/d/tan(d*x+c)^(3/ 2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(566\) vs. \(2(190)=380\).
Time = 11.38 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.98 \[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {b B (a+i a \tan (c+d x))^{5/2}}{a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (-\frac {3 (2 a+5 i b) B (a+i a \tan (c+d x))^{5/2}}{2 d \sqrt {\tan (c+d x)}}+\frac {2 \left (\frac {3 i a^3 (2 a+5 i b) B \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} (-1)^{3/4} \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right )+\frac {5}{4} \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}+\frac {1}{2} i \sqrt {1+i \tan (c+d x)} \tan ^{\frac {3}{2}}(c+d x)\right )}{2 d \sqrt {1+i \tan (c+d x)}}+\frac {a \left (\frac {3}{8} i a^2 (10 i a-23 b) B+\frac {3}{2} a^2 (2 a+5 i b) B\right ) \left (-\frac {4 i \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{\sqrt {i a \tan (c+d x)}}+\frac {4 i a^{3/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+i \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}+\frac {i \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{\sqrt {1+i \tan (c+d x)} \sqrt {i a \tan (c+d x)}}\right )}{d}\right )}{a}\right )}{3 a} \]
Integrate[((a + I*a*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/ Tan[c + d*x]^(5/2),x]
-((b*B*(a + I*a*Tan[c + d*x])^(5/2))/(a*d*Tan[c + d*x]^(3/2))) + (2*((-3*( 2*a + (5*I)*b)*B*(a + I*a*Tan[c + d*x])^(5/2))/(2*d*Sqrt[Tan[c + d*x]]) + (2*((((3*I)/2)*a^3*(2*a + (5*I)*b)*B*Sqrt[a + I*a*Tan[c + d*x]]*((-3*(-1)^ (3/4)*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]])/4 + (5*Sqrt[1 + I*Tan[c + d* x]]*Sqrt[Tan[c + d*x]])/4 + (I/2)*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(3 /2)))/(d*Sqrt[1 + I*Tan[c + d*x]]) + (a*(((3*I)/8)*a^2*((10*I)*a - 23*b)*B + (3*a^2*(2*a + (5*I)*b)*B)/2)*(((-4*I)*Sqrt[2]*a*ArcTanh[(Sqrt[2]*Sqrt[I *a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[I*a *Tan[c + d*x]] + ((4*I)*a^(3/2)*ArcSinh[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sq rt[1 + I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + I*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]] + ( I*Sqrt[a]*ArcSinh[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sqrt[Tan[c + d*x]]*Sqrt[ a + I*a*Tan[c + d*x]])/(Sqrt[1 + I*Tan[c + d*x]]*Sqrt[I*a*Tan[c + d*x]]))) /d))/a))/(3*a)
Time = 1.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4076, 27, 3042, 4076, 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.
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {2}{3} \int \frac {3 (i \tan (c+d x) a+a)^{3/2} ((a+3 i b) B+i a \tan (c+d x) B)}{2 \tan ^{\frac {3}{2}}(c+d x)}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a)^{3/2} ((a+3 i b) B+i a \tan (c+d x) B)}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a)^{3/2} ((a+3 i b) B+i a \tan (c+d x) B)}{\tan (c+d x)^{3/2}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle 2 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 a (i a-2 b) B-a^2 B \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 a (i a-2 b) B-a^2 B \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 a (i a-2 b) B-a^2 B \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 4084 |
| \(\displaystyle 2 a B (-3 b+2 i a) \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-i a B \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a B (-3 b+2 i a) \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-i a B \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle -\frac {4 i a^3 B (-3 b+2 i a) \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-i a B \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -i a B \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+\frac {(2-2 i) a^{3/2} B (-3 b+2 i a) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle -\frac {i a^3 B \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {(2-2 i) a^{3/2} B (-3 b+2 i a) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle -\frac {2 i a^3 B \int \frac {1}{1-\frac {i a \tan (c+d x)}{i \tan (c+d x) a+a}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}+\frac {(2-2 i) a^{3/2} B (-3 b+2 i a) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 (-1)^{3/4} a^{5/2} B \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {(2-2 i) a^{3/2} B (-3 b+2 i a) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a B (a+3 i b) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\) |
(2*(-1)^(3/4)*a^(5/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqr t[a + I*a*Tan[c + d*x]]])/d + ((2 - 2*I)*a^(3/2)*((2*I)*a - 3*b)*B*ArcTanh [((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2* a*(a + (3*I)*b)*B*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (b* B*(a + I*a*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(3/2))
3.2.78.3.1 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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-a^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1)) Int[ (a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b *d*(n + 1)))*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] && GtQ[m, 1] && LtQ[n, -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[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. 617 vs. \(2 (156 ) = 312\).
Time = 0.14 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.25
| method | result | size |
| derivativedivides | \(\frac {a B \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (6 i \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 ) \sqrt {-i a}\, a^{2} \left (\tan ^{2}\left (d x +c \right )\right )-i \sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )+2 i \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 ) \sqrt {-i a}\, a \left (\tan ^{2}\left (d x +c \right )\right )-14 i \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, b -12 \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 ) \sqrt {-i a}\, a b \left (\tan ^{2}\left (d x +c \right )\right )-\sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )-2 \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 ) \sqrt {-i a}\, a \left (\tan ^{2}\left (d x +c \right )\right )-4 \sqrt {i a}\, \sqrt {-i a}\, a \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-2 b \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{2 d \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(618\) |
| default | \(\frac {a B \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (6 i \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 ) \sqrt {-i a}\, a^{2} \left (\tan ^{2}\left (d x +c \right )\right )-i \sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )+2 i \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 ) \sqrt {-i a}\, a \left (\tan ^{2}\left (d x +c \right )\right )-14 i \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, b -12 \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 ) \sqrt {-i a}\, a b \left (\tan ^{2}\left (d x +c \right )\right )-\sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )-2 \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 ) \sqrt {-i a}\, a \left (\tan ^{2}\left (d x +c \right )\right )-4 \sqrt {i a}\, \sqrt {-i a}\, a \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-2 b \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{2 d \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(618\) |
| parts | \(\frac {B \left (-i \sqrt {i a}\, \sqrt {2}\, \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 \tan \left (d x +c \right )-i \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 ) \sqrt {-i a}\, a \tan \left (d x +c \right )-\sqrt {i a}\, \sqrt {2}\, \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 ) \tan \left (d x +c \right ) a -4 \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 ) \sqrt {-i a}\, 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}\, \sqrt {-i a}\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2}}{d \sqrt {\tan \left (d x +c \right )}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}-\frac {b B a \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 i \sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )+12 i \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 ) \sqrt {-i a}\, a \left (\tan ^{2}\left (d x +c \right )\right )-3 \sqrt {i a}\, \sqrt {2}\, \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 \left (\tan ^{2}\left (d x +c \right )\right )+14 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-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}\, \sqrt {-i a}\right )}{2 d \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(757\) |
int((a+I*a*tan(d*x+c))^(5/2)*(3/2*b*B/a+B*tan(d*x+c))/tan(d*x+c)^(5/2),x,m ethod=_RETURNVERBOSE)
1/2/d*a*B*(a*(1+I*tan(d*x+c)))^(1/2)*(6*I*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*ta n(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*(-I*a)^(1/2)* a^2*tan(d*x+c)^2-I*(I*a)^(1/2)*2^(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*tan(d* x+c)^2+2*I*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))*(-I*a)^(1/2)*a*tan(d*x+c)^2-14*I*(I*a)^(1/2) *(-I*a)^(1/2)*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*b-12*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))*(-I*a)^(1/2)*a*b*tan(d*x+c)^2-(I*a)^(1/2)*2^(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*tan(d*x+c)^2-2*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))*(-I*a)^(1/2)*a*tan(d*x +c)^2-4*(I*a)^(1/2)*(-I*a)^(1/2)*a*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan(d*x+c )))^(1/2)-2*b*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/ 2))/tan(d*x+c)^(3/2)/(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. 900 vs. \(2 (146) = 292\).
Time = 0.28 (sec) , antiderivative size = 900, normalized size of antiderivative = 4.74 \[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\text {Too large to display} \]
integrate((a+I*a*tan(d*x+c))^(5/2)*(3/2*b*B/a+B*tan(d*x+c))/tan(d*x+c)^(5/ 2),x, algorithm="fricas")
1/2*(2*sqrt(2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt( -(-4*I*B^2*a^5 + 12*B^2*a^4*b + 9*I*B^2*a^3*b^2)/d^2)*log((sqrt(2)*d*sqrt( -(-4*I*B^2*a^5 + 12*B^2*a^4*b + 9*I*B^2*a^3*b^2)/d^2)*e^(I*d*x + I*c) + sq rt(2)*(2*I*B*a^2 - 3*B*a*b + (2*I*B*a^2 - 3*B*a*b)*e^(2*I*d*x + 2*I*c))*sq rt(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)))*e^(-I*d*x - I*c)/(2*I*B*a^2 - 3*B*a*b)) - 2*sqrt(2)*(d *e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-(-4*I*B^2*a^5 + 12*B^2*a^4*b + 9*I*B^2*a^3*b^2)/d^2)*log(-(sqrt(2)*d*sqrt(-(-4*I*B^2*a^5 + 12*B^2*a^4*b + 9*I*B^2*a^3*b^2)/d^2)*e^(I*d*x + I*c) - sqrt(2)*(2*I*B*a^2 - 3*B*a*b + (2*I*B*a^2 - 3*B*a*b)*e^(2*I*d*x + 2*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 )))*e^(-I*d*x - I*c)/(2*I*B*a^2 - 3*B*a*b)) + 4*sqrt(2)*(B*a*b*e^(3*I*d*x + 3*I*c) - (I*B*a^2 - 4*B*a*b)*e^(5*I*d*x + 5*I*c) - (-I*B*a^2 + 3*B*a*b)* 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)) + sqrt(4*I*B^2*a^5/d^2)*(d*e^(4*I*d* x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*(B*a^2*e^(2*I*d*x + 2*I*c) + B*a^2)*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(4*I*B^2*a^5/d^2)*d*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(B*a^2)) - sqrt(4*I*B^2*a^5/d^2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*(B*a^2*e^(2*I*d*x +...
\[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\frac {B \left (\int \frac {2 a^{3} \sqrt {i a \tan {\left (c + d x \right )} + a}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \left (- 2 a^{3} \sqrt {i a \tan {\left (c + d x \right )} + a} \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int \frac {4 i a^{3} \sqrt {i a \tan {\left (c + d x \right )} + a}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx + \int \frac {3 a^{2} b \sqrt {i a \tan {\left (c + d x \right )} + a}}{\tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \left (- \frac {3 a^{2} b \sqrt {i a \tan {\left (c + d x \right )} + a}}{\sqrt {\tan {\left (c + d x \right )}}}\right )\, dx + \int \frac {6 i a^{2} b \sqrt {i a \tan {\left (c + d x \right )} + a}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right )}{2 a} \]
B*(Integral(2*a**3*sqrt(I*a*tan(c + d*x) + a)/tan(c + d*x)**(3/2), x) + In tegral(-2*a**3*sqrt(I*a*tan(c + d*x) + a)*sqrt(tan(c + d*x)), x) + Integra l(4*I*a**3*sqrt(I*a*tan(c + d*x) + a)/sqrt(tan(c + d*x)), x) + Integral(3* a**2*b*sqrt(I*a*tan(c + d*x) + a)/tan(c + d*x)**(5/2), x) + Integral(-3*a* *2*b*sqrt(I*a*tan(c + d*x) + a)/sqrt(tan(c + d*x)), x) + Integral(6*I*a**2 *b*sqrt(I*a*tan(c + d*x) + a)/tan(c + d*x)**(3/2), x))/(2*a)
\[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (2 \, B \tan \left (d x + c\right ) + \frac {3 \, B b}{a}\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{2 \, \tan \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((a+I*a*tan(d*x+c))^(5/2)*(3/2*b*B/a+B*tan(d*x+c))/tan(d*x+c)^(5/ 2),x, algorithm="maxima")
1/2*integrate((2*B*tan(d*x + c) + 3*B*b/a)*(I*a*tan(d*x + c) + a)^(5/2)/ta n(d*x + c)^(5/2), x)
Exception generated. \[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\text {Exception raised: TypeError} \]
integrate((a+I*a*tan(d*x+c))^(5/2)*(3/2*b*B/a+B*tan(d*x+c))/tan(d*x+c)^(5/ 2),x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Non regular value [0] was discarded and replaced randomly by 0=[-52]Warning, replacing -52 by -83, a substitu tion vari
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\left (B\,\mathrm {tan}\left (c+d\,x\right )+\frac {3\,B\,b}{2\,a}\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{5/2}} \,d x \]