Integrand size = 25, antiderivative size = 337 \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {i (i a-b)^{5/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {5 a \left (a^2-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{8 \sqrt {b} d}-\frac {i (i a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}} \]
-I*(I*a-b)^(5/2)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1 /2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-I*(I*a+b)^(5/2)*arctanh((I*a+b)^( 1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^ (1/2)/d+5/8*a*(a^2-8*b^2)*arctanh(b^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c) )^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d/b^(1/2)+1/3*b^2*(a+b*tan(d*x+ c))^(1/2)/d/cot(d*x+c)^(5/2)+13/12*a*b*(a+b*tan(d*x+c))^(1/2)/d/cot(d*x+c) ^(3/2)+1/8*(11*a^2-8*b^2)*(a+b*tan(d*x+c))^(1/2)/d/cot(d*x+c)^(1/2)
Time = 4.48 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-24 (-1)^{3/4} (-a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+24 (-1)^{3/4} (a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+3 \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}+26 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}+8 b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}+\frac {15 a^{3/2} \left (a^2-8 b^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}\right )}{24 d} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-24*(-1)^(3/4)*(-a + I*b)^(5/2)*Ar cTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x ]]] + 24*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[ Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(11*a^2 - 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 26*a*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan [c + d*x]] + 8*b^2*Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]] + (15*a^(3/ 2)*(a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b *Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(24*d)
Time = 1.99 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.93, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4729, 3042, 4049, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4138, 2035, 2257, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan (c+d x)^{3/2} (a+b \tan (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (13 a b^2 \tan ^2(c+d x)+6 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (6 a^2-5 b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (13 a b^2 \tan ^2(c+d x)+6 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (6 a^2-5 b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \int \frac {\tan (c+d x)^{3/2} \left (13 a b^2 \tan (c+d x)^2+6 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (6 a^2-5 b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {\int -\frac {3 \sqrt {\tan (c+d x)} \left (13 a^2 b^2-\left (11 a^2-8 b^2\right ) \tan ^2(c+d x) b^2-8 a \left (a^2-3 b^2\right ) \tan (c+d x) b\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{2 b}+\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \int \frac {\sqrt {\tan (c+d x)} \left (13 a^2 b^2-\left (11 a^2-8 b^2\right ) \tan ^2(c+d x) b^2-8 a \left (a^2-3 b^2\right ) \tan (c+d x) b\right )}{\sqrt {a+b \tan (c+d x)}}dx}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \int \frac {\sqrt {\tan (c+d x)} \left (13 a^2 b^2-\left (11 a^2-8 b^2\right ) \tan (c+d x)^2 b^2-8 a \left (a^2-3 b^2\right ) \tan (c+d x) b\right )}{\sqrt {a+b \tan (c+d x)}}dx}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \frac {16 \left (3 a^2-b^2\right ) \tan (c+d x) b^3-5 a \left (a^2-8 b^2\right ) \tan ^2(c+d x) b^2+a \left (11 a^2-8 b^2\right ) b^2}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{b}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \frac {16 \left (3 a^2-b^2\right ) \tan (c+d x) b^3-5 a \left (a^2-8 b^2\right ) \tan ^2(c+d x) b^2+a \left (11 a^2-8 b^2\right ) b^2}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{2 b}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \frac {16 \left (3 a^2-b^2\right ) \tan (c+d x) b^3-5 a \left (a^2-8 b^2\right ) \tan (c+d x)^2 b^2+a \left (11 a^2-8 b^2\right ) b^2}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{2 b}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \frac {16 \left (3 a^2-b^2\right ) \tan (c+d x) b^3-5 a \left (a^2-8 b^2\right ) \tan ^2(c+d x) b^2+a \left (11 a^2-8 b^2\right ) b^2}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{2 b d}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \frac {16 \left (3 a^2-b^2\right ) \tan (c+d x) b^3-5 a \left (a^2-8 b^2\right ) \tan ^2(c+d x) b^2+a \left (11 a^2-8 b^2\right ) b^2}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\sqrt {\tan (c+d x)}}{b d}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int \left (\frac {16 \left (\left (3 a^2-b^2\right ) \tan (c+d x) b^3+a \left (a^2-3 b^2\right ) b^2\right )}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}-\frac {5 a b^2 \left (a^2-8 b^2\right )}{\sqrt {a+b \tan (c+d x)}}\right )d\sqrt {\tan (c+d x)}}{b d}-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 b}\right )+\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {13 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {b \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}+\frac {-5 a b^{3/2} \left (a^2-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+8 i b^2 (-b+i a)^{5/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+8 i b^2 (b+i a)^{5/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b d}\right )}{4 b}\right )\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((b^2*Tan[c + d*x]^(5/2)*Sqrt[a + b* Tan[c + d*x]])/(3*d) + ((13*a*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x] ])/(2*d) - (3*(((8*I)*(I*a - b)^(5/2)*b^2*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 5*a*b^(3/2)*(a^2 - 8*b^2)*ArcTanh[(S qrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (8*I)*b^2*(I*a + b) ^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]] ])/(b*d) - (b*(11*a^2 - 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] )/d))/(4*b))/6)
3.9.60.3.1 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[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
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[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 16.29 (sec) , antiderivative size = 29752, normalized size of antiderivative = 88.28
Leaf count of result is larger than twice the leaf count of optimal. 4861 vs. \(2 (269) = 538\).
Time = 1.41 (sec) , antiderivative size = 9755, normalized size of antiderivative = 28.95 \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]