Integrand size = 23, antiderivative size = 193 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{3 d}-\frac {2 a^2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}{3 d} \] Output:
1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d+1/ 2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2*( a+b)*(a^2-4*a*b+b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1 /2)/d-16/3*a^2*b*cot(d*x+c)^(1/2)/d-2/3*a^2*cot(d*x+c)^(1/2)*(b+a*cot(d*x+ c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.54 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {72 a^2 b \sqrt {\cot (c+d x)}+8 a^3 \cot ^{\frac {3}{2}}(c+d x)-8 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-3 \sqrt {2} b \left (-3 a^2+b^2\right ) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{12 d} \] Input:
Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]
Output:
-1/12*(72*a^2*b*Sqrt[Cot[c + d*x]] + 8*a^3*Cot[c + d*x]^(3/2) - 8*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 3*Sqrt[2]*b*(-3*a^2 + b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d* x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])) /d
Time = 0.81 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.15, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4156, 3042, 4049, 27, 3042, 4113, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^3}{\sqrt {\cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle -\frac {2}{3} \int \frac {-8 a^2 b \cot ^2(c+d x)+3 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (a^2-3 b^2\right )}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \int \frac {-8 a^2 b \cot ^2(c+d x)+3 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (a^2-3 b^2\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{3} \int \frac {-8 a^2 b \tan \left (c+d x+\frac {\pi }{2}\right )^2-3 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b \left (a^2-3 b^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {3 b \left (3 a^2-b^2\right )+3 a \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {3 b \left (3 a^2-b^2\right )-3 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{3} \left (-\frac {2 \int -\frac {3 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {16 a^2 b \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
Input:
Int[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]
Output:
(-2*a^2*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x]))/(3*d) + ((-16*a^2*b*Sqrt[ Cot[c + d*x]])/d + (6*(((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]* Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt [2]))/2 - ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d)/3
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, 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[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[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[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_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(167)=334\).
Time = 0.68 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.83
| method | result | size |
| derivativedivides | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}-9 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}+6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+9 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b -3 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+72 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{12 d}\) | \(546\) |
| default | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}-9 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}+6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+9 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b -3 \tan \left (d x +c \right )^{\frac {3}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+72 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{12 d}\) | \(546\) |
Input:
int(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(3*tan(d*x+c)^(3/2)*2^(1/2)*ln(-(t an(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c) -1))*a^3-9*tan(d*x+c)^(3/2)*2^(1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/ 2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a*b^2+6*tan(d*x+c)^(3/2)*2^ (1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+18*tan(d*x+c)^(3/2)*2^(1/2)*a rctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-18*tan(d*x+c)^(3/2)*2^(1/2)*arctan (1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-6*tan(d*x+c)^(3/2)*2^(1/2)*arctan(1+2^( 1/2)*tan(d*x+c)^(1/2))*b^3+6*tan(d*x+c)^(3/2)*2^(1/2)*arctan(-1+2^(1/2)*ta n(d*x+c)^(1/2))*a^3+18*tan(d*x+c)^(3/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+ c)^(1/2))*a^2*b-18*tan(d*x+c)^(3/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^( 1/2))*a*b^2-6*tan(d*x+c)^(3/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)) *b^3+9*tan(d*x+c)^(3/2)*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1 )/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*a^2*b-3*tan(d*x+c)^(3/2)*2^(1/2 )*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+ c)^(1/2)+1))*b^3+72*tan(d*x+c)*a^2*b+8*a^3)
Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (167) = 334\).
Time = 0.10 (sec) , antiderivative size = 933, normalized size of antiderivative = 4.83 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
1/6*(6*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^ 4 + 6*a*b^5 + b^6)/d^2)*arctan(-(2*sqrt(1/2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^ 3)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^ 3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a^2*b ^4 - b^6))*tan(d*x + c) + 6*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*arctan(-(2*sqrt(1/2)*(a^3 - 3 *a^2*b - 3*a*b^2 + b^3)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3 *a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^6 + 6*a^5* b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6 *a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6))*tan(d*x + c) - 3*sqrt(1/2)*d*sqrt((a^6 - 6 *a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*log(a^3 - 3*a^2*b - 3*a*b^2 + b^3 + 2*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) + (a^3 - 3*a^2*b - 3*a*b^2 + b^3)*tan(d*x + c))*tan(d*x + c) + 3*sqrt(1/2)*d*sqrt( (a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)* log(a^3 - 3*a^2*b - 3*a*b^2 + b^3 - 2*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3* a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c...
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(5/2)*(a+b*tan(d*x+c))**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.14 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {72 \, a^{2} b}{\sqrt {\tan \left (d x + c\right )}} - \frac {8 \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/12*(6*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2 ) + 2/sqrt(tan(d*x + c)))) + 6*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arc tan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 3*sqrt(2)*(a^3 - 3*a^ 2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 3*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c) ) + 1/tan(d*x + c) + 1) - 72*a^2*b/sqrt(tan(d*x + c)) - 8*a^3/tan(d*x + c) ^(3/2))/d
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)^3*cot(d*x + c)^(5/2), x)
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \] Input:
int(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3,x)
Output:
int(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3, x)
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3}d x \right ) b^{3}+3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) a^{3} \] Input:
int(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x)
Output:
int(sqrt(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x)**3,x)*b**3 + 3*int(sqr t(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x)**2,x)*a*b**2 + 3*int(sqrt(cot (c + d*x))*cot(c + d*x)**2*tan(c + d*x),x)*a**2*b + int(sqrt(cot(c + d*x)) *cot(c + d*x)**2,x)*a**3