Integrand size = 25, antiderivative size = 221 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {i \sqrt {i a-b} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i \sqrt {i a+b} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 b \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3 a d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]
I*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*(I*a-b)^(1 /2)*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+I*arctanh((I*a+b)^(1/2)*tan(d*x+c) ^(1/2)/(a+b*tan(d*x+c))^(1/2))*(I*a+b)^(1/2)*cot(d*x+c)^(1/2)*tan(d*x+c)^( 1/2)/d-2/3*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/d-2/3*b*cot(d*x+c)^(1/2 )*(a+b*tan(d*x+c))^(1/2)/a/d
Time = 0.76 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {(-1)^{3/4} \sqrt {-a+i b} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(-1)^{3/4} \sqrt {a+i b} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (a+b \tan (c+d x))^{3/2}}{3 a d \tan ^{\frac {3}{2}}(c+d x)}\right ) \]
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((-1)^(3/4)*Sqrt[-a + I*b]*ArcTan [((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) /d) + ((-1)^(3/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[ c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*(a + b*Tan[c + d*x])^(3/2))/( 3*a*d*Tan[c + d*x]^(3/2)))
Time = 0.87 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4729, 3042, 4051, 27, 3042, 4132, 27, 2011, 3042, 4058, 610, 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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} \sqrt {a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {a+b \tan (c+d x)}}{\tan ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {a+b \tan (c+d x)}}{\tan (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4051 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3} \int -\frac {-2 b \tan ^2(c+d x)-3 a \tan (c+d x)+b}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \int \frac {-2 b \tan ^2(c+d x)-3 a \tan (c+d x)+b}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \int \frac {-2 b \tan (c+d x)^2-3 a \tan (c+d x)+b}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-\frac {2 \int \frac {3 \left (a^2+b \tan (c+d x) a\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-\frac {3 \int \frac {a^2+b \tan (c+d x) a}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-3 \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}}dx-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-3 \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}}dx-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-\frac {3 \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}\right )-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {3 \int \left (\frac {i a-b}{2 (i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {i a+b}{2 \sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {a+b \tan (c+d x)}}\right )d\tan (c+d x)}{d}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {2 b \sqrt {a+b \tan (c+d x)}}{a d \sqrt {\tan (c+d x)}}-\frac {3 \left (-i \sqrt {-b+i a} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-i \sqrt {b+i a} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{d}\right )\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*Sqrt[a + b*Tan[c + d*x]])/(3*d* Tan[c + d*x]^(3/2)) + ((-3*((-I)*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[ Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I *a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/d - (2*b*Sqrt[a + b*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]]))/3)
3.9.41.3.1 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[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[e^(m + 1/2) Int[ExpandIntegrand[1/(Sqrt[e*x]*Sqrt[c + d*x] ), x^(m + 1/2)*((c + d*x)^(n + 1/2)/(a + b*x^2)), x], x], x] /; FreeQ[{a, b , c, d, e}, x] && IGtQ[n + 1/2, 0] && ILtQ[m - 1/2, 0]
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 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/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 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[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1712\) vs. \(2(179)=358\).
Time = 36.58 (sec) , antiderivative size = 1713, normalized size of antiderivative = 7.75
1/12/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+ c)))^(5/2)*(1-cos(d*x+c))*((csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c )-cot(d*x+c))-a)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(3*csc(d*x+c)^2* (b+(a^2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*ln(-1/(1-cos(d*x+c))*(csc(d*x+c) *a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c ))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/ 2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a ))*(1-cos(d*x+c))^2*(-b+(a^2+b^2)^(1/2))^(1/2)-3*csc(d*x+c)^2*(b+(a^2+b^2) ^(1/2))^(1/2)*(a^2+b^2)^(1/2)*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d* x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d* x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^( 1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(1-cos(d* x+c))^2*(-b+(a^2+b^2)^(1/2))^(1/2)-3*csc(d*x+c)^2*(b+(a^2+b^2)^(1/2))^(1/2 )*b*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc (d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1 -cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x +c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(1-cos(d*x+c))^2*(-b+(a^2+b^2)^(1/2 ))^(1/2)+3*csc(d*x+c)^2*(b+(a^2+b^2)^(1/2))^(1/2)*b*ln(1/(1-cos(d*x+c))*(- csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c )*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x...
Leaf count of result is larger than twice the leaf count of optimal. 2867 vs. \(2 (173) = 346\).
Time = 0.48 (sec) , antiderivative size = 2867, normalized size of antiderivative = 12.97 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
1/24*(3*a*d*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2)*log((((a^4*b + 4*a^2*b^3)* d*tan(d*x + c)^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4*b + 4*a^2*b^3)*d + (a^4*d^3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c)^2 - 4*(a^3*b + 2*a*b^3)*d^3*tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(-(d^2*sqrt(-a ^2/d^4) + b)/d^2) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4 *b + 2*a^2*b^3)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - ( a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) + 3*a*d*s qrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2)*log(-(((a^4*b + 4*a^2*b^3)*d*tan(d*x + c)^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4*b + 4*a^2*b^3 )*d + (a^4*d^3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c)^2 - 4*(a^3*b + 2*a*b^3)*d^3*tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(-(d^2*sqrt(-a^2/d^4) + b) /d^2) + 2*((a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b + 2*a^2*b ^3)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2* b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(b*tan(d*x + c) + a)/sq rt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 3*a*d*sqrt(-(d^2*sq rt(-a^2/d^4) + b)/d^2)*log((((a^4*b + 4*a^2*b^3)*d*tan(d*x + c)^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4*b + 4*a^2*b^3)*d + (a^4*d^ 3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c)^2 - 4*(a^3*b + 2*a*b^3)*d^3 *tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2) - 2*...
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Timed out} \]
\[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]