Integrand size = 21, antiderivative size = 120 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {1}{3 d (a+a \sec (c+d x))^{3/2}}-\frac {3}{2 a d \sqrt {a+a \sec (c+d x)}} \] Output:
2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-1/4*arctanh(1/2*(a+a*s ec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(3/2)/d-1/3/d/(a+a*sec(d*x+c)) ^(3/2)-3/2/a/d/(a+a*sec(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.50 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right )-2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\sec (c+d x)\right )}{3 d (a (1+\sec (c+d x)))^{3/2}} \] Input:
Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]
Output:
(Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sec[c + d*x])/2] - 2*Hypergeometric 2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]])/(3*d*(a*(1 + Sec[c + d*x]))^(3/2))
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 25, 4368, 25, 27, 96, 25, 27, 169, 27, 174, 73, 219, 220}
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.
\(\displaystyle \int \frac {\cot (c+d x)}{(a \sec (c+d x)+a)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4368 |
\(\displaystyle \frac {a^2 \int -\frac {\cos (c+d x)}{a (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \int \frac {\cos (c+d x)}{a (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \int \frac {\cos (c+d x)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{d}\) |
\(\Big \downarrow \) 96 |
\(\displaystyle -\frac {a \left (\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}-\frac {\int -\frac {a \cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a^2}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a \left (\frac {\int \frac {a \cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a^2}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (\frac {\int \frac {\cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {a \cos (c+d x) (4-3 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {\cos (c+d x) (4-3 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+4 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {a \left (\frac {\frac {\frac {2 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {8 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a \left (\frac {\frac {\frac {8 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\frac {a \left (\frac {\frac {\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {3}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{3 a (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
Input:
Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]
Output:
-((a*(1/(3*a*(a + a*Sec[c + d*x])^(3/2)) + (((-8*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (Sqrt[2]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqr t[2]*Sqrt[a])])/Sqrt[a])/(2*a) + 3/(a*Sqrt[a + a*Sec[c + d*x]]))/(2*a)))/d )
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(d*b^(m - 1))^(-1) Subst[Int[(-a + b*x)^((m - 1)/2 )*((a + b*x)^((m - 1)/2 + n)/x), x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(95)=190\).
Time = 1.01 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \left (3 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}}-3 \left (1-\cos \left (d x +c \right )\right )^{4} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{4}-5 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}+16 \left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{2}+60 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}{2}\right )-58 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}+15 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right )\right )}{60 d \,a^{2}}\) | \(304\) |
Input:
int(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/60/d/a^2*(-2*a/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*((1-cos(d*x+c)) ^2*csc(d*x+c)^2-1)^(1/2)*(3*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(5/2)-3*(1-c os(d*x+c))^4*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*csc(d*x+c)^4-5*((1-co s(d*x+c))^2*csc(d*x+c)^2-1)^(3/2)+16*(1-cos(d*x+c))^2*((1-cos(d*x+c))^2*cs c(d*x+c)^2-1)^(1/2)*csc(d*x+c)^2+60*2^(1/2)*arctan(1/2*2^(1/2)*((1-cos(d*x +c))^2*csc(d*x+c)^2-1)^(1/2))-58*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)+1 5*arctan(1/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (95) = 190\).
Time = 0.16 (sec) , antiderivative size = 485, normalized size of antiderivative = 4.04 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) + 12 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, {\left (11 \, \cos \left (d x + c\right )^{2} + 9 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {3 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 12 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, {\left (11 \, \cos \left (d x + c\right )^{2} + 9 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \] Input:
integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[1/24*(3*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqr t(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*co s(d*x + c) - a)/(cos(d*x + c) - 1)) + 12*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c)) *sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(11*cos(d*x + c)^2 + 9*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), 1/12*(3*sqrt (2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a) *sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a) ) - 12*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sq rt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(11*cos(d*x + c)^2 + 9*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d* x + c)))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)]
\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(d*x+c)/(a+a*sec(d*x+c))**(3/2),x)
Output:
Integral(cot(c + d*x)/(a*(sec(c + d*x) + 1))**(3/2), x)
\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate(cot(d*x + c)/(a*sec(d*x + c) + a)^(3/2), x)
Time = 0.61 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.37 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {24 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\sqrt {2} {\left ({\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{6} + 9 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{7}\right )}}{a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{12 \, d} \] Input:
integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
Output:
1/12*(3*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt (-a)*a*sgn(cos(d*x + c))) - 24*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/ 2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a*sgn(cos(d*x + c))) - sqrt(2)*((-a*tan(1/ 2*d*x + 1/2*c)^2 + a)^(3/2)*a^6 + 9*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^ 7)/(a^9*sgn(cos(d*x + c))))/d
Timed out. \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(cot(c + d*x)/(a + a/cos(c + d*x))^(3/2),x)
Output:
int(cot(c + d*x)/(a + a/cos(c + d*x))^(3/2), x)
\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x))/(sec(c + d*x)**2 + 2*se c(c + d*x) + 1),x))/a**2