Integrand size = 23, antiderivative size = 64 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {2 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{d} \] Output:
-2*a^(3/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d-2*a*cot(d*x +c)*(a+a*sec(d*x+c))^(1/2)/d
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.59 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 \cot (c+d x) \sqrt {\frac {1}{1+\sec (c+d x)}} (a (1+\sec (c+d x)))^{3/2} \left (\sqrt {\cos (c+d x)} \sqrt {\frac {1}{1+\cos (c+d x)}}+\arcsin \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\cos (c+d x)}}}\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \] Input:
Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]
Output:
(-2*Cot[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*(a*(1 + Sec[c + d*x]))^(3/2 )*(Sqrt[Cos[c + d*x]]*Sqrt[(1 + Cos[c + d*x])^(-1)] + ArcSin[Tan[(c + d*x) /2]/Sqrt[(1 + Cos[c + d*x])^(-1)]]*Tan[(c + d*x)/2]))/d
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4375, 264, 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.
\(\displaystyle \int \cot ^2(c+d x) (a \sec (c+d x)+a)^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 4375 |
\(\displaystyle -\frac {2 a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a)}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {2 a \left (\cot (c+d x) \sqrt {a \sec (c+d x)+a}-a \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2 a \left (\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )+\cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]
Output:
(-2*a*(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]] + C ot[c + d*x]*Sqrt[a + a*Sec[c + d*x]]))/d
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d) Subst[Int[x^m*((2 + a*x^2 )^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] ]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I ntegerQ[n - 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(190\) vs. \(2(56)=112\).
Time = 1.01 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.98
method | result | size |
default | \(\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+2 \sqrt {2}\, \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right )+\left (2 \cos \left (d x +c \right )-2\right ) \cot \left (d x +c \right )\right )}{d \left (1+\cos \left (d x +c \right )\right )}\) | \(191\) |
Input:
int(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d*a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))*((1+cos(d*x+c))*2^(1/2)*(-2* cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+cot(d*x+c))/ (cot(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+csc(d*x+c)^2-1)^(1/2))+2*2^(1/2)*(1+ cos(d*x+c))*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c )))^(1/2)*cot(d*x+c)+(2*cos(d*x+c)-2)*cot(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 4.12 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\left [\frac {\sqrt {-a} a \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, d \sin \left (d x + c\right )}, -\frac {a^{\frac {3}{2}} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{d \sin \left (d x + c\right )}\right ] \] Input:
integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[1/2*(sqrt(-a)*a*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a* cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c)), -(a^(3/2)*arctan(2* sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/ (2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*a*sqrt((a*cos( d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c))]
\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**2*(a+a*sec(d*x+c))**(3/2),x)
Output:
Integral((a*(sec(c + d*x) + 1))**(3/2)*cot(c + d*x)**2, x)
\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)^(3/2)*cot(d*x + c)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (56) = 112\).
Time = 0.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.08 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {\frac {\sqrt {-a} a^{2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{{\left | a \right |}} + \frac {2 \, \sqrt {2} \sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{{\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a}}{d} \] Input:
integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
Output:
(sqrt(-a)*a^2*log(abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d *x + 1/2*c)^2 + a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*d* x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + 4*sqrt(2)*abs(a) - 6 *a))*sgn(cos(d*x + c))/abs(a) + 2*sqrt(2)*sqrt(-a)*a^2*sgn(cos(d*x + c))/( (sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a))/d
Timed out. \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \] Input:
int(cot(c + d*x)^2*(a + a/cos(c + d*x))^(3/2),x)
Output:
int(cot(c + d*x)^2*(a + a/cos(c + d*x))^(3/2), x)
\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{2} \sec \left (d x +c \right )d x +\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{2}d x \right ) \] Input:
int(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**2*sec(c + d*x),x) + in t(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**2,x))