Integrand size = 25, antiderivative size = 294 \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{9/2}}-\frac {4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{9/2}}+\frac {4 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a^2 d (e \cot (c+d x))^{9/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \] Output:
2/3*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(9/2)-4*cos(d*x+c)*cot(d*x+c)^3/a^2/ d/(e*cot(d*x+c))^(9/2)-4*cos(d*x+c)*cot(d*x+c)^4*EllipticE(cos(c+1/4*Pi+d* x),2^(1/2))/a^2/d/(e*cot(d*x+c))^(9/2)/sin(2*d*x+2*c)^(1/2)+1/2*arctan(-1+ 2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(9/2)/tan(d*x+c)^(9 /2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/a^2/d/(e*cot(d*x+c))^(9 /2)/tan(d*x+c)^(9/2)-1/2*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c)))* 2^(1/2)/a^2/d/(e*cot(d*x+c))^(9/2)/tan(d*x+c)^(9/2)
\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx \] Input:
Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2),x]
Output:
Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2), x]
Time = 0.73 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4388, 3042, 4376, 3042, 4374, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{9/2}}dx\) |
| \(\Big \downarrow \) 4388 |
| \(\displaystyle \frac {\int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(\sec (c+d x) a+a)^2}dx}{\tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{\tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int (a-a \sec (c+d x))^2 \sqrt {\tan (c+d x)}dx}{a^4 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2dx}{a^4 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \frac {\int \left (\sec ^2(c+d x) \sqrt {\tan (c+d x)} a^2-2 \sec (c+d x) \sqrt {\tan (c+d x)} a^2+\sqrt {\tan (c+d x)} a^2\right )dx}{a^4 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {4 a^2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}+\frac {4 a^2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}}{a^4 \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}\) |
Input:
Int[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2),x]
Output:
(-((a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + (a^2*ArcTan [1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + (a^2*Log[1 - Sqrt[2]*Sqrt[ Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (a^2*Log[1 + Sqrt[2]*Sqrt[T an[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (4*a^2*Cos[c + d*x]*Elliptic E[c - Pi/4 + d*x, 2]*Sqrt[Tan[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a ^2*Tan[c + d*x]^(3/2))/(3*d) - (4*a^2*Cos[c + d*x]*Tan[c + d*x]^(3/2))/d)/ (a^4*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2))
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x _)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[m]
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.40
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sqrt {-\frac {2 \sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \left (i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (3 \cot \left (d x +c \right )+3 \csc \left (d x +c \right )\right )+i \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-3 \cot \left (d x +c \right )-3 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-24 \cot \left (d x +c \right )-24 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (12 \cot \left (d x +c \right )+12 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-3 \cot \left (d x +c \right )-3 \csc \left (d x +c \right )\right )+\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-3 \cot \left (d x +c \right )-3 \csc \left (d x +c \right )\right )+20 \cot \left (d x +c \right )-24 \csc \left (d x +c \right )+4 \sec \left (d x +c \right ) \csc \left (d x +c \right )\right )}{12 a^{2} d \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, e^{4}}\) | \(706\) |
Input:
int(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/12/a^2/d*2^(1/2)*(-2*sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(e*co t(d*x+c))^(1/2)/(-sin(d*x+c)*cos(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/e^4*(I*(-c ot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d* x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2 *I,1/2*2^(1/2))*(3*cot(d*x+c)+3*csc(d*x+c))+I*(-cot(d*x+c)+csc(d*x+c)+1)^( 1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*El lipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-3*cot(d *x+c)-3*csc(d*x+c))+(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d *x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticE((-cot(d*x+c)+csc(d *x+c)+1)^(1/2),1/2*2^(1/2))*(-24*cot(d*x+c)-24*csc(d*x+c))+(-cot(d*x+c)+cs c(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x +c))^(1/2)*EllipticF((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2*2^(1/2))*(12*cot (d*x+c)+12*csc(d*x+c))+(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*cs c(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+c sc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-3*cot(d*x+c)-3*csc(d*x+c))+(-c ot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d* x+c)+cot(d*x+c))^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2 *I,1/2*2^(1/2))*(-3*cot(d*x+c)-3*csc(d*x+c))+20*cot(d*x+c)-24*csc(d*x+c)+4 *sec(d*x+c)*csc(d*x+c))
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cot(d*x+c))**(9/2)/(a+a*sec(d*x+c))**2,x)
Output:
Timed out
\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
Output:
integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)^2), x)
\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")
Output:
integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)^2), x)
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{9/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \] Input:
int(1/((e*cot(c + d*x))^(9/2)*(a + a/cos(c + d*x))^2),x)
Output:
int(cos(c + d*x)^2/(a^2*(e*cot(c + d*x))^(9/2)*(cos(c + d*x) + 1)^2), x)
\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}+2 \cot \left (d x +c \right )^{5} \sec \left (d x +c \right )+\cot \left (d x +c \right )^{5}}d x \right )}{a^{2} e^{5}} \] Input:
int(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x)
Output:
(sqrt(e)*int(sqrt(cot(c + d*x))/(cot(c + d*x)**5*sec(c + d*x)**2 + 2*cot(c + d*x)**5*sec(c + d*x) + cot(c + d*x)**5),x))/(a**2*e**5)