Integrand size = 25, antiderivative size = 321 \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{7/2}}-\frac {2 \cot ^3(c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{a^2 d (e \cot (c+d x))^{7/2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)} \]
2*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(7/2)+2*cot(d*x+c)^3*csc(d*x+c)*(sin(c +1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticF(cos(c+1/4*Pi+d*x),2^(1/2 ))*sin(2*d*x+2*c)^(1/2)/a^2/d/(e*cot(d*x+c))^(7/2)+1/2*arctan(-1+2^(1/2)*t an(d*x+c)^(1/2))/a^2/d/(e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d*x+c)^(7/2)+1/2*a rctan(1+2^(1/2)*tan(d*x+c)^(1/2))/a^2/d/(e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d *x+c)^(7/2)-1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a^2/d/(e*cot(d*x +c))^(7/2)*2^(1/2)/tan(d*x+c)^(7/2)+1/4*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan( d*x+c))/a^2/d/(e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d*x+c)^(7/2)
\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx \]
Time = 0.69 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.74, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \cot (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4388 |
| \(\displaystyle \frac {\int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(\sec (c+d x) a+a)^2}dx}{\tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{\tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int \frac {(a-a \sec (c+d x))^2}{\sqrt {\tan (c+d x)}}dx}{a^4 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^4 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \frac {\int \left (\frac {\sec ^2(c+d x) a^2}{\sqrt {\tan (c+d x)}}-\frac {2 \sec (c+d x) a^2}{\sqrt {\tan (c+d x)}}+\frac {a^2}{\sqrt {\tan (c+d x)}}\right )dx}{a^4 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 a^2 \sqrt {\tan (c+d x)}}{d}-\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}}{a^4 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
(-((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) - (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[Tan[c + d*x]]) + (2*a ^2*Sqrt[Tan[c + d*x]])/d)/(a^4*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))
3.3.53.3.1 Defintions of rubi rules used
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 = 10.28 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.99
-1/2/a^2/d*2^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^4*(I*(csc(d*x+c)-cot( d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c) )^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))* ((1-cos(d*x+c))*(-cot(d*x+c)+csc(d*x+c)-1)*(csc(d*x+c)-cot(d*x+c)+1)*csc(d *x+c))^(1/2)-I*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c ))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1 )^(1/2),1/2+1/2*I,1/2*2^(1/2))*((1-cos(d*x+c))*(-cot(d*x+c)+csc(d*x+c)-1)* (csc(d*x+c)-cot(d*x+c)+1)*csc(d*x+c))^(1/2)-6*(csc(d*x+c)-cot(d*x+c)+1)^(1 /2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*Elli pticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*((1-cos(d*x+c))*(-cot(d *x+c)+csc(d*x+c)-1)*(csc(d*x+c)-cot(d*x+c)+1)*csc(d*x+c))^(1/2)+(csc(d*x+c )-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc( d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^( 1/2))*((1-cos(d*x+c))*(-cot(d*x+c)+csc(d*x+c)-1)*(csc(d*x+c)-cot(d*x+c)+1) *csc(d*x+c))^(1/2)+(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d *x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+ c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*((1-cos(d*x+c))*(-cot(d*x+c)+csc(d*x+c) -1)*(csc(d*x+c)-cot(d*x+c)+1)*csc(d*x+c))^(1/2)+4*((1-cos(d*x+c))^3*csc(d* x+c)^3+cot(d*x+c)-csc(d*x+c))^(1/2)*(-cot(d*x+c)+csc(d*x+c)))/(-e/(1-cos(d *x+c))*((1-cos(d*x+c))^2*csc(d*x+c)-sin(d*x+c)))^(7/2)/(1-cos(d*x+c))^3...
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]