Integrand size = 25, antiderivative size = 413 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \]
2*cot(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)-12/5*cos(d*x+c)*cot(d*x+c)/a^2/d/( e*cot(d*x+c))^(1/2)-4/5*cot(d*x+c)^3/a^2/d/(e*cot(d*x+c))^(1/2)+4/5*cot(d* x+c)^2*csc(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)+12/5*cos(d*x+c)*(sin(c+1/4*Pi +d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))/a^2/ d/(e*cot(d*x+c))^(1/2)/sin(2*d*x+2*c)^(1/2)+1/2*arctan(-1+2^(1/2)*tan(d*x+ c)^(1/2))/a^2/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)+1/2*arctan(1 +2^(1/2)*tan(d*x+c)^(1/2))/a^2/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^( 1/2)+1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a^2/d*2^(1/2)/(e*cot(d* x+c))^(1/2)/tan(d*x+c)^(1/2)-1/4*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)) /a^2/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx \]
Time = 0.81 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.76, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 \sqrt {e \cot (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 4388 |
| \(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{(\sec (c+d x) a+a)^2}dx}{\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )}}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int \frac {(a-a \sec (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)}dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \frac {\int \left (\frac {\sec ^2(c+d x) a^2}{\tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \sec (c+d x) a^2}{\tan ^{\frac {7}{2}}(c+d x)}+\frac {a^2}{\tan ^{\frac {7}{2}}(c+d x)}\right )dx}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {a^2 \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {4 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2}{d \sqrt {\tan (c+d x)}}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {12 a^2 \cos (c+d x)}{5 d \sqrt {\tan (c+d x)}}+\frac {4 a^2 \sec (c+d x)}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {12 a^2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{5 d \sqrt {\sin (2 c+2 d x)}}}{a^4 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
(-((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)/(5*d*Tan[c + d*x]^(5 /2)) + (4*a^2*Sec[c + d*x])/(5*d*Tan[c + d*x]^(5/2)) + (2*a^2)/(d*Sqrt[Tan [c + d*x]]) - (12*a^2*Cos[c + d*x])/(5*d*Sqrt[Tan[c + d*x]]) - (12*a^2*Cos [c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[Tan[c + d*x]])/(5*d*Sqrt[Sin[2 *c + 2*d*x]]))/(a^4*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])
3.3.50.3.1 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 = 8.55 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \left (5 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-5 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+24 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-12 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-5 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 \left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-2 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}\right )}{10 a^{2} d \sqrt {-\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )}}\) | \(705\) |
1/10/a^2/d*2^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(5*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))- 5*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)*(c ot(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))+24*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*c ot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticE((csc(d*x+c)-cot(d *x+c)+1)^(1/2),1/2*2^(1/2))-12*(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)*EllipticF((csc(d*x+ c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-5*(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))-5*(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))+2* (1-cos(d*x+c))^4*csc(d*x+c)^4-2*(1-cos(d*x+c))^2*csc(d*x+c)^2)/(-e/(1-cos( d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)-sin(d*x+c)))^(1/2)/((1-cos(d*x+c))*(( 1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)/((1-cos(d*x+c))^3*csc(d* x+c)^3+cot(d*x+c)-csc(d*x+c))^(1/2)
Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )} + 2 \sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a^{2}} \]
Integral(1/(sqrt(e*cot(c + d*x))*sec(c + d*x)**2 + 2*sqrt(e*cot(c + d*x))* sec(c + d*x) + sqrt(e*cot(c + d*x))), x)/a**2
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]