Integrand size = 25, antiderivative size = 222 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \] Output:
-4*a^2/d/e/(e*csc(d*x+c))^(1/2)-2/3*a^2*cos(d*x+c)/d/e/(e*csc(d*x+c))^(1/2 )+a^2*sec(d*x+c)/d/e/(e*csc(d*x+c))^(1/2)+2*a^2*arctan(sin(d*x+c)^(1/2))/d /e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+2*a^2*arctanh(sin(d*x+c)^(1/2))/d /e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)-1/3*a^2*InverseJacobiAM(1/2*c-1/4 *Pi+1/2*d*x,2^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.90 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.74 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {e \csc (c+d x)} \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (3-6 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},\csc ^2(c+d x)\right )+3 \sqrt {-\cot ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\csc ^2(c+d x)\right )+\sqrt {-\cot ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {1}{4},\csc ^2(c+d x)\right ) \sin ^2(c+d x)\right ) \tan (c+d x)}{3 d e^2} \] Input:
Integrate[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]
Output:
(2*a^2*Cos[(c + d*x)/2]^4*Sqrt[e*Csc[c + d*x]]*Sec[ArcCsc[Csc[c + d*x]]/2] ^4*(3 - 6*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[-1/4, 1, 3/4, Csc[c + d*x ]^2] + 3*Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Csc[c + d* x]^2] + Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[-3/4, 3/2, 1/4, Csc[c + d* x]^2]*Sin[c + d*x]^2)*Tan[c + d*x])/(3*d*e^2)
Time = 0.66 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4366, 3042, 4360, 3042, 3352, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^2}{(e \csc (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^2 \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int (-\cos (c+d x) a-a)^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (\sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) a^2+2 \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x) a^2+\sin ^{\frac {3}{2}}(c+d x) a^2\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}-\frac {4 a^2 \sqrt {\sin (c+d x)}}{d}-\frac {a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 a^2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}+\frac {a^2 \sqrt {\sin (c+d x)} \sec (c+d x)}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
Input:
Int[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]
Output:
((2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]] )/d - (a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) - (4*a^2*Sqrt[Sin[c + d *x]])/d - (2*a^2*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(3*d) + (a^2*Sec[c + d*x ]*Sqrt[Sin[c + d*x]])/d)/(e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.10
| method | result | size |
| parts | \(\frac {a^{2} \sqrt {2}\, \left (i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )-\sqrt {2}\, \cos \left (d x +c \right )\right )}{3 d e \sqrt {e \csc \left (d x +c \right )}}-\frac {a^{2} \sqrt {2}\, \left (i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )-\sqrt {2}\, \sec \left (d x +c \right )\right )}{2 d \sqrt {e \csc \left (d x +c \right )}\, e}+\frac {a^{2} \left (-2 \cos \left (d x +c \right )+2\right ) \left (\arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )-\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )+2 \cos \left (d x +c \right )-2\right )}{d e \sqrt {e \csc \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(467\) |
| default | \(-\frac {a^{2} \sqrt {2}\, \left (\sqrt {2}\, \left (2 \cos \left (d x +c \right )+12-3 \sec \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \left (-6 \cot \left (d x +c \right )-6 \csc \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (6 \cot \left (d x +c \right )+6 \csc \left (d x +c \right )\right )+i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \left (-6 \cot \left (d x +c \right )-6 \csc \left (d x +c \right )\right )+i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \left (-6 \cot \left (d x +c \right )-6 \csc \left (d x +c \right )\right )+i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (13 \cot \left (d x +c \right )+13 \csc \left (d x +c \right )\right )\right )}{6 d \sqrt {e \csc \left (d x +c \right )}\, e}\) | \(616\) |
Input:
int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*a^2/d*2^(1/2)/e/(e*csc(d*x+c))^(1/2)*(I*(1+I*cot(d*x+c)-I*csc(d*x+c))^ (1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1 /2)*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))*(csc(d*x+c) +cot(d*x+c))-2^(1/2)*cos(d*x+c))-1/2*a^2/d*2^(1/2)/(e*csc(d*x+c))^(1/2)/e* (I*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2) *(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c ))^(1/2),1/2*2^(1/2))*(csc(d*x+c)+cot(d*x+c))-2^(1/2)*sec(d*x+c))+a^2/d*(- 2*cos(d*x+c)+2)*(arctan(sin(d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1 +cos(d*x+c)))*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*sin(d*x+c)-arctanh(sin(d *x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))*(sin(d*x+c)/(1+ cos(d*x+c))^2)^(1/2)*sin(d*x+c)+2*cos(d*x+c)-2)/e/(e*csc(d*x+c))^(1/2)/(-1 +cos(d*x+c))^2
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.26 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[-1/12*(6*a^2*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*s qrt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e))*cos(d*x + c) + 3*a^2*sq rt(-e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(cos(d *x + c)^4 - 9*cos(d*x + c)^2 + (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sq rt(-e)*sqrt(e/sin(d*x + c)) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 7 2*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) - 2*I*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassPInverse(4, 0, cos (d*x + c) + I*sin(d*x + c)) + 2*I*a^2*sqrt(-2*I*e)*cos(d*x + c)*weierstras sPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 4*(2*a^2*cos(d*x + c)^2 + 12*a^2*cos(d*x + c) - 3*a^2)*sqrt(e/sin(d*x + c))*sin(d*x + c))/(d*e^2*co s(d*x + c)), -1/12*(6*a^2*sqrt(e)*arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) - e))*cos(d*x + c) - 3*a^2*sqrt(e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 - (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt(e)*sqrt(e/sin(d*x + c)) - 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*si n(d*x + c) + 8)) - 2*I*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + 2*I*a^2*sqrt(-2*I*e)*cos(d*x + c)*wei erstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 4*(2*a^2*cos(d*x + c)^2 + 12*a^2*cos(d*x + c) - 3*a^2)*sqrt(e/sin(d*x + c))*sin(d*x + c))...
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=a^{2} \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \] Input:
integrate((a+a*sec(d*x+c))**2/(e*csc(d*x+c))**(3/2),x)
Output:
a**2*(Integral((e*csc(c + d*x))**(-3/2), x) + Integral(2*sec(c + d*x)/(e*c sc(c + d*x))**(3/2), x) + Integral(sec(c + d*x)**2/(e*csc(c + d*x))**(3/2) , x))
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)^2/(e*csc(d*x + c))^(3/2), x)
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^2/(e*csc(d*x + c))^(3/2), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(3/2),x)
Output:
int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(3/2), x)
\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx=\frac {\sqrt {e}\, a^{2} \left (\int \frac {\sqrt {\csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{2}}d x +\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\csc \left (d x +c \right )^{2}}d x +2 \left (\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{\csc \left (d x +c \right )^{2}}d x \right )\right )}{e^{2}} \] Input:
int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x)
Output:
(sqrt(e)*a**2*(int(sqrt(csc(c + d*x))/csc(c + d*x)**2,x) + int((sqrt(csc(c + d*x))*sec(c + d*x)**2)/csc(c + d*x)**2,x) + 2*int((sqrt(csc(c + d*x))*s ec(c + d*x))/csc(c + d*x)**2,x)))/e**2