Integrand size = 25, antiderivative size = 250 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{15 a^2 d} \]
-4/15*e*cos(d*x+c)*(e*csc(d*x+c))^(1/2)/a^2/d+16/45*e*cot(d*x+c)*csc(d*x+c )*(e*csc(d*x+c))^(1/2)/a^2/d-2/9*e*cot(d*x+c)^3*csc(d*x+c)*(e*csc(d*x+c))^ (1/2)/a^2/d-4/5*e*csc(d*x+c)^2*(e*csc(d*x+c))^(1/2)/a^2/d-2/9*e*cot(d*x+c) *csc(d*x+c)^3*(e*csc(d*x+c))^(1/2)/a^2/d+4/9*e*csc(d*x+c)^4*(e*csc(d*x+c)) ^(1/2)/a^2/d+4/15*e*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1 /2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)* sin(d*x+c)^(1/2)/a^2/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.36 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.99 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \sec (c+d x) \left (\frac {16 \sqrt {2} e^{i (c-d x)} \sqrt {\frac {i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left (3-3 e^{2 i (c+d x)}+e^{2 i d x} \left (1+e^{2 i c}\right ) \sqrt {1-e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )\right ) \sec (c+d x)}{d \left (1+e^{2 i c}\right ) \csc ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (24 \cos (d x) \sec (c)+(8+13 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right ) \tan (c+d x)}{d}\right )}{45 a^2 (1+\sec (c+d x))^2} \]
(Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*Sec[c + d*x]*((16*Sqrt[2]*E^(I* (c - d*x))*Sqrt[(I*E^(I*(c + d*x)))/(-1 + E^((2*I)*(c + d*x)))]*(3 - 3*E^( (2*I)*(c + d*x)) + E^((2*I)*d*x)*(1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])*Sec[c + d*x] )/(d*(1 + E^((2*I)*c))*Csc[c + d*x]^(3/2)) - (2*(24*Cos[d*x]*Sec[c] + (8 + 13*Cos[c + d*x])*Sec[(c + d*x)/2]^4)*Tan[c + d*x])/d))/(45*a^2*(1 + Sec[c + d*x])^2)
Time = 0.85 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4366, 3042, 4360, 3042, 3354, 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.
| \(\displaystyle \int \frac {(e \csc (c+d x))^{3/2}}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{(\sec (c+d x) a+a)^2 \sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x)}{(-\cos (c+d x) a-a)^2 \sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x) (a-a \cos (c+d x))^2}{\sin ^{\frac {11}{2}}(c+d x)}dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{11/2}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}+\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {4 a^2}{5 d \sin ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2}{9 d \sin ^{\frac {9}{2}}(c+d x)}-\frac {4 a^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{15 d}-\frac {2 a^2 \cos ^3(c+d x)}{9 d \sin ^{\frac {9}{2}}(c+d x)}+\frac {16 a^2 \cos (c+d x)}{45 d \sin ^{\frac {5}{2}}(c+d x)}-\frac {2 a^2 \cos (c+d x)}{9 d \sin ^{\frac {9}{2}}(c+d x)}-\frac {4 a^2 \cos (c+d x)}{15 d \sqrt {\sin (c+d x)}}\right )}{a^4}\) |
(e*Sqrt[e*Csc[c + d*x]]*((-4*a^2*EllipticE[(c - Pi/2 + d*x)/2, 2])/(15*d) + (4*a^2)/(9*d*Sin[c + d*x]^(9/2)) - (2*a^2*Cos[c + d*x])/(9*d*Sin[c + d*x ]^(9/2)) - (2*a^2*Cos[c + d*x]^3)/(9*d*Sin[c + d*x]^(9/2)) - (4*a^2)/(5*d* Sin[c + d*x]^(5/2)) + (16*a^2*Cos[c + d*x])/(45*d*Sin[c + d*x]^(5/2)) - (4 *a^2*Cos[c + d*x])/(15*d*Sqrt[Sin[c + d*x]]))*Sqrt[Sin[c + d*x]])/a^4
3.4.1.3.1 Defintions of rubi rules used
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_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[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 = 9.25 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.44
1/45/a^2/d*2^(1/2)*(12*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x +c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I -cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*cos(d*x+c)^3-6*(-I*(I-cot(d*x+ c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c) -csc(d*x+c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^( 1/2))*cos(d*x+c)^3+36*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+ c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I- cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*cos(d*x+c)^2-18*(-I*(I-cot(d*x+ c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c) -csc(d*x+c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^( 1/2))*cos(d*x+c)^2+36*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+ c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I- cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*cos(d*x+c)-18*(-I*(I-cot(d*x+c) +csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-c sc(d*x+c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/ 2))*cos(d*x+c)+12*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-c sc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE((-I*(I-cot( d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-6*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1 /2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2 )*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-6*2^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.73 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (3 \, {\left (e \cos \left (d x + c\right )^{2} + 2 \, e \cos \left (d x + c\right ) + e\right )} \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (e \cos \left (d x + c\right )^{2} + 2 \, e \cos \left (d x + c\right ) + e\right )} \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (6 \, e \cos \left (d x + c\right )^{3} + 12 \, e \cos \left (d x + c\right )^{2} + 19 \, e \cos \left (d x + c\right ) + 8 \, e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{45 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
-2/45*(3*(e*cos(d*x + c)^2 + 2*e*cos(d*x + c) + e)*sqrt(2*I*e)*weierstrass Zeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*( e*cos(d*x + c)^2 + 2*e*cos(d*x + c) + e)*sqrt(-2*I*e)*weierstrassZeta(4, 0 , weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) + (6*e*cos(d*x + c)^3 + 12*e*cos(d*x + c)^2 + 19*e*cos(d*x + c) + 8*e)*sqrt(e/sin(d*x + c)))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
\[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Timed out. \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]