Integrand size = 25, antiderivative size = 240 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d} \]
-4*a^2*e*(e*csc(d*x+c))^(1/2)/d-2*a^2*e*cos(d*x+c)*(e*csc(d*x+c))^(1/2)/d- 2*a^2*e*sec(d*x+c)*(e*csc(d*x+c))^(1/2)/d-2*a^2*e*arctan(sin(d*x+c)^(1/2)) *(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+2*a^2*e*arctanh(sin(d*x+c)^(1/2)) *(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+5*a^2*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)/d+3*a^2*e*sin(d*x+c)*(e*csc( d*x+c))^(1/2)*tan(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.91 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.81 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \left (3 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}+3 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}-6 \sqrt {\csc (c+d x)}-6 \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+5 \sqrt {-\cot ^2(c+d x)} \sqrt {\csc (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\csc ^2(c+d x)\right )\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right )}{3 d \csc ^{\frac {3}{2}}(c+d x)} \]
(2*a^2*Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*(3*ArcTan[Sqrt[Csc[c + d* x]]]*Sqrt[Cos[c + d*x]^2] + 3*ArcTanh[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x ]^2] - 6*Sqrt[Csc[c + d*x]] - 6*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 5*Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*Hypergeometric2F1[3/4, 3/2, 7/4 , Csc[c + d*x]^2])*Sec[c + d*x]*Sec[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*Csc[c + d*x]^(3/2))
Time = 0.63 (sec) , antiderivative size = 177, 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, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \csc (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2 \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(\sec (c+d x) a+a)^2}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(-\cos (c+d x) a-a)^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \sin \left (c+d x-\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {\sec ^2(c+d x) a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {2 \sec (c+d x) a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {a^2}{\sin ^{\frac {3}{2}}(c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}-\frac {4 a^2}{d \sqrt {\sin (c+d x)}}-\frac {5 a^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {\sin (c+d x)}}+\frac {3 a^2 \sin ^{\frac {3}{2}}(c+d x) \sec (c+d x)}{d}-\frac {2 a^2 \sec (c+d x)}{d \sqrt {\sin (c+d x)}}\right )\) |
e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]*((-2*a^2*ArcTan[Sqrt[Sin[c + d*x ]]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/d - (5*a^2*EllipticE[(c - Pi/ 2 + d*x)/2, 2])/d - (4*a^2)/(d*Sqrt[Sin[c + d*x]]) - (2*a^2*Cos[c + d*x])/ (d*Sqrt[Sin[c + d*x]]) - (2*a^2*Sec[c + d*x])/(d*Sqrt[Sin[c + d*x]]) + (3* a^2*Sec[c + d*x]*Sin[c + d*x]^(3/2))/d)
3.3.88.3.1 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_.)*(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 = 13.20 (sec) , antiderivative size = 1069, normalized size of antiderivative = 4.45
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1069\) |
| default | \(\text {Expression too large to display}\) | \(1231\) |
a^2/d*2^(1/2)*e*(e*csc(d*x+c))^(1/2)*(2*(-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)-(- 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*(-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)*Ellipti cE((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-(-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))- 2^(1/2))+1/2*a^2/d*2^(1/2)*e*(e*csc(d*x+c))^(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)*EllipticE((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))* cos(d*x+c)-3*(-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)+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)*EllipticE((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-3*(-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)))...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.04 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\left [-\frac {2 \, a^{2} \sqrt {-e} e \arctan \left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) + e\right )}}\right ) \cos \left (d x + c\right ) - a^{2} \sqrt {-e} e \cos \left (d x + c\right ) \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) + 10 \, a^{2} \sqrt {2 i \, e} e \cos \left (d x + c\right ) {\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 ) + 10 \, a^{2} \sqrt {-2 i \, e} e \cos \left (d x + c\right ) {\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 ) + 4 \, {\left (5 \, a^{2} e \cos \left (d x + c\right )^{2} + 4 \, a^{2} e \cos \left (d x + c\right ) - a^{2} e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}, \frac {2 \, a^{2} e^{\frac {3}{2}} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) - e\right )}}\right ) \cos \left (d x + c\right ) + a^{2} e^{\frac {3}{2}} \cos \left (d x + c\right ) \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) - 10 \, a^{2} \sqrt {2 i \, e} e \cos \left (d x + c\right ) {\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 ) - 10 \, a^{2} \sqrt {-2 i \, e} e \cos \left (d x + c\right ) {\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 ) - 4 \, {\left (5 \, a^{2} e \cos \left (d x + c\right )^{2} + 4 \, a^{2} e \cos \left (d x + c\right ) - a^{2} e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}\right ] \]
[-1/4*(2*a^2*sqrt(-e)*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) - a^2*sqr t(-e)*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)*s qrt(-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)*sin(d*x + c) + 8)) + 10*a^2*sqrt(2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weier strassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 10*a^2*sqrt(-2*I*e) *e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) + 4*(5*a^2*e*cos(d*x + c)^2 + 4*a^2*e*cos(d*x + c) - a^2*e)*sqrt(e/sin(d*x + c)))/(d*cos(d*x + c)), 1/4*(2*a^2*e^(3/2)*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) + a^2*e^(3/2)*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*c os(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)*sin(d*x + c) + 8)) - 10*a^2*sqrt(2*I*e)*e*cos (d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I *sin(d*x + c))) - 10*a^2*sqrt(-2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 4*(5*a^2*e...
Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2} \,d x \]