Integrand size = 25, antiderivative size = 270 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {2 a^2 e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {7 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d} \]
-2/3*a^2*e^2*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/d-4/3*a^2*e^2*csc(d*x+c)*(e*c sc(d*x+c))^(1/2)/d-2/3*a^2*e^2*csc(d*x+c)*sec(d*x+c)*(e*csc(d*x+c))^(1/2)/ d+2*a^2*e^2*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^2*arctanh(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/ 2)/d-7/3*a^2*e^2*(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)*EllipticF(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+5/3*a^2*e^2*(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 = 13.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.72 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 e^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {e \csc (c+d x)} \left (-7+6 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}-6 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+4 \csc ^2(c+d x)+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x)+7 \sqrt {-\cot ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\csc ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \tan (c+d x)}{3 d} \]
-1/3*(a^2*e^2*Cos[(c + d*x)/2]^4*Sqrt[e*Csc[c + d*x]]*(-7 + 6*ArcTan[Sqrt[ Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] - 6*ArcTanh[Sqrt[Cs c[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 4*Csc[c + d*x]^2 + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2 + 7*Sqrt[-Cot[c + d*x]^2]*Hypergeome tric2F1[1/4, 1/2, 5/4, Csc[c + d*x]^2])*Sec[ArcCsc[Csc[c + d*x]]/2]^4*Tan[ c + d*x])/d
Time = 0.65 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.70, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (e \csc (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2 \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(\sec (c+d x) a+a)^2}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(-\cos (c+d x) a-a)^2 \sec ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \sin \left (c+d x-\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {\sec ^2(c+d x) a^2}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {2 \sec (c+d x) a^2}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {a^2}{\sin ^{\frac {5}{2}}(c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}-\frac {4 a^2}{3 d \sin ^{\frac {3}{2}}(c+d x)}+\frac {7 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 a^2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2 \sec (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}+\frac {5 a^2 \sqrt {\sin (c+d x)} \sec (c+d x)}{3 d}\right )\) |
e^2*Sqrt[e*Csc[c + d*x]]*((2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/d + (2*a^2*Ar cTanh[Sqrt[Sin[c + d*x]]])/d + (7*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3 *d) - (4*a^2)/(3*d*Sin[c + d*x]^(3/2)) - (2*a^2*Cos[c + d*x])/(3*d*Sin[c + d*x]^(3/2)) - (2*a^2*Sec[c + d*x])/(3*d*Sin[c + d*x]^(3/2)) + (5*a^2*Sec[ c + d*x]*Sqrt[Sin[c + d*x]])/(3*d))*Sqrt[Sin[c + d*x]]
3.3.87.3.1 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 = 11.70 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.25
| method | result | size |
| default | \(\frac {a^{2} e^{2} \sqrt {2}\, \left (6 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+6 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-5 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-7 \sqrt {2}\, \cos \left (d x +c \right )+3 \sqrt {2}\right ) \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \csc \left (d x +c \right )}\, \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\) | \(607\) |
| parts | \(\text {Expression too large to display}\) | \(1052\) |
1/6*a^2*e^2/d*2^(1/2)*(6*I*sin(d*x+c)*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)*EllipticPi((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2+1/2*I,1/2*2^ (1/2))+6*I*sin(d*x+c)*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)*Ellipt icPi((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2-1/2*I,1/2*2^(1/2))-5*I*sin(d *x+c)*cos(d*x+c)*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-cs c(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*cos(d*x+c)*sin(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)*EllipticPi((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2 +1/2*I,1/2*2^(1/2))-6*cos(d*x+c)*sin(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)*EllipticPi((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2-1/2*I,1/2*2^(1/2) )-7*2^(1/2)*cos(d*x+c)+3*2^(1/2))*(cos(d*x+c)+1)*(e*csc(d*x+c))^(1/2)*sec( d*x+c)*csc(d*x+c)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 806, normalized size of antiderivative = 2.99 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Too large to display} \]
[-1/12*(6*a^2*sqrt(-e)*e^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)*sin(d* x + c) - 3*a^2*sqrt(-e)*e^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*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)*sin(d*x + c) + 8))*sin(d*x + c) + 14*I*a^2*sqrt(2*I*e)*e^2*cos (d*x + c)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 14*I*a^2*sqrt(-2*I*e)*e^2*cos(d*x + c)*sin(d*x + c)*weierstrassPIn verse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 4*(7*a^2*e^2*cos(d*x + c)^2 + 4*a^2*e^2*cos(d*x + c) - 3*a^2*e^2)*sqrt(e/sin(d*x + c)))/(d*cos(d*x + c) *sin(d*x + c)), -1/12*(6*a^2*e^(5/2)*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 )*sin(d*x + c) - 3*a^2*e^(5/2)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*c os(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)*sin(d*x + c) + 8))*sin(d*x + c) + 14*I*a^2*sqrt(2*I*e)*e^2*c os(d*x + c)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d* x + c)) - 14*I*a^2*sqrt(-2*I*e)*e^2*cos(d*x + c)*sin(d*x + c)*weierstra...
Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2} \,d x \]