\(\int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx\) [293]

Optimal result

Integrand size = 25, antiderivative size = 155 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {4 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a d} \]

[Out]

-4/21*e^2*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a/d+2/7*e^2*cot(d*x+c)*csc(d*x+c)^2*(e*csc(d*x+c))^(1/2)/a/d-2/7*e^2
*csc(d*x+c)^3*(e*csc(d*x+c))^(1/2)/a/d-4/21*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)/a/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3963, 3957, 2918, 2644, 30, 2647, 2716, 2720} \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {4 e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {e \csc (c+d x)}}{21 a d} \]

[In]

Int[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]

[Out]

(-4*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a*d) + (2*e^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])
/(7*a*d) - (2*e^2*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a*d) + (4*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi
/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 3963

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[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]

Rubi steps \begin{align*} \text {integral}& = \left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x)) \sin ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\left (\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) \sin ^{\frac {5}{2}}(c+d x)} \, dx\right ) \\ & = \frac {\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a}-\frac {\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a} \\ & = \frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {\left (2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{7 a}+\frac {\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{9/2}} \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {\left (2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a} \\ & = -\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {4 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)}}{21 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{5/2} \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left ((2+\cos (c+d x)-2 \cos (2 (c+d x))-\cos (3 (c+d x))) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+2 (4+2 \cos (c+d x)+\cos (2 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sin ^{\frac {5}{2}}(c+d x)}{168 a d} \]

[In]

Integrate[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]

[Out]

-1/168*(Csc[(c + d*x)/2]^2*(e*Csc[c + d*x])^(5/2)*Sec[(c + d*x)/2]^4*((2 + Cos[c + d*x] - 2*Cos[2*(c + d*x)] -
Cos[3*(c + d*x)])*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + 2*(4 + 2*Cos[c + d*x] + Cos[2*(c + d*x)])*Sqrt[Sin[c
+ d*x]])*Sin[c + d*x]^(5/2))/(a*d)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.68 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {\sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (8 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-3 \left (1-\cos \left (d x +c \right )\right )^{6} \csc \left (d x +c \right )^{6}-5 \left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-9 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-7\right ) \csc \left (d x +c \right )^{2}}{84 a d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\)	\(360\)

[In]

int((e*csc(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/84/a/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+c)))^(5/2)*(1-cos(d*x+c))^2*(8*I*(-I*(
I-cot(d*x+c)+csc(d*x+c)))^(1/2)*2^(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))*(-cot(d*x+c)+csc(d*x+c))-3*(1-cos(d*x+c))^6*csc(
d*x+c)^6-5*(1-cos(d*x+c))^4*csc(d*x+c)^4-9*(1-cos(d*x+c))^2*csc(d*x+c)^2-7)/((1-cos(d*x+c))^2*csc(d*x+c)^2+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+csc(d*x+c
)-cot(d*x+c))^(1/2)*csc(d*x+c)^2

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.05 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left ({\left (i \, e^{2} \cos \left (d x + c\right ) + i \, e^{2}\right )} \sqrt {2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (-i \, e^{2} \cos \left (d x + c\right ) - i \, e^{2}\right )} \sqrt {-2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (2 \, e^{2} \cos \left (d x + c\right )^{2} + 2 \, e^{2} \cos \left (d x + c\right ) + 3 \, e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{21 \, {\left (a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]

[In]

integrate((e*csc(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-2/21*((I*e^2*cos(d*x + c) + I*e^2)*sqrt(2*I*e)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*
x + c)) + (-I*e^2*cos(d*x + c) - I*e^2)*sqrt(-2*I*e)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) - I*s
in(d*x + c)) + (2*e^2*cos(d*x + c)^2 + 2*e^2*cos(d*x + c) + 3*e^2)*sqrt(e/sin(d*x + c)))/((a*d*cos(d*x + c) +
a*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((e*csc(d*x+c))**(5/2)/(a+a*sec(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((e*csc(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate((e*csc(d*x + c))^(5/2)/(a*sec(d*x + c) + a), x)

Giac [F]

\[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((e*csc(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((e*csc(d*x + c))^(5/2)/(a*sec(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

int((e/sin(c + d*x))^(5/2)/(a + a/cos(c + d*x)),x)

[Out]

int((cos(c + d*x)*(e/sin(c + d*x))^(5/2))/(a*(cos(c + d*x) + 1)), x)