\(\int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx\) [299]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=-\frac {2 \cos (c+d x)}{21 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{21 a d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \sin ^2(c+d x)}{5 a d e^3 \sqrt {e \csc (c+d x)}} \]

[Out]

-2/21*cos(d*x+c)/a/d/e^3/(e*csc(d*x+c))^(1/2)+2/7*cos(d*x+c)^3/a/d/e^3/(e*csc(d*x+c))^(1/2)+2/5*sin(d*x+c)^2/a
/d/e^3/(e*csc(d*x+c))^(1/2)+4/21*(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))/a/d/e^3/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 149, 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, 2648, 2649, 2720} \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x)}{21 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \sin ^2(c+d x)}{5 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{21 a d e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

Int[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]

[Out]

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

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 2648

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

Rule 2649

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(b*Sin[e +
f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Sin[e + f*x])^
n*(a*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m
, 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}& = \frac {\int \frac {\sin ^{\frac {7}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int \frac {\cos (c+d x) \sin ^{\frac {7}{2}}(c+d x)}{-a-a \cos (c+d x)} \, dx}{e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \cos (c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{a e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {\int \cos ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{a e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{7 a e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\text {Subst}\left (\int x^{3/2} \, dx,x,\sin (c+d x)\right )}{a d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x)}{21 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \sin ^2(c+d x)}{5 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x)}{21 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{21 a d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \sin ^2(c+d x)}{5 a d e^3 \sqrt {e \csc (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\frac {\sqrt {e \csc (c+d x)} \left (80 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}+126 \sin (c+d x)+10 \sin (2 (c+d x))-42 \sin (3 (c+d x))+15 \sin (4 (c+d x))\right )}{420 a d e^4} \]

[In]

Integrate[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]

[Out]

(Sqrt[e*Csc[c + d*x]]*(80*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + 126*Sin[c + d*x] + 10*Sin[2
*(c + d*x)] - 42*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)]))/(420*a*d*e^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 10.01 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {\sqrt {2}\, \left (-10 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \cos \left (d x +c \right )+15 \sqrt {2}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )-10 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}-21 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sin \left (d x +c \right )-5 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}+21 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )^{3}}{105 a d \,e^{3} \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\)	\(316\)

[In]

int(1/(e*csc(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/105/a/d*2^(1/2)*(-10*I*(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)+15*2^(1/2)*cos(
d*x+c)^3*sin(d*x+c)-10*I*(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))-21*cos(d*x+c)^2*2^(1/2)*si
n(d*x+c)-5*cos(d*x+c)*sin(d*x+c)*2^(1/2)+21*2^(1/2)*sin(d*x+c))/e^3/(e*csc(d*x+c))^(1/2)/(cos(d*x+c)-1)^2/(cos
(d*x+c)+1)^2*sin(d*x+c)^3

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\frac {2 \, {\left ({\left (15 \, \cos \left (d x + c\right )^{3} - 21 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 21\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 5 i \, \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{105 \, a d e^{4}} \]

[In]

integrate(1/(e*csc(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

2/105*((15*cos(d*x + c)^3 - 21*cos(d*x + c)^2 - 5*cos(d*x + c) + 21)*sqrt(e/sin(d*x + c))*sin(d*x + c) + 5*I*s
qrt(2*I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(-2*I*e)*weierstrassPInverse(4,
0, cos(d*x + c) - I*sin(d*x + c)))/(a*d*e^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*csc(d*x+c))**(7/2)/(a+a*sec(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*csc(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate(1/((e*csc(d*x + c))^(7/2)*(a*sec(d*x + c) + a)), x)

Giac [F]

\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*csc(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((e*csc(d*x + c))^(7/2)*(a*sec(d*x + c) + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{7/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

int(1/((a + a/cos(c + d*x))*(e/sin(c + d*x))^(7/2)),x)

[Out]

int(cos(c + d*x)/(a*(e/sin(c + d*x))^(7/2)*(cos(c + d*x) + 1)), x)