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

Optimal. Leaf size=155 \[ -\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{21 a d} \]

[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

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Rubi [A]  time = 0.22, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3878, 3872, 2839, 2564, 30, 2567, 2636, 2641} \[ -\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{21 a d} \]

Antiderivative was successfully verified.

[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 2564

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 2567

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 2636

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 2641

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

Rule 2839

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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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 3878

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*} \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx &=\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 ) \operatorname {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)} F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a d}\\ \end {align*}

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Mathematica [A]  time = 1.04, size = 131, normalized size = 0.85 \[ -\frac {\sin ^{\frac {5}{2}}(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{5/2} \left (2 \sqrt {\sin (c+d x)} (2 \cos (c+d x)+\cos (2 (c+d x))+4)+(\cos (c+d x)-2 \cos (2 (c+d x))-\cos (3 (c+d x))+2) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{168 a d} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )} e^{2} \csc \left (d x + c\right )^{2}}{a \sec \left (d x + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*csc(d*x + c))*e^2*csc(d*x + c)^2/(a*sec(d*x + c) + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.33, size = 465, normalized size = 3.00 \[ -\frac {\left (-1+\cos \left (d x +c \right )\right )^{3} \left (2 i \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+4 i \cos \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+2 i \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-2 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-2 \cos \left (d x +c \right ) \sqrt {2}-3 \sqrt {2}\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {2}}{21 a d \sin \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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