\(\int (c \csc (a+b x))^{7/2} \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 103 \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]

[Out]

-2/5*c*cos(b*x+a)*(c*csc(b*x+a))^(5/2)/b-6/5*c^3*cos(b*x+a)*(c*csc(b*x+a))^(1/2)/b+6/5*c^4*(sin(1/2*a+1/4*Pi+1
/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticE(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))/b/(c*csc(b*x+a))^(1/2)
/sin(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856, 2719} \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b} \]

[In]

Int[(c*Csc[a + b*x])^(7/2),x]

[Out]

(-6*c^3*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/(5*b) - (2*c*Cos[a + b*x]*(c*Csc[a + b*x])^(5/2))/(5*b) - (6*c^4*El
lipticE[(a - Pi/2 + b*x)/2, 2])/(5*b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])

Rule 2719

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

Rule 3853

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

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}+\frac {1}{5} \left (3 c^2\right ) \int (c \csc (a+b x))^{3/2} \, dx \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {1}{5} \left (3 c^4\right ) \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {\left (3 c^4\right ) \int \sqrt {\sin (a+b x)} \, dx}{5 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.65 \[ \int (c \csc (a+b x))^{7/2} \, dx=\frac {(c \csc (a+b x))^{7/2} \left (24 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {7}{2}}(a+b x)-10 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{20 b} \]

[In]

Integrate[(c*Csc[a + b*x])^(7/2),x]

[Out]

((c*Csc[a + b*x])^(7/2)*(24*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(7/2) - 10*Sin[2*(a + b*x)] + 3*S
in[4*(a + b*x)]))/(20*b)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.86 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41

method	result	size
default	\(-\frac {c^{3} \sqrt {c \csc \left (x b +a \right )}\, \left (\left (-6 \cos \left (x b +a \right )-6\right ) \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\left (3 \cos \left (x b +a \right )+3\right ) \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (3+\csc \left (x b +a \right ) \cot \left (x b +a \right )\right )\right ) \sqrt {2}}{5 b}\)	\(248\)

[In]

int((c*csc(b*x+a))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/b*c^3*(c*csc(b*x+a))^(1/2)*((-6*cos(b*x+a)-6)*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(-I*(I+cot(b*x+a)-csc(
b*x+a)))^(1/2)*(I*(csc(b*x+a)-cot(b*x+a)))^(1/2)*EllipticE((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))+(
3*cos(b*x+a)+3)*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(-I*(I+cot(b*x+a)-csc(b*x+a)))^(1/2)*(I*(csc(b*x+a)-cot(b
*x+a)))^(1/2)*EllipticF((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))+2^(1/2)*(3+csc(b*x+a)*cot(b*x+a)))*2
^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.46 \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {3 \, {\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sqrt {2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, {\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sqrt {-2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{3} - 4 \, c^{3} \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\sin \left (b x + a\right )}}}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]

[In]

integrate((c*csc(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

-1/5*(3*(c^3*cos(b*x + a)^2 - c^3)*sqrt(2*I*c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) +
I*sin(b*x + a))) + 3*(c^3*cos(b*x + a)^2 - c^3)*sqrt(-2*I*c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, c
os(b*x + a) - I*sin(b*x + a))) + 2*(3*c^3*cos(b*x + a)^3 - 4*c^3*cos(b*x + a))*sqrt(c/sin(b*x + a)))/(b*cos(b*
x + a)^2 - b)

Sympy [F]

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

[In]

integrate((c*csc(b*x+a))**(7/2),x)

[Out]

Integral((c*csc(a + b*x))**(7/2), x)

Maxima [F]

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

[In]

integrate((c*csc(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^(7/2), x)

Giac [F]

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

[In]

integrate((c*csc(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(7/2), x)

Mupad [F(-1)]

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

[In]

int((c/sin(a + b*x))^(7/2),x)

[Out]

int((c/sin(a + b*x))^(7/2), x)