\(\int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx\) [113]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {14 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}} \]

[Out]

14/15*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))/b-14/45*cos(2*b*x+2*a
)/b/sin(2*b*x+2*a)^(5/2)-1/9*csc(b*x+a)^2/b/sin(2*b*x+2*a)^(5/2)-14/15*cos(2*b*x+2*a)/b/sin(2*b*x+2*a)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4385, 2716, 2719} \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {14 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)} \]

[In]

Int[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2),x]

[Out]

(-14*EllipticE[a - Pi/4 + b*x, 2])/(15*b) - (14*Cos[2*a + 2*b*x])/(45*b*Sin[2*a + 2*b*x]^(5/2)) - Csc[a + b*x]
^2/(9*b*Sin[2*a + 2*b*x]^(5/2)) - (14*Cos[2*a + 2*b*x])/(15*b*Sqrt[Sin[2*a + 2*b*x]])

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 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 4385

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Sin[a + b*
x])^m*((g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1))), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a
+ b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b,
2] &&  !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {14}{9} \int \frac {1}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx \\ & = -\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {14}{15} \int \frac {1}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx \\ & = -\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}}-\frac {14}{15} \int \sqrt {\sin (2 a+2 b x)} \, dx \\ & = -\frac {14 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.80 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {336 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\frac {(-9+98 \cos (2 (a+b x))-28 \cos (4 (a+b x))-42 \cos (6 (a+b x))+21 \cos (8 (a+b x))) \csc ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 (a+b x))}}{360 b} \]

[In]

Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2),x]

[Out]

-1/360*(336*EllipticE[a - Pi/4 + b*x, 2] + ((-9 + 98*Cos[2*(a + b*x)] - 28*Cos[4*(a + b*x)] - 42*Cos[6*(a + b*
x)] + 21*Cos[8*(a + b*x)])*Csc[a + b*x]^2)/Sin[2*(a + b*x)]^(5/2))/b

Maple [F(-1)]

Timed out.

\[\int \frac {\csc \left (x b +a \right )^{2}}{\sin \left (2 x b +2 a \right )^{\frac {7}{2}}}d x\]

[In]

int(csc(b*x+a)^2/sin(2*b*x+2*a)^(7/2),x)

[Out]

int(csc(b*x+a)^2/sin(2*b*x+2*a)^(7/2),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 3.26 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx=-\frac {168 \, \sqrt {2 i} {\left (i \, \cos \left (b x + a\right )^{7} - 2 i \, \cos \left (b x + a\right )^{5} + i \, \cos \left (b x + a\right )^{3}\right )} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {-2 i} {\left (-i \, \cos \left (b x + a\right )^{7} + 2 i \, \cos \left (b x + a\right )^{5} - i \, \cos \left (b x + a\right )^{3}\right )} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {2 i} {\left (-i \, \cos \left (b x + a\right )^{7} + 2 i \, \cos \left (b x + a\right )^{5} - i \, \cos \left (b x + a\right )^{3}\right )} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 168 \, \sqrt {-2 i} {\left (i \, \cos \left (b x + a\right )^{7} - 2 i \, \cos \left (b x + a\right )^{5} + i \, \cos \left (b x + a\right )^{3}\right )} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + \sqrt {2} {\left (336 \, \cos \left (b x + a\right )^{8} - 840 \, \cos \left (b x + a\right )^{6} + 644 \, \cos \left (b x + a\right )^{4} - 126 \, \cos \left (b x + a\right )^{2} - 9\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{360 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )} \]

[In]

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

[Out]

-1/360*(168*sqrt(2*I)*(I*cos(b*x + a)^7 - 2*I*cos(b*x + a)^5 + I*cos(b*x + a)^3)*elliptic_e(arcsin(cos(b*x + a
) + I*sin(b*x + a)), -1)*sin(b*x + a) + 168*sqrt(-2*I)*(-I*cos(b*x + a)^7 + 2*I*cos(b*x + a)^5 - I*cos(b*x + a
)^3)*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + 168*sqrt(2*I)*(-I*cos(b*x + a)^7 + 2
*I*cos(b*x + a)^5 - I*cos(b*x + a)^3)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + 168
*sqrt(-2*I)*(I*cos(b*x + a)^7 - 2*I*cos(b*x + a)^5 + I*cos(b*x + a)^3)*elliptic_f(arcsin(cos(b*x + a) - I*sin(
b*x + a)), -1)*sin(b*x + a) + sqrt(2)*(336*cos(b*x + a)^8 - 840*cos(b*x + a)^6 + 644*cos(b*x + a)^4 - 126*cos(
b*x + a)^2 - 9)*sqrt(cos(b*x + a)*sin(b*x + a)))/((b*cos(b*x + a)^7 - 2*b*cos(b*x + a)^5 + b*cos(b*x + a)^3)*s
in(b*x + a))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^(7/2)),x)

[Out]

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^(7/2)), x)