\(\int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\) [174]

Optimal result

Integrand size = 22, antiderivative size = 46 \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=-\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}-\frac {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]

[Out]

1/2*(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-cos(b*x+a)^2/b/sin(2*
b*x+2*a)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4380, 2719} \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=-\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b}-\frac {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]

[In]

Int[Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2),x]

[Out]

-1/2*EllipticE[a - Pi/4 + b*x, 2]/b - Cos[a + b*x]^2/(b*Sqrt[Sin[2*a + 2*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 4380

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Cos[a + b*
x])^m*((g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1))), x] + Dist[e^2*((m + 2*p + 2)/(4*g^2*(p + 1))), Int[(e*Cos[a
+ b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d
/b, 2] &&  !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && Inte
gersQ[2*m, 2*p]

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=-\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\cot (a+b x) \sqrt {\sin (2 (a+b x))}}{2 b} \]

[In]

Integrate[Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2),x]

[Out]

-1/2*(EllipticE[a - Pi/4 + b*x, 2] + Cot[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 11.49 (sec) , antiderivative size = 107396897, normalized size of antiderivative = 2334715.15

method	result	size
default	\(\text {Expression too large to display}\)	\(107396897\)

[In]

int(cos(b*x+a)^2/sin(2*b*x+2*a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.39 \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\frac {-i \, \sqrt {2 i} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + i \, \sqrt {-2 i} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + i \, \sqrt {2 i} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {-2 i} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - 2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right )}{4 \, b \sin \left (b x + a\right )} \]

[In]

integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^(3/2),x, algorithm="fricas")

[Out]

1/4*(-I*sqrt(2*I)*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + I*sqrt(-2*I)*elliptic_e
(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + I*sqrt(2*I)*elliptic_f(arcsin(cos(b*x + a) + I*sin(
b*x + a)), -1)*sin(b*x + a) - I*sqrt(-2*I)*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a)
- 2*sqrt(2)*sqrt(cos(b*x + a)*sin(b*x + a))*cos(b*x + a))/(b*sin(b*x + a))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(b*x+a)**2/sin(2*b*x+2*a)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^(3/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^2/sin(2*b*x + 2*a)^(3/2), x)

Giac [F]

\[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^(3/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^2/sin(2*b*x + 2*a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \]

[In]

int(cos(a + b*x)^2/sin(2*a + 2*b*x)^(3/2),x)

[Out]

int(cos(a + b*x)^2/sin(2*a + 2*b*x)^(3/2), x)