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

Optimal. Leaf size=46 \[ -\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}-\frac {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]

[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)

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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4380, 2719} \begin {gather*} -\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)}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx &=-\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*}

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Mathematica [A]
time = 0.13, size = 39, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 17.61, size = 106313389, normalized size = 2311160.63

method	result	size
default	\(\text {Expression too large to display}\)	\(106313389\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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