\(\int \frac {x \cos (a+b x)}{\sqrt {\sin (a+b x)}} \, dx\) [347]

Optimal result

Integrand size = 18, antiderivative size = 38 \[ \int \frac {x \cos (a+b x)}{\sqrt {\sin (a+b x)}} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b^2}+\frac {2 x \sqrt {\sin (a+b x)}}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3524, 2719} \[ \int \frac {x \cos (a+b x)}{\sqrt {\sin (a+b x)}} \, dx=\frac {2 x \sqrt {\sin (a+b x)}}{b}-\frac {4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b^2} \]

[In]

Int[(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]],x]

[Out]

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

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 3524

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n +
1)*(Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.57 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.26 \[ \int \frac {x \cos (a+b x)}{\sqrt {\sin (a+b x)}} \, dx=\frac {2 \sqrt {\sin (a+b x)} \left (3 b x-6 \tan \left (\frac {1}{2} (a+b x)\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} \tan \left (\frac {1}{2} (a+b x)\right )\right )}{3 b^2} \]

[In]

Integrate[(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]],x]

[Out]

(2*Sqrt[Sin[a + b*x]]*(3*b*x - 6*Tan[(a + b*x)/2] + 2*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sq
rt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(3*b^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 308, normalized size of antiderivative = 8.11

method	result	size
risch	\(-\frac {i \left (x b +2 i\right ) \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) \sqrt {2}\, {\mathrm e}^{-i \left (x b +a \right )}}{b^{2} \sqrt {-i \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{-i \left (x b +a \right )}}}-\frac {2 \left (\frac {2 i \left (i-i {\mathrm e}^{2 i \left (x b +a \right )}\right )}{\sqrt {{\mathrm e}^{i \left (x b +a \right )} \left (i-i {\mathrm e}^{2 i \left (x b +a \right )}\right )}}-\frac {\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (x b +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (x b +a \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-i {\mathrm e}^{3 i \left (x b +a \right )}+i {\mathrm e}^{i \left (x b +a \right )}}}\right ) \sqrt {2}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{i \left (x b +a \right )}}\, {\mathrm e}^{-i \left (x b +a \right )}}{b^{2} \sqrt {-i \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{-i \left (x b +a \right )}}}\)	\(308\)

[In]

int(x*cos(b*x+a)/sin(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-I*(x*b+2*I)*(exp(I*(b*x+a))^2-1)/b^2*2^(1/2)/(-I*(exp(I*(b*x+a))^2-1)/exp(I*(b*x+a)))^(1/2)/exp(I*(b*x+a))-2/
b^2*(2*I*(I-I*exp(I*(b*x+a))^2)/(exp(I*(b*x+a))*(I-I*exp(I*(b*x+a))^2))^(1/2)-(exp(I*(b*x+a))+1)^(1/2)*(-2*exp
(I*(b*x+a))+2)^(1/2)*(-exp(I*(b*x+a)))^(1/2)/(-I*exp(I*(b*x+a))^3+I*exp(I*(b*x+a)))^(1/2)*(-2*EllipticE((exp(I
*(b*x+a))+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(I*(b*x+a))+1)^(1/2),1/2*2^(1/2))))*2^(1/2)/(-I*(exp(I*(b*x+a))^
2-1)/exp(I*(b*x+a)))^(1/2)*(-I*(exp(I*(b*x+a))^2-1)*exp(I*(b*x+a)))^(1/2)/exp(I*(b*x+a))

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \cos (a+b x)}{\sqrt {\sin (a+b x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)**(1/2),x)

[Out]

Integral(x*cos(a + b*x)/sqrt(sin(a + b*x)), x)

Maxima [F]

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

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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