\(\int \frac {x \sin (a+b x)}{\sqrt {\cos (a+b x)}} \, dx\) [331]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int \frac {x \sin (a+b x)}{\sqrt {\cos (a+b x)}} \, dx=-\frac {2 x \sqrt {\cos (a+b x)}}{b}+\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, 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 = {3525, 2719} \[ \int \frac {x \sin (a+b x)}{\sqrt {\cos (a+b x)}} \, dx=\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2}-\frac {2 x \sqrt {\cos (a+b x)}}{b} \]

[In]

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

[Out]

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

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 3525

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(-x^(m - n
+ 1))*(Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] + Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cos[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 {\cos (a+b x)}}{b}+\frac {2 \int \sqrt {\cos (a+b x)} \, dx}{b} \\ & = -\frac {2 x \sqrt {\cos (a+b x)}}{b}+\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b^2} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {x \sin (a+b x)}{\sqrt {\cos (a+b x)}} \, dx=-\frac {2 \sqrt {\cos (a+b x)} \left (b x-2 \tan (a+b x)+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\tan ^2(a+b x)\right ) \sqrt [4]{\sec ^2(a+b x)} \tan (a+b x)\right )}{b^2} \]

[In]

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

[Out]

(-2*Sqrt[Cos[a + b*x]]*(b*x - 2*Tan[a + b*x] + Hypergeometric2F1[1/4, 1/2, 3/2, -Tan[a + b*x]^2]*(Sec[a + b*x]
^2)^(1/4)*Tan[a + b*x]))/b^2

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 310, normalized size of antiderivative = 9.39

method	result	size
risch	\(-\frac {\left (x b +2 i\right ) \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) \sqrt {2}\, {\mathrm e}^{-i \left (x b +a \right )}}{b^{2} \sqrt {\left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{-i \left (x b +a \right )}}}-\frac {2 i \left (-\frac {2 \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{\sqrt {\left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{i \left (x b +a \right )}}}+\frac {i \sqrt {-i \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (x b +a \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 i \left (x b +a \right )}+{\mathrm e}^{i \left (x b +a \right )}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{i \left (x b +a \right )}}\, {\mathrm e}^{-i \left (x b +a \right )}}{b^{2} \sqrt {\left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{-i \left (x b +a \right )}}}\)	\(310\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*sin(b*x+a)/cos(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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