\(\int x \sqrt {\cos (a+b x)} \, dx\) [75]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int x \sqrt {\cos (a+b x)} \, dx=\text {Int}\left (x \sqrt {\cos (a+b x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \sqrt {\cos (a+b x)} \, dx=\int x \sqrt {\cos (a+b x)} \, dx \]

[In]

Int[x*Sqrt[Cos[a + b*x]],x]

[Out]

Defer[Int][x*Sqrt[Cos[a + b*x]], x]

Rubi steps \begin{align*} \text {integral}& = \int x \sqrt {\cos (a+b x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 26.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int x \sqrt {\cos (a+b x)} \, dx=\int x \sqrt {\cos (a+b x)} \, dx \]

[In]

Integrate[x*Sqrt[Cos[a + b*x]],x]

[Out]

Integrate[x*Sqrt[Cos[a + b*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83

\[\int x \left (\sqrt {\cos }\left (b x +a \right )\right )d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*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 [N/A]

Not integrable

Time = 6.36 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sqrt {\cos (a+b x)} \, dx=\int x \sqrt {\cos {\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(x*sqrt(cos(a + b*x)), x)

Maxima [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sqrt {\cos (a+b x)} \, dx=\int { x \sqrt {\cos \left (b x + a\right )} \,d x } \]

[In]

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

[Out]

integrate(x*sqrt(cos(b*x + a)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sqrt {\cos (a+b x)} \, dx=\int { x \sqrt {\cos \left (b x + a\right )} \,d x } \]

[In]

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

[Out]

integrate(x*sqrt(cos(b*x + a)), x)

Mupad [N/A]

Not integrable

Time = 13.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sqrt {\cos (a+b x)} \, dx=\int x\,\sqrt {\cos \left (a+b\,x\right )} \,d x \]

[In]

int(x*cos(a + b*x)^(1/2),x)

[Out]

int(x*cos(a + b*x)^(1/2), x)