\(\int \sqrt {c+d x} \csc (a+b x) \, dx\) [65]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\text {Int}\left (\sqrt {c+d x} \csc (a+b x),x\right ) \]

[Out]

Unintegrable(csc(b*x+a)*(d*x+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int \sqrt {c+d x} \csc (a+b x) \, dx \]

[In]

Int[Sqrt[c + d*x]*Csc[a + b*x],x]

[Out]

Defer[Int][Sqrt[c + d*x]*Csc[a + b*x], x]

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

Mathematica [N/A]

Not integrable

Time = 15.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int \sqrt {c+d x} \csc (a+b x) \, dx \]

[In]

Integrate[Sqrt[c + d*x]*Csc[a + b*x],x]

[Out]

Integrate[Sqrt[c + d*x]*Csc[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

int(csc(b*x+a)*(d*x+c)^(1/2),x)

[Out]

int(csc(b*x+a)*(d*x+c)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int { \sqrt {d x + c} \csc \left (b x + a\right ) \,d x } \]

[In]

integrate(csc(b*x+a)*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x + c)*csc(b*x + a), x)

Sympy [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int \sqrt {c + d x} \csc {\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)*(d*x+c)**(1/2),x)

[Out]

Integral(sqrt(c + d*x)*csc(a + b*x), x)

Maxima [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int { \sqrt {d x + c} \csc \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate(sqrt(d*x + c)*csc(b*x + a), x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int { \sqrt {d x + c} \csc \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate(sqrt(d*x + c)*csc(b*x + a), x)

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \sqrt {c+d x} \csc (a+b x) \, dx=\int \frac {\sqrt {c+d\,x}}{\sin \left (a+b\,x\right )} \,d x \]

[In]

int((c + d*x)^(1/2)/sin(a + b*x),x)

[Out]

int((c + d*x)^(1/2)/sin(a + b*x), x)