\(\int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx\) [66]

Optimal result

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

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

[In]

Int[Csc[a + b*x]/Sqrt[c + d*x],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.48 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\csc {\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx=\int { \frac {\csc \left (b x + a\right )}{\sqrt {d x + c}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx=\int { \frac {\csc \left (b x + a\right )}{\sqrt {d x + c}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc (a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {1}{\sin \left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]

[In]

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

[Out]

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