\(\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx\) [226]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\text {Int}\left (\csc (c+d x) (a+b \sin (c+d x))^n,x\right ) \]

[Out]

Unintegrable(csc(d*x+c)*(a+b*sin(d*x+c))^n,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 19, 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 = {} \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx \]

[In]

Int[Csc[c + d*x]*(a + b*Sin[c + d*x])^n,x]

[Out]

Defer[Int][Csc[c + d*x]*(a + b*Sin[c + d*x])^n, x]

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

Mathematica [N/A]

Not integrable

Time = 2.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx \]

[In]

Integrate[Csc[c + d*x]*(a + b*Sin[c + d*x])^n,x]

[Out]

Integrate[Csc[c + d*x]*(a + b*Sin[c + d*x])^n, x]

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

int(csc(d*x+c)*(a+b*sin(d*x+c))^n,x)

[Out]

int(csc(d*x+c)*(a+b*sin(d*x+c))^n,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ) \,d x } \]

[In]

integrate(csc(d*x+c)*(a+b*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*sin(d*x + c) + a)^n*csc(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 4.77 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{n} \csc {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ) \,d x } \]

[In]

integrate(csc(d*x+c)*(a+b*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^n*csc(d*x + c), x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ) \,d x } \]

[In]

integrate(csc(d*x+c)*(a+b*sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^n*csc(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 6.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \csc (c+d x) (a+b \sin (c+d x))^n \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^n}{\sin \left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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