\(\int \frac {\csc ^3(a+b x)}{c+d x} \, dx\) [36]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\csc ^3(a+b x)}{c+d x},x\right ) \]

[Out]

Unintegrable(csc(b*x+a)^3/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x)}{c+d x} \, dx \]

[In]

Int[Csc[a + b*x]^3/(c + d*x),x]

[Out]

Defer[Int][Csc[a + b*x]^3/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^3(a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 24.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^3(a+b x)}{c+d x} \, dx \]

[In]

Integrate[Csc[a + b*x]^3/(c + d*x),x]

[Out]

Integrate[Csc[a + b*x]^3/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\csc ^{3}\left (b x +a \right )}{d x +c}d x\]

[In]

int(csc(b*x+a)^3/(d*x+c),x)

[Out]

int(csc(b*x+a)^3/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^3/(d*x+c),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^3/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

integrate(csc(b*x+a)**3/(d*x+c),x)

[Out]

Integral(csc(a + b*x)**3/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 1791, normalized size of antiderivative = 111.94 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^3/(d*x+c),x, algorithm="maxima")

[Out]

(((b*d*x + b*c)*cos(3*b*x + 3*a) + (b*d*x + b*c)*cos(b*x + a) - d*sin(3*b*x + 3*a) + d*sin(b*x + a))*cos(4*b*x
+ 4*a) + (b*d*x + b*c - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - 2*d*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) - 2*((b*d*x
+ b*c)*cos(b*x + a) + d*sin(b*x + a))*cos(2*b*x + 2*a) + (b*d*x + b*c)*cos(b*x + a) + (b^2*d^2*x^2 + 2*b^2*c*
d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*
c^2)*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*
d*x + b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2
+ 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x
+ 4*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*integrate(1/2*(b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2 + 2*d^2)*sin(b*x + a)/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2
*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^
3)*sin(b*x + a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)), x) + (b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2
*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*
b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*
a))*cos(4*b*x + 4*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*integrate(1/2*(b^2*d^2*x^2 +
2*b^2*c*d*x + b^2*c^2 + 2*d^2)*sin(b*x + a)/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^
3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2
*d*x + b^2*c^3)*sin(b*x + a)^2 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)), x)
+ (d*cos(3*b*x + 3*a) - d*cos(b*x + a) + (b*d*x + b*c)*sin(3*b*x + 3*a) + (b*d*x + b*c)*sin(b*x + a))*sin(4*b
*x + 4*a) + (2*d*cos(2*b*x + 2*a) - 2*(b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(3*b*x + 3*a) + 2*(d*cos(b*x + a)
- (b*d*x + b*c)*sin(b*x + a))*sin(2*b*x + 2*a) + d*sin(b*x + a))/(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*
d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)
^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*
b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*
b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^2*
x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))

Giac [N/A]

Not integrable

Time = 2.77 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^3/(d*x+c),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^3/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{c+d x} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/(sin(a + b*x)^3*(c + d*x)),x)

[Out]

int(1/(sin(a + b*x)^3*(c + d*x)), x)