\(\int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx\) [239]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

CannotIntegrate(csc(b*x+a)^2*sec(b*x+a)/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2,x]

[Out]

Defer[Int][(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 21.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

[In]

Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2,x]

[Out]

Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {\csc \left (x b +a \right )^{2} \sec \left (x b +a \right )}{\left (d x +c \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(csc(b*x + a)^2*sec(b*x + a)/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 2.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(csc(a + b*x)**2*sec(a + b*x)/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 1008, normalized size of antiderivative = 45.82 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 4.25 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 25.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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