\(\int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx\) [314]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.90 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 1865, normalized size of antiderivative = 84.77 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 + (2*(b*d*x + b*c)*cos(2*b*x + 2*a) -
d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*d*x + b*c)*cos(2*b*x + 2*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(2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + d^2
)*sin(2*b*x + 2*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(2*b*x + 2*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(2
*b*x + 2*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(2*b*x + 2*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(sin(b*x + a)/((d*x
+ c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x + 2*(d*x + c)*cos(b*x + a) + c), 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(sin(b*x + a)/((d*x + c)*cos(
b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x - 2*(d*x + c)*cos(b*x + a) + c), x) + (d*cos(2*b*x + 2*a) + 2*(b*d
*x + b*c)*sin(2*b*x + 2*a) + d)*sin(4*b*x + 4*a) + d*sin(2*b*x + 2*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)*s
in(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 = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^3\,\sin \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/(cos(a + b*x)^3*sin(a + b*x)*(c + d*x)),x)

[Out]

int(1/(cos(a + b*x)^3*sin(a + b*x)*(c + d*x)), x)