\(\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx\) [315]
Optimal result
Integrand size = 22, antiderivative size = 22 \[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2},x\right )
\]
[Out]
CannotIntegrate(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,x)
Rubi [N/A]
Not integrable
Time = 0.25 (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 = {} \[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx
\]
[In]
Int[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2,x]
[Out]
Defer[Int][(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2, x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 5.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
\[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx
\]
[In]
Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2,x]
[Out]
Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2, x]
Maple [N/A] (verified)
Not integrable
Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {\csc \left (x b +a \right ) \sec \left (x b +a \right )^{3}}{\left (d x +c \right )^{2}}d x\]
[In]
int(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,x)
[Out]
int(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,x)
Fricas [N/A]
Not integrable
Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59
\[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x }
\]
[In]
integrate(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")
[Out]
integral(csc(b*x + a)*sec(b*x + a)^3/(d^2*x^2 + 2*c*d*x + c^2), x)
Sympy [N/A]
Not integrable
Time = 4.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc {\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx
\]
[In]
integrate(csc(b*x+a)*sec(b*x+a)**3/(d*x+c)**2,x)
[Out]
Integral(csc(a + b*x)*sec(a + b*x)**3/(c + d*x)**2, x)
Maxima [N/A]
Not integrable
Time = 5.43 (sec) , antiderivative size = 2516, normalized size of antiderivative = 114.36
\[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x }
\]
[In]
integrate(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,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^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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)^2 + 4*(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(4*b
*x + 4*a)*sin(2*b*x + 2*a) + 4*(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 + 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))*cos(4*b*x + 4*a) + 4*(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)*co
s(2*b*x + 2*a))*integrate(2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 3*d^2)*sin(2*b*x + 2*a)/(b^2*d^4*x^4 + 4*b^
2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2
+ 4*b^2*c^3*d*x + b^2*c^4)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^
3*d*x + b^2*c^4)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b
^2*c^4)*cos(2*b*x + 2*a)), x) - (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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)^2 +
4*(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(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(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 + 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))*cos(4*b*x + 4*
a) + 4*(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))*integrate(sin(b*x + a)/(d^2
*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^2*x^2 + 2*c*d*x + c^2)*sin(b*x + a)^2 + c^2 + 2
*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + (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(4*b*x + 4*a)^2 + 4*(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(4*
b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)*sin(2*b*x + 2*a) +
4*(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 + 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))
*cos(4*b*x + 4*a) + 4*(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))*integrate(si
n(b*x + a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^2*x^2 + 2*c*d*x + c^2)*sin(b*x +
a)^2 + c^2 - 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + 2*(d*cos(2*b*x + 2*a) + (b*d*x + b*c)*sin(2*b*x
+ 2*a) + d)*sin(4*b*x + 4*a) + 2*d*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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)*sin(2*b*x +
2*a) + 4*(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 + 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))*cos(4*b*x + 4*a) + 4*(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))
Giac [F(-1)]
Timed out. \[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\text {Timed out}
\]
[In]
integrate(csc(b*x+a)*sec(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")
[Out]
Timed out
Mupad [N/A]
Not integrable
Time = 25.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
\[
\int \frac {\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^3\,\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x
\]
[In]
int(1/(cos(a + b*x)^3*sin(a + b*x)*(c + d*x)^2),x)
[Out]
int(1/(cos(a + b*x)^3*sin(a + b*x)*(c + d*x)^2), x)