\(\int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [274]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

Int[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]

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

Mathematica [N/A]

Not integrable

Time = 14.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

Integrate[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]

[Out]

Integrate[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

\[\int \frac {\sec \left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}d x\]

[In]

int(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x)

[Out]

int(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(sec(d*x + c)/(a*f^2*x^2 + 2*a*e*f*x + a*e^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*x + c)), x)

Sympy [N/A]

Not integrable

Time = 8.53 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sec {\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]

[In]

integrate(sec(d*x+c)/(f*x+e)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) + 2*e*f*x + f**2*x**2*sin(c + d*x) + f*
*2*x**2), x)/a

Maxima [N/A]

Not integrable

Time = 5.17 (sec) , antiderivative size = 2018, normalized size of antiderivative = 77.62 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-(2*(d*f*x + d*e)*cos(d*x + c)^2 + 2*(d*f*x + d*e)*sin(d*x + c)^2 - (2*f*cos(d*x + c) + (d*f*x + d*e)*sin(d*x
+ c))*cos(2*d*x + 2*c) - 2*f*cos(d*x + c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 +
(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a
*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*
e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*
d^2*e^3)*sin(2*d*x + 2*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)
^2 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2
+ 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^
2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*integrate(1/2*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 + 12*f^2)*cos(d*x + c)
/(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4 + (a*d^2*f^4*x^4 + 4*a
*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4)*cos(d*x + c)^2 + (a*d^2*f^4*x^4 + 4*a*d^2*
e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4)*sin(d*x + c)^2 + 2*(a*d^2*f^4*x^4 + 4*a*d^2*e*f
^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4)*sin(d*x + c)), x) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2
*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d
*x + 2*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + 4*(a*d^2*f^
3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^3*x^3 + 3*a*
d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d
^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)^2 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 +
2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*integrate(1/2*cos(d*x + c)/(a*f^2*x^2
+ 2*a*e*f*x + a*e^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*cos(d*x + c)^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*
x + c)^2 - 2*(a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*x + c)), x) + ((d*f*x + d*e)*cos(d*x + c) - 2*f*sin(d*x + c
) - 2*f)*sin(2*d*x + 2*c) + (d*f*x + d*e)*sin(d*x + c))/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x +
a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c)^2 + 4*(a*d^2*f
^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^
2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*
e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*
sin(d*x + c)^2 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + 2*(a*d^2*f^3*x^3 + 3*a*d
^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*
x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))

Giac [N/A]

Not integrable

Time = 10.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sec(d*x+c)/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)/((f*x + e)^2*(a*sin(d*x + c) + a)), x)

Mupad [N/A]

Not integrable

Time = 3.90 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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