\(\int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [280]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 17.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(sec(d*x + c)^2/(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 = 8.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]

[In]

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

[Out]

Integral(sec(c + d*x)**2/(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 = 21.58 (sec) , antiderivative size = 4597, normalized size of antiderivative = 164.18 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-1/3*(12*f^2*cos(2*d*x + 2*c)*cos(d*x + c) - 4*(d*f^2*x + d*e*f)*cos(3*d*x + 3*c)^2 + 6*f^2*cos(d*x + c) - 4*(
d*f^2*x + d*e*f)*cos(d*x + c)^2 - 4*(d*f^2*x + d*e*f)*sin(3*d*x + 3*c)^2 - 4*(d*f^2*x + d*e*f)*sin(d*x + c)^2
+ 2*(3*f^2*cos(3*d*x + 3*c) - 3*f^2*sin(2*d*x + 2*c) + (4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + 3*f^2)*cos(d
*x + c) + (d*f^2*x + d*e*f)*sin(3*d*x + 3*c) + (d*f^2*x + d*e*f)*sin(d*x + c))*cos(4*d*x + 4*c) + 2*(4*d^2*f^2
*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + 6*f^2*cos(2*d*x + 2*c) + 3*f^2 - 4*(d*f^2*x + d*e*f)*cos(d*x + c) + 8*(d^2*f^
2*x^2 + 2*d^2*e*f*x + d^2*e^2)*sin(d*x + c))*cos(3*d*x + 3*c) + 3*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3
*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4 + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^
3*e^3*f*x + a*d^3*e^4)*cos(4*d*x + 4*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d
^3*e^3*f*x + a*d^3*e^4)*cos(3*d*x + 3*c)^2 + 8*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*
d^3*e^3*f*x + a*d^3*e^4)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^
2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*cos(d*x + c)^2 + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2
+ 4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(4*d*x + 4*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^
2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(3*d*x + 3*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x
^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(d*x + c)^2 - 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2
+ 4*a*d^3*e^3*f*x + a*d^3*e^4 + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x +
a*d^3*e^4)*sin(3*d*x + 3*c) + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x +
a*d^3*e^4)*sin(d*x + c))*cos(4*d*x + 4*c) + 4*((a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*
d^3*e^3*f*x + a*d^3*e^4)*cos(3*d*x + 3*c) + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3
*e^3*f*x + a*d^3*e^4)*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*
x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4 + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f
*x + a*d^3*e^4)*sin(d*x + c))*sin(3*d*x + 3*c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 +
4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(d*x + c))*integrate(1/3*(5*d^2*f^3*x^2 + 10*d^2*e*f^2*x + 5*d^2*e^2*f + 24*f^
3)*cos(d*x + c)/(a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*x^3 + 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4
*f*x + a*d^3*e^5 + (a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*x^3 + 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*
e^4*f*x + a*d^3*e^5)*cos(d*x + c)^2 + (a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*x^3 + 10*a*d^3*e^3
*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5)*sin(d*x + c)^2 + 2*(a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f
^3*x^3 + 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5)*sin(d*x + c)), x) - 3*(a*d^3*f^5*x^4 + 4*a*d^3*e*
f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f + (a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3
*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(4*d*x + 4*c)^2 + 4*(a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*
a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(3*d*x + 3*c)^2 + 8*(a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3
+ 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^5*x^4 + 4
*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(d*x + c)^2 + (a*d^3*f^5*x^4 + 4*
a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(4*d*x + 4*c)^2 + 4*(a*d^3*f^5*x^4
+ 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(3*d*x + 3*c)^2 + 4*(a*d^3*f^
5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(d*x + c)^2 - 2*(a*d^3*f
^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f + 2*(a*d^3*f^5*x^4 + 4*a*d^
3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(3*d*x + 3*c) + 2*(a*d^3*f^5*x^4 + 4*a
*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(d*x + c))*cos(4*d*x + 4*c) + 4*((a
*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(3*d*x + 3*c) + (
a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*cos(d*x + c))*sin(4
*d*x + 4*c) + 4*(a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f + 2
*(a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*sin(d*x + c))*sin
(3*d*x + 3*c) + 4*(a*d^3*f^5*x^4 + 4*a*d^3*e*f^4*x^3 + 6*a*d^3*e^2*f^3*x^2 + 4*a*d^3*e^3*f^2*x + a*d^3*e^4*f)*
sin(d*x + c))*integrate(cos(d*x + c)/(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3 + (a*d*f^3*x^3 +
3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*cos(d*x + c)^2 + (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x +
a*d*e^3)*sin(d*x + c)^2 - 2*(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*sin(d*x + c)), x) + 2*(2
*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^2*e^2 + 3*f^2*cos(2*d*x + 2*c) + 3*f^2*sin(3*d*x + 3*c) + 3*f^2 - (d*f^2*x +
d*e*f)*cos(3*d*x + 3*c) - (d*f^2*x + d*e*f)*cos(d*x + c) + (4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + 3*f^2)*s
in(d*x + c))*sin(4*d*x + 4*c) - 2*(d*f^2*x + d*e*f - 6*f^2*sin(2*d*x + 2*c) + 8*(d^2*f^2*x^2 + 2*d^2*e*f*x + d
^2*e^2)*cos(d*x + c) + 4*(d*f^2*x + d*e*f)*sin(d*x + c))*sin(3*d*x + 3*c) + 6*(2*f^2*sin(d*x + c) + f^2)*sin(2
*d*x + 2*c) - 2*(d*f^2*x + d*e*f)*sin(d*x + c))/(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a
*d^3*e^3*f*x + a*d^3*e^4 + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*
e^4)*cos(4*d*x + 4*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3
*e^4)*cos(3*d*x + 3*c)^2 + 8*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^
3*e^4)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^
3*f*x + a*d^3*e^4)*cos(d*x + c)^2 + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x
+ a*d^3*e^4)*sin(4*d*x + 4*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*
x + a*d^3*e^4)*sin(3*d*x + 3*c)^2 + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f
*x + a*d^3*e^4)*sin(d*x + c)^2 - 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x
+ a*d^3*e^4 + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(3*
d*x + 3*c) + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*sin(d*x
+ c))*cos(4*d*x + 4*c) + 4*((a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^
3*e^4)*cos(3*d*x + 3*c) + (a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e
^4)*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*
f*x + a*d^3*e^4 + 2*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*si
n(d*x + c))*sin(3*d*x + 3*c) + 4*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x +
a*d^3*e^4)*sin(d*x + c))

Giac [N/A]

Not integrable

Time = 71.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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