3.274 \(\int \frac{\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\)
Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\sec (c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right ) \]
[Out]
Unintegrable[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]
________________________________________________________________________________________
Rubi [A] time = 0.0456112, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[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*} \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\\ \end{align*}
Mathematica [A] time = 22.1101, size = 0, normalized size = 0. \[ \int \frac{\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[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 [A] time = 3.25, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( dx+c \right ) }{ \left ( fx+e \right ) ^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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))
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (d x + c\right )}{a f^{2} x^{2} + 2 \, a e f x + a e^{2} +{\left (a f^{2} x^{2} + 2 \, a e f x + a e^{2}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)