3.273 \(\int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\)
Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\sec (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]
[Out]
Unintegrable[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]
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Rubi [A] time = 0.0453772, 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 = {}
\[ \int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
Defer[Int][Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end{align*}
Mathematica [A] time = 13.8387, size = 0, normalized size = 0. \[ \int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
Integrate[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]
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Maple [A] time = 1.774, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( dx+c \right ) }{ \left ( fx+e \right ) \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)/(a+a*sin(d*x+c)),x)
[Out]
int(sec(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)
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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)/(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 - (f*cos(d*x + c) + (d*f*x + d*e)*sin(d*x +
c))*cos(2*d*x + 2*c) - f*cos(d*x + c) - (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*
e*f*x + a*d^2*e^2)*cos(2*d*x + 2*c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(d*x + c)^2 + 4*(a*d^
2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*
e^2)*sin(2*d*x + 2*c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c)^2 - 2*(a*d^2*f^2*x^2 + 2*
a*d^2*e*f*x + a*d^2*e^2 + 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a*
d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c))*integrate(1/2*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 + 4*
f^2)*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(d*x + c)^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)^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)*sin(d*x + c
)), x) - (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(2*d*x +
2*c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(d*x + c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d
^2*e^2)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(2*d*x + 2*c)^2 + 4*(a*
d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c)^2 - 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2 + 2*(a*
d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a
*d^2*e^2)*sin(d*x + c))*integrate(1/2*cos(d*x + c)/(a*f*x + (a*f*x + a*e)*cos(d*x + c)^2 + (a*f*x + a*e)*sin(d
*x + c)^2 + a*e - 2*(a*f*x + a*e)*sin(d*x + c)), x) + ((d*f*x + d*e)*cos(d*x + c) - f*sin(d*x + c) - f)*sin(2*
d*x + 2*c) + (d*f*x + d*e)*sin(d*x + c))/(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2
*e*f*x + a*d^2*e^2)*cos(2*d*x + 2*c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(d*x + c)^2 + 4*(a*d
^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cos(d*x + c)*sin(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2
*e^2)*sin(2*d*x + 2*c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c)^2 - 2*(a*d^2*f^2*x^2 + 2
*a*d^2*e*f*x + a*d^2*e^2 + 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c))*cos(2*d*x + 2*c) + 4*(a
*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sin(d*x + c))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (d x + c\right )}{a f x + a e +{\left (a f x + a e\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)/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
integral(sec(d*x + c)/(a*f*x + a*e + (a*f*x + a*e)*sin(d*x + c)), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec{\left (c + d x \right )}}{e \sin{\left (c + d x \right )} + e + f x \sin{\left (c + d x \right )} + f x}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)
[Out]
Integral(sec(c + d*x)/(e*sin(c + d*x) + e + f*x*sin(c + d*x) + f*x), x)/a
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )}{{\left (f x + e\right )}{\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)/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
integrate(sec(d*x + c)/((f*x + e)*(a*sin(d*x + c) + a)), x)