3.42 \(\int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx\)
Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+b \sec (e+f x))^2},x\right ) \]
[Out]
Unintegrable[1/((c + d*x)*(a + b*Sec[e + f*x])^2), x]
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Rubi [A] time = 0.0579287, 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 = {}
\[ \int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)*(a + b*Sec[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)*(a + b*Sec[e + f*x])^2), x]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx &=\int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx\\ \end{align*}
Mathematica [A] time = 28.1104, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])^2), x]
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Maple [A] time = 1.746, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\sec \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*x+c)/(a+b*sec(f*x+e))^2,x)
[Out]
int(1/(d*x+c)/(a+b*sec(f*x+e))^2,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(1/(d*x+c)/(a+b*sec(f*x+e))^2,x, algorithm="maxima")
[Out]
(2*a*b^3*d*sin(f*x + e) + ((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*cos(2*f*x + 2*e)^2*log(d*x + c) + 4*((
a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*cos(f*x + e)^2*log(d*x + c) + ((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*
b^2)*c*f)*log(d*x + c)*sin(2*f*x + 2*e)^2 + 4*((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*log(d*x + c)*sin(f
*x + e)^2 + 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*cos(f*x + e)*log(d*x + c) - 2*(a*b^3*d*sin(f*x + e
) - 2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*cos(f*x + e)*log(d*x + c) - ((a^4 - a^2*b^2)*d*f*x + (a^4
- a^2*b^2)*c*f)*log(d*x + c))*cos(2*f*x + 2*e) - ((a^6 - a^4*b^2)*d^2*f*x + (a^6 - a^4*b^2)*c*d*f + ((a^6 - a^
4*b^2)*d^2*f*x + (a^6 - a^4*b^2)*c*d*f)*cos(2*f*x + 2*e)^2 + 4*((a^4*b^2 - a^2*b^4)*d^2*f*x + (a^4*b^2 - a^2*b
^4)*c*d*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^2*f*x + (a^6 - a^4*b^2)*c*d*f)*sin(2*f*x + 2*e)^2 + 4*((a^5*b -
a^3*b^3)*d^2*f*x + (a^5*b - a^3*b^3)*c*d*f)*sin(2*f*x + 2*e)*sin(f*x + e) + 4*((a^4*b^2 - a^2*b^4)*d^2*f*x +
(a^4*b^2 - a^2*b^4)*c*d*f)*sin(f*x + e)^2 + 2*((a^6 - a^4*b^2)*d^2*f*x + (a^6 - a^4*b^2)*c*d*f + 2*((a^5*b - a
^3*b^3)*d^2*f*x + (a^5*b - a^3*b^3)*c*d*f)*cos(f*x + e))*cos(2*f*x + 2*e) + 4*((a^5*b - a^3*b^3)*d^2*f*x + (a^
5*b - a^3*b^3)*c*d*f)*cos(f*x + e))*integrate(-2*(a*b^3*d*sin(f*x + e) - 2*((2*a^2*b^2 - b^4)*d*f*x + (2*a^2*b
^2 - b^4)*c*f)*cos(f*x + e)^2 - 2*((2*a^2*b^2 - b^4)*d*f*x + (2*a^2*b^2 - b^4)*c*f)*sin(f*x + e)^2 - (a*b^3*d*
sin(f*x + e) + ((2*a^3*b - a*b^3)*d*f*x + (2*a^3*b - a*b^3)*c*f)*cos(f*x + e))*cos(2*f*x + 2*e) - ((2*a^3*b -
a*b^3)*d*f*x + (2*a^3*b - a*b^3)*c*f)*cos(f*x + e) + (a*b^3*d*cos(f*x + e) + a^2*b^2*d - ((2*a^3*b - a*b^3)*d*
f*x + (2*a^3*b - a*b^3)*c*f)*sin(f*x + e))*sin(2*f*x + 2*e))/((a^6 - a^4*b^2)*d^2*f*x^2 + 2*(a^6 - a^4*b^2)*c*
d*f*x + (a^6 - a^4*b^2)*c^2*f + ((a^6 - a^4*b^2)*d^2*f*x^2 + 2*(a^6 - a^4*b^2)*c*d*f*x + (a^6 - a^4*b^2)*c^2*f
)*cos(2*f*x + 2*e)^2 + 4*((a^4*b^2 - a^2*b^4)*d^2*f*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*d*f*x + (a^4*b^2 - a^2*b^4)*
c^2*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^2*f*x^2 + 2*(a^6 - a^4*b^2)*c*d*f*x + (a^6 - a^4*b^2)*c^2*f)*sin(2*
f*x + 2*e)^2 + 4*((a^5*b - a^3*b^3)*d^2*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d*f*x + (a^5*b - a^3*b^3)*c^2*f)*sin(2*f
*x + 2*e)*sin(f*x + e) + 4*((a^4*b^2 - a^2*b^4)*d^2*f*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*d*f*x + (a^4*b^2 - a^2*b^4
)*c^2*f)*sin(f*x + e)^2 + 2*((a^6 - a^4*b^2)*d^2*f*x^2 + 2*(a^6 - a^4*b^2)*c*d*f*x + (a^6 - a^4*b^2)*c^2*f + 2
*((a^5*b - a^3*b^3)*d^2*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d*f*x + (a^5*b - a^3*b^3)*c^2*f)*cos(f*x + e))*cos(2*f*x
+ 2*e) + 4*((a^5*b - a^3*b^3)*d^2*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d*f*x + (a^5*b - a^3*b^3)*c^2*f)*cos(f*x + e)
), x) + ((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*log(d*x + c) + 2*(a*b^3*d*cos(f*x + e) + a^2*b^2*d + 2*(
(a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*log(d*x + c)*sin(f*x + e))*sin(2*f*x + 2*e))/((a^6 - a^4*b^2)*d^2
*f*x + (a^6 - a^4*b^2)*c*d*f + ((a^6 - a^4*b^2)*d^2*f*x + (a^6 - a^4*b^2)*c*d*f)*cos(2*f*x + 2*e)^2 + 4*((a^4*
b^2 - a^2*b^4)*d^2*f*x + (a^4*b^2 - a^2*b^4)*c*d*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^2*f*x + (a^6 - a^4*b^2
)*c*d*f)*sin(2*f*x + 2*e)^2 + 4*((a^5*b - a^3*b^3)*d^2*f*x + (a^5*b - a^3*b^3)*c*d*f)*sin(2*f*x + 2*e)*sin(f*x
+ e) + 4*((a^4*b^2 - a^2*b^4)*d^2*f*x + (a^4*b^2 - a^2*b^4)*c*d*f)*sin(f*x + e)^2 + 2*((a^6 - a^4*b^2)*d^2*f*
x + (a^6 - a^4*b^2)*c*d*f + 2*((a^5*b - a^3*b^3)*d^2*f*x + (a^5*b - a^3*b^3)*c*d*f)*cos(f*x + e))*cos(2*f*x +
2*e) + 4*((a^5*b - a^3*b^3)*d^2*f*x + (a^5*b - a^3*b^3)*c*d*f)*cos(f*x + e))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} d x + a^{2} c +{\left (b^{2} d x + b^{2} c\right )} \sec \left (f x + e\right )^{2} + 2 \,{\left (a b d x + a b c\right )} \sec \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)/(a+b*sec(f*x+e))^2,x, algorithm="fricas")
[Out]
integral(1/(a^2*d*x + a^2*c + (b^2*d*x + b^2*c)*sec(f*x + e)^2 + 2*(a*b*d*x + a*b*c)*sec(f*x + e)), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \sec{\left (e + f x \right )}\right )^{2} \left (c + d x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)/(a+b*sec(f*x+e))**2,x)
[Out]
Integral(1/((a + b*sec(e + f*x))**2*(c + d*x)), x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (b \sec \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)/(a+b*sec(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)*(b*sec(f*x + e) + a)^2), x)