3.19 \(\int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx\)
Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a \sec (e+f x)+a)^2},x\right ) \]
[Out]
Unintegrable[1/((c + d*x)*(a + a*Sec[e + f*x])^2), x]
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Rubi [A] time = 0.0551167, 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 = {}
\[ \int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)*(a + a*Sec[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)*(a + a*Sec[e + f*x])^2), x]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx &=\int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx\\ \end{align*}
Mathematica [A] time = 15.5166, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])^2), x]
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Maple [A] time = 2.137, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+a\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+a*sec(f*x+e))^2,x)
[Out]
int(1/(d*x+c)/(a+a*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+a*sec(f*x+e))^2,x, algorithm="maxima")
[Out]
1/3*(3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*cos(3*f*x + 3*e)^2*log(d*x + c) + 3*(d^3*f^3*
x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c)*sin(3*f*x + 3*e)^2 + 3*(2*d^3*f*x + 2*c*d^2*f +
9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*cos(2*f*x + 2*e)^2 + 3*(2*d^3*f*x +
2*c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*cos(f*x + e)^2 + 3*(2*d^
3*f*x + 2*c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*sin(2*f*x + 2*e)
^2 + 3*(2*d^3*f*x + 2*c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*sin(
f*x + e)^2 + 2*((d^3*f*x + c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))
*cos(2*f*x + 2*e) + (d^3*f*x + c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x +
c))*cos(f*x + e) + 3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c) + 2*(3*d^3*f^2*x^
2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*sin(2*f*x + 2*e) + (9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^
3)*sin(f*x + e))*cos(3*f*x + 3*e) + 2*(d^3*f*x + c*d^2*f + 3*(2*d^3*f*x + 2*c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2
*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*cos(f*x + e) + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^
3*x + c^3*f^3)*log(d*x + c) + 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 2*d^3)*sin(f*x + e))*cos(2*f*x
+ 2*e) + 2*(d^3*f*x + c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*cos(
f*x + e) - 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + (a^2*d^4*f^3*x^3 +
3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*
d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x
^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*
d^2*f^3*x + a^2*c^3*d*f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x
+ a^2*c^3*d*f^3)*sin(2*f*x + 2*e)^2 + 18*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c
^3*d*f^3)*sin(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2
*c^3*d*f^3)*sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 +
3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^4*
f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e) + 6*(a^2*d
^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*
x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(2*f*x + 2*e) + 6*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f
^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e) + 6*((a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2
*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*sin(2*f*x + 2*e) + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*
x + a^2*c^3*d*f^3)*sin(f*x + e))*sin(3*f*x + 3*e))*integrate(2/3*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2
- 6*d^3)*sin(f*x + e)/(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^
2*c^4*f^3 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*
cos(f*x + e)^2 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*
f^3)*sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a
^2*c^4*f^3)*cos(f*x + e)), x) + 3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c) - 2*(
5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3 + 2*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*c
os(2*f*x + 2*e) + (9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^3)*cos(f*x + e) - (d^3*f*x + c*d^2*f + 9
*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*sin(2*f*x + 2*e) - (d^3*f*x + c*d^2*f
+ 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(d*x + c))*sin(f*x + e))*sin(3*f*x + 3*e) -
2*(9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^3 + 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 2*d
^3)*cos(f*x + e) - 3*(2*d^3*f*x + 2*c*d^2*f + 9*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*log(
d*x + c))*sin(f*x + e))*sin(2*f*x + 2*e) - 4*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*sin(f*x + e))
/(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3
*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2
+ 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c
^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x +
a^2*c^3*d*f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d
*f^3)*sin(2*f*x + 2*e)^2 + 18*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*si
n(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*
sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f
^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^4*f^3*x^3 + 3*
a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e) + 6*(a^2*d^4*f^3*x^3 +
3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*
c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(2*f*x + 2*e) + 6*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a
^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e) + 6*((a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3
*x + a^2*c^3*d*f^3)*sin(2*f*x + 2*e) + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*
d*f^3)*sin(f*x + e))*sin(3*f*x + 3*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 (a^{2} d x + a^{2} c\right )} \sec \left (f x + e\right )^{2} + 2 \,{\left (a^{2} d x + a^{2} 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+a*sec(f*x+e))^2,x, algorithm="fricas")
[Out]
integral(1/(a^2*d*x + a^2*c + (a^2*d*x + a^2*c)*sec(f*x + e)^2 + 2*(a^2*d*x + a^2*c)*sec(f*x + e)), x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c \sec ^{2}{\left (e + f x \right )} + 2 c \sec{\left (e + f x \right )} + c + d x \sec ^{2}{\left (e + f x \right )} + 2 d x \sec{\left (e + f x \right )} + d x}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)/(a+a*sec(f*x+e))**2,x)
[Out]
Integral(1/(c*sec(e + f*x)**2 + 2*c*sec(e + f*x) + c + d*x*sec(e + f*x)**2 + 2*d*x*sec(e + f*x) + d*x), x)/a**
2
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (a \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+a*sec(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)*(a*sec(f*x + e) + a)^2), x)