3.279 \(\int \frac {\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\)
Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sec ^2(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]
[Out]
Unintegrable(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)
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Rubi [A] time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \[ \int \frac {\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
Defer[Int][Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac {\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end {align*}
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Mathematica [A] time = 20.56, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
Integrate[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]
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fricas [A] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (d x + c\right )^{2}}{a f x + a e + {\left (a f x + a e\right )} \sin \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
integral(sec(d*x + c)^2/(a*f*x + a*e + (a*f*x + a*e)*sin(d*x + c)), x)
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
integrate(sec(d*x + c)^2/((f*x + e)*(a*sin(d*x + c) + a)), x)
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maple [A] time = 3.90, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)
[Out]
int(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/3*(4*f^2*cos(2*d*x + 2*c)*cos(d*x + c) - 2*(d*f^2*x + d*e*f)*cos(3*d*x + 3*c)^2 + 2*f^2*cos(d*x + c) - 2*(d
*f^2*x + d*e*f)*cos(d*x + c)^2 - 2*(d*f^2*x + d*e*f)*sin(3*d*x + 3*c)^2 - 2*(d*f^2*x + d*e*f)*sin(d*x + c)^2 +
(2*f^2*cos(3*d*x + 3*c) - 2*f^2*sin(2*d*x + 2*c) + 2*(4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + 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 + 2*f^2*cos(2*d*x + 2*c) + f^2 - 2*(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^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*
f*x + a*d^3*e^3 + (a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*
d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c)^2 + 8*(a*d^3*f^3*x^3 + 3*a*d^3
*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2
+ 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(d*x + c)^2 + (a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*
e^3)*sin(4*d*x + 4*c)^2 + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(3*d*x + 3*c)
^2 + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x + c)^2 - 2*(a*d^3*f^3*x^3 + 3
*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3 + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^
3*e^3)*sin(3*d*x + 3*c) + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x + c))*co
s(4*d*x + 4*c) + 4*((a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c) + (a*d^
3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(a*d^3*f^3*x^3
+ 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3 + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x +
a*d^3*e^3)*sin(d*x + c))*sin(3*d*x + 3*c) + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3
)*sin(d*x + c))*integrate(1/6*(5*d^2*f^3*x^2 + 10*d^2*e*f^2*x + 5*d^2*e^2*f + 12*f^3)*cos(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(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(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)*sin(d*x + c)), x) - 3*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a
*d^3*e^2*f^2*x + a*d^3*e^3*f + (a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*cos(4*d*x
+ 4*c)^2 + 4*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*cos(3*d*x + 3*c)^2 + 8*(a*
d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^
4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*cos(d*x + c)^2 + (a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x
^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*sin(4*d*x + 4*c)^2 + 4*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*
f^2*x + a*d^3*e^3*f)*sin(3*d*x + 3*c)^2 + 4*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3
*f)*sin(d*x + c)^2 - 2*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f + 2*(a*d^3*f^4*x^3
+ 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*sin(3*d*x + 3*c) + 2*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^
2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*sin(d*x + c))*cos(4*d*x + 4*c) + 4*((a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 +
3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*cos(3*d*x + 3*c) + (a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a
*d^3*e^3*f)*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*
e^3*f + 2*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*sin(d*x + c))*sin(3*d*x + 3*c)
+ 4*(a*d^3*f^4*x^3 + 3*a*d^3*e*f^3*x^2 + 3*a*d^3*e^2*f^2*x + a*d^3*e^3*f)*sin(d*x + c))*integrate(1/2*cos(d*x
+ c)/(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*cos(d*x + c)^2 + (a*d*f^2*x
^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c)^2 - 2*(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c)), x) + (4*d^
2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + 2*f^2*cos(2*d*x + 2*c) + 2*f^2*sin(3*d*x + 3*c) + 2*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) + 2*(4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 + f^2)*sin(
d*x + c))*sin(4*d*x + 4*c) - (d*f^2*x + d*e*f - 4*f^2*sin(2*d*x + 2*c) + 16*(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) + 2*(2*f^2*sin(d*x + c) + f^2)*sin(2*d*x
+ 2*c) - (d*f^2*x + d*e*f)*sin(d*x + c))/(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3 + (
a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^3*f^3*x^3 + 3*a*d
^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c)^2 + 8*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^
3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c)*cos(d*x + c) + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x
+ a*d^3*e^3)*cos(d*x + c)^2 + (a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(4*d*x + 4*
c)^2 + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(3*d*x + 3*c)^2 + 4*(a*d^3*f^3*x
^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x + c)^2 - 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 +
3*a*d^3*e^2*f*x + a*d^3*e^3 + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(3*d*x +
3*c) + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x + c))*cos(4*d*x + 4*c) + 4*
((a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(3*d*x + 3*c) + (a*d^3*f^3*x^3 + 3*a*d^3
*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^
2 + 3*a*d^3*e^2*f*x + a*d^3*e^3 + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x
+ c))*sin(3*d*x + 3*c) + 4*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*sin(d*x + c))
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cos(c + d*x)^2*(e + f*x)*(a + a*sin(c + d*x))),x)
[Out]
int(1/(cos(c + d*x)^2*(e + f*x)*(a + a*sin(c + d*x))), x)
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)**2/(f*x+e)/(a+a*sin(d*x+c)),x)
[Out]
Integral(sec(c + d*x)**2/(e*sin(c + d*x) + e + f*x*sin(c + d*x) + f*x), x)/a
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