3.171 \(\int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+b \sin (e+f x))^2},x\right ) \]
[Out]
Unintegrable[1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]
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Rubi [A] time = 0.0592222, 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 = {}
\[ \int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)*(a + b*Sin[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx &=\int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx\\ \end{align*}
Mathematica [A] time = 32.7934, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]
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Maple [A] time = 3.118, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\sin \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*sin(f*x+e))^2,x)
[Out]
int(1/(d*x+c)/(a+b*sin(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*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
(2*a*b*cos(2*f*x + 2*e)*cos(f*x + e) + 2*a*b*cos(f*x + e) - ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + ((a
^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f
)*cos(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)*sin(2*f*x + 2*e) + ((a^2*b^2 -
b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*sin(f*
x + e)^2 - 2*((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + 2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*si
n(f*x + e))*cos(2*f*x + 2*e) + 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))*integrate(-2*(a*b
*d*cos(f*x + e) + 2*(a^2*d*f*x + a^2*c*f)*cos(f*x + e)^2 + 2*(a^2*d*f*x + a^2*c*f)*sin(f*x + e)^2 + (a*b*d*cos
(f*x + e) - (a*b*d*f*x + a*b*c*f)*sin(f*x + e))*cos(2*f*x + 2*e) + (a*b*d*sin(f*x + e) + b^2*d + (a*b*d*f*x +
a*b*c*f)*cos(f*x + e))*sin(2*f*x + 2*e) + (a*b*d*f*x + a*b*c*f)*sin(f*x + e))/((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(
a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*f + ((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2
*b^2 - b^4)*c^2*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d^2*f*x^2 + 2*(a^4 - a^2*b^2)*c*d*f*x + (a^4 - a^2*
b^2)*c^2*f)*cos(f*x + e)^2 + 4*((a^3*b - a*b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)
*cos(f*x + e)*sin(2*f*x + 2*e) + ((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*
f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d^2*f*x^2 + 2*(a^4 - a^2*b^2)*c*d*f*x + (a^4 - a^2*b^2)*c^2*f)*sin(
f*x + e)^2 - 2*((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*f + 2*((a^3*b - a*
b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)*sin(f*x + e))*cos(2*f*x + 2*e) + 4*((a^3*b
- a*b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)*sin(f*x + e)), x) + 2*(a*b*sin(f*x +
e) + b^2)*sin(2*f*x + 2*e))/((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 -
b^4)*c*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*cos(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)*sin(2*f*x + 2*e) + ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*
f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*sin(f*x + e)^2 - 2*((a^2*b^2 - b^4)*d*
f*x + (a^2*b^2 - b^4)*c*f + 2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))*cos(2*f*x + 2*e) + 4
*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{2} + b^{2}\right )} d x -{\left (b^{2} d x + b^{2} c\right )} \cos \left (f x + e\right )^{2} +{\left (a^{2} + b^{2}\right )} c + 2 \,{\left (a b d x + a b c\right )} \sin \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*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
integral(1/((a^2 + b^2)*d*x - (b^2*d*x + b^2*c)*cos(f*x + e)^2 + (a^2 + b^2)*c + 2*(a*b*d*x + a*b*c)*sin(f*x +
e)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)/(a+b*sin(f*x+e))**2,x)
[Out]
Timed out
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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 \sin \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*sin(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)*(b*sin(f*x + e) + a)^2), x)