3.131 \(\int \frac {1}{(c+d x) (a+a \cos (e+f x))} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x) (a \cos (e+f x)+a)},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(a+a*cos(f*x+e)),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{(c+d x) (a+a \cos (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)*(a + a*Cos[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + a*Cos[e + f*x])), x]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x) (a+a \cos (e+f x))} \, dx &=\int \frac {1}{(c+d x) (a+a \cos (e+f x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 2.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a+a \cos (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])), x]

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fricas [A]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a d x + a c + {\left (a d x + a c\right )} \cos \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*cos(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (a*d*x + a*c)*cos(f*x + e)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x + c\right )} {\left (a \cos \left (f x + e\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*cos(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(a*cos(f*x + e) + a)), x)

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maple [A]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a +a \cos \left (f x +e \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)/(a+a*cos(f*x+e)),x)

[Out]

int(1/(d*x+c)/(a+a*cos(f*x+e)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (\frac {{\left (a d^{2} f x + a c d f + {\left (a d^{2} f x + a c d f\right )} \cos \left (f x + e\right )^{2} + {\left (a d^{2} f x + a c d f\right )} \sin \left (f x + e\right )^{2} + 2 \, {\left (a d^{2} f x + a c d f\right )} \cos \left (f x + e\right )\right )} \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x}}{a f} + \sin \left (f x + e\right )\right )}}{a d f x + a c f + {\left (a d f x + a c f\right )} \cos \left (f x + e\right )^{2} + {\left (a d f x + a c f\right )} \sin \left (f x + e\right )^{2} + 2 \, {\left (a d f x + a c f\right )} \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*cos(f*x+e)),x, algorithm="maxima")

[Out]

2*((a*d^2*f*x + a*c*d*f + (a*d^2*f*x + a*c*d*f)*cos(f*x + e)^2 + (a*d^2*f*x + a*c*d*f)*sin(f*x + e)^2 + 2*(a*d
^2*f*x + a*c*d*f)*cos(f*x + e))*integrate(sin(f*x + e)/(a*d^2*f*x^2 + 2*a*c*d*f*x + a*c^2*f + (a*d^2*f*x^2 + 2
*a*c*d*f*x + a*c^2*f)*cos(f*x + e)^2 + (a*d^2*f*x^2 + 2*a*c*d*f*x + a*c^2*f)*sin(f*x + e)^2 + 2*(a*d^2*f*x^2 +
2*a*c*d*f*x + a*c^2*f)*cos(f*x + e)), x) + sin(f*x + e))/(a*d*f*x + a*c*f + (a*d*f*x + a*c*f)*cos(f*x + e)^2
+ (a*d*f*x + a*c*f)*sin(f*x + e)^2 + 2*(a*d*f*x + a*c*f)*cos(f*x + e))

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (a+a\,\cos \left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*cos(e + f*x))*(c + d*x)),x)

[Out]

int(1/((a + a*cos(e + f*x))*(c + d*x)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c \cos {\left (e + f x \right )} + c + d x \cos {\left (e + f x \right )} + d x}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*cos(f*x+e)),x)

[Out]

Integral(1/(c*cos(e + f*x) + c + d*x*cos(e + f*x) + d*x), x)/a

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