3.2.11 \(\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx\) [111]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+a \sin (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+a*sin(f*x+e)),x)

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + a*Sin[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx &=\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx\\ \end {align*}

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Mathematica [A]
time = 3.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right )^{2} \left (a +a \sin \left (f x +e \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+a*sin(f*x+e)),x)

[Out]

int(1/(d*x+c)^2/(a+a*sin(f*x+e)),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

-2*(2*(a*d^3*f*x^2 + 2*a*c*d^2*f*x + a*c^2*d*f + (a*d^3*f*x^2 + 2*a*c*d^2*f*x + a*c^2*d*f)*cos(f*x + e)^2 + (a
*d^3*f*x^2 + 2*a*c*d^2*f*x + a*c^2*d*f)*sin(f*x + e)^2 + 2*(a*d^3*f*x^2 + 2*a*c*d^2*f*x + a*c^2*d*f)*sin(f*x +
e))*integrate(cos(f*x + e)/(a*d^3*f*x^3 + 3*a*c*d^2*f*x^2 + 3*a*c^2*d*f*x + a*c^3*f + (a*d^3*f*x^3 + 3*a*c*d^
2*f*x^2 + 3*a*c^2*d*f*x + a*c^3*f)*cos(f*x + e)^2 + (a*d^3*f*x^3 + 3*a*c*d^2*f*x^2 + 3*a*c^2*d*f*x + a*c^3*f)*
sin(f*x + e)^2 + 2*(a*d^3*f*x^3 + 3*a*c*d^2*f*x^2 + 3*a*c^2*d*f*x + a*c^3*f)*sin(f*x + e)), x) + cos(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)*sin(f*x + e))

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + (a*d^2*x^2 + 2*a*c*d*x + a*c^2)*sin(f*x + e)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{2} \sin {\left (e + f x \right )} + c^{2} + 2 c d x \sin {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sin {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+a*sin(f*x+e)),x)

[Out]

Integral(1/(c**2*sin(e + f*x) + c**2 + 2*c*d*x*sin(e + f*x) + 2*c*d*x + d**2*x**2*sin(e + f*x) + d**2*x**2), x
)/a

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(a*sin(f*x + e) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*sin(e + f*x))*(c + d*x)^2),x)

[Out]

int(1/((a + a*sin(e + f*x))*(c + d*x)^2), x)

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