3.584 \(\int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{b^2 (1-n) \text{Unintegrable}\left ((b x)^{-n} \cos ^{n-2}(a x),x\right )}{a^2 c^2}-\frac{b (b x)^{1-n} \cos ^{n-1}(a x)}{a^2 \left (a c^2 x \sin (a x)+c^2 \cos (a x)\right )} \]

[Out]

-((b*(b*x)^(1 - n)*Cos[a*x]^(-1 + n))/(a^2*(c^2*Cos[a*x] + a*c^2*x*Sin[a*x]))) + (b^2*(1 - n)*Unintegrable[Cos
[a*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)

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Rubi [A]  time = 0.146015, 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 = {} \[ \int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2,x]

[Out]

-((b*(b*x)^(1 - n)*Cos[a*x]^(-1 + n))/(a^2*(c^2*Cos[a*x] + a*c^2*x*Sin[a*x]))) + (b^2*(1 - n)*Defer[Int][Cos[a
*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)

Rubi steps

\begin{align*} \int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx &=-\frac{b (b x)^{1-n} \cos ^{-1+n}(a x)}{a^2 \left (c^2 \cos (a x)+a c^2 x \sin (a x)\right )}+\frac{\left (b^2 (1-n)\right ) \int (b x)^{-n} \cos ^{-2+n}(a x) \, dx}{a^2 c^2}\\ \end{align*}

Mathematica [A]  time = 4.91937, size = 0, normalized size = 0. \[ \int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2,x]

[Out]

Integrate[((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2, x]

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Maple [A]  time = 0.837, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx \right ) ^{2-n} \left ( \cos \left ( ax \right ) \right ) ^{n}}{ \left ( c\cos \left ( ax \right ) +acx\sin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^(2-n)*cos(a*x)^n/(c*cos(a*x)+a*c*x*sin(a*x))^2,x)

[Out]

int((b*x)^(2-n)*cos(a*x)^n/(c*cos(a*x)+a*c*x*sin(a*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{{\left (a c x \sin \left (a x\right ) + c \cos \left (a x\right )\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(2-n)*cos(a*x)^n/(c*cos(a*x)+a*c*x*sin(a*x))^2,x, algorithm="maxima")

[Out]

integrate((b*x)^(-n + 2)*cos(a*x)^n/(a*c*x*sin(a*x) + c*cos(a*x))^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{a^{2} c^{2} x^{2} + 2 \, a c^{2} x \cos \left (a x\right ) \sin \left (a x\right ) -{\left (a^{2} c^{2} x^{2} - c^{2}\right )} \cos \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(2-n)*cos(a*x)^n/(c*cos(a*x)+a*c*x*sin(a*x))^2,x, algorithm="fricas")

[Out]

integral((b*x)^(-n + 2)*cos(a*x)^n/(a^2*c^2*x^2 + 2*a*c^2*x*cos(a*x)*sin(a*x) - (a^2*c^2*x^2 - c^2)*cos(a*x)^2
), 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((b*x)**(2-n)*cos(a*x)**n/(c*cos(a*x)+a*c*x*sin(a*x))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{{\left (a c x \sin \left (a x\right ) + c \cos \left (a x\right )\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(2-n)*cos(a*x)^n/(c*cos(a*x)+a*c*x*sin(a*x))^2,x, algorithm="giac")

[Out]

integrate((b*x)^(-n + 2)*cos(a*x)^n/(a*c*x*sin(a*x) + c*cos(a*x))^2, x)