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=79 \[ \frac {b^2 (1-n) \text {Int}\left ((b x)^{-n} \cos ^{n-2}(a x),x\right )}{a^2 c^2}-\frac {b (b x)^{1-n} \cos ^{n-1}(a x)}{a^2 \left (a c^2 x \sin (a x)+c^2 \cos (a x)\right )} \]

[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 = 0.15, 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 = {} \[ \int \frac {(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx \]

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*}

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Mathematica [A]  time = 4.97, size = 0, normalized size = 0.00 \[ \int \frac {(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx \]

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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fricas [A]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{a^{2} c^{2} x^{2} + 2 \, a c^{2} x \cos \left (a x\right ) \sin \left (a x\right ) - {\left (a^{2} c^{2} x^{2} - c^{2}\right )} \cos \left (a x\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{{\left (a c x \sin \left (a x\right ) + c \cos \left (a x\right )\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 2.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x \right )^{2-n} \left (\cos ^{n}\left (a x \right )\right )}{\left (c \cos \left (a x \right )+a c x \sin \left (a x \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x\right )^{-n + 2} \cos \left (a x\right )^{n}}{{\left (a c x \sin \left (a x\right ) + c \cos \left (a x\right )\right )}^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a\,x\right )}^n\,{\left (b\,x\right )}^{2-n}}{{\left (c\,\cos \left (a\,x\right )+a\,c\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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