3.176 \(\int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=93 \[ \text {Int}\left (\frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2},x\right )+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {\cos (a+b x)}{d (c+d x)} \]

[Out]

CannotIntegrate(cot(b*x+a)*csc(b*x+a)/(d*x+c)^2,x)+cos(b*x+a)/d/(d*x+c)+b*cos(a-b*c/d)*Si(b*c/d+b*x)/d^2+b*Ci(
b*c/d+b*x)*sin(a-b*c/d)/d^2

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Rubi [A]  time = 0.26, 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 = {} \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2,x]

[Out]

Cos[a + b*x]/(d*(c + d*x)) + (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^2 + (b*Cos[a - (b*c)/d]*SinInte
gral[(b*c)/d + b*x])/d^2 + Defer[Int][(Cot[a + b*x]*Csc[a + b*x])/(c + d*x)^2, x]

Rubi steps

\begin {align*} \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx &=-\int \frac {\cos (a+b x)}{(c+d x)^2} \, dx+\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\cos (a+b x)}{d (c+d x)}+\frac {b \int \frac {\sin (a+b x)}{c+d x} \, dx}{d}+\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\cos (a+b x)}{d (c+d x)}+\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\cos (a+b x)}{d (c+d x)}+\frac {b \text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d^2}+\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 4.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2,x]

[Out]

Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2, x]

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(cos(b*x + a)*cot(b*x + a)^2/(d^2*x^2 + 2*c*d*x + c^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(cos(b*x + a)*cot(b*x + a)^2/(d*x + c)^2, x)

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\cot ^{2}\left (b x +a \right )\right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*cot(b*x+a)^2/(d*x+c)^2,x)

[Out]

int(cos(b*x+a)*cot(b*x+a)^2/(d*x+c)^2,x)

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maxima [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(cos(b*x+a)*cot(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(a + b*x)*cot(a + b*x)^2)/(c + d*x)^2,x)

[Out]

int((cos(a + b*x)*cot(a + b*x)^2)/(c + d*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*cot(b*x+a)**2/(d*x+c)**2,x)

[Out]

Integral(cos(a + b*x)*cot(a + b*x)**2/(c + d*x)**2, x)

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