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

Optimal. Leaf size=88 \[ \text {Int}\left (\frac {\csc (a+b x)}{(c+d x)^2},x\right )-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {\sin (a+b x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.02, size = 0, normalized size = 0.00 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(b*x + a)*cot(b*x + a)/(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 \[ \int \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((I*exp_integral_e(2, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(2, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) + 2*(d^2*x + c*d)*integrate(sin(b*x + a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^
2*x^2 + 2*c*d*x + c^2)*sin(b*x + a)^2 + c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + 2*(d^2*x + c*d)*
integrate(sin(b*x + a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^2*x^2 + 2*c*d*x + c^
2)*sin(b*x + a)^2 + c^2 - 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + (exp_integral_e(2, (I*b*d*x + I*b*c)
/d) + exp_integral_e(2, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d))/(d^2*x + c*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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