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

Optimal. Leaf size=70 \[ \text {Int}\left (\frac {\csc (a+b x)}{c+d x},x\right )-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d) - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + Defer[I
nt][Csc[a + b*x]/(c + d*x), x]

Rubi steps

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

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Mathematica [A]  time = 8.35, size = 0, normalized size = 0.00 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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