3.267 \(\int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=26 \[ -\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]

[Out]

-(a*x) - (Cot[c + d*x]*(a + b*Sec[c + d*x]))/d

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Rubi [A]  time = 0.0264657, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) - (Cot[c + d*x]*(a + b*Sec[c + d*x]))/d

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 0.0203991, size = 43, normalized size = 1.65 \[ -\frac{a \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}-\frac{b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*Csc[c + d*x])/d) - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d

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Maple [A]  time = 0.035, size = 35, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( a \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -{\frac{b}{\sin \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a*(-cot(d*x+c)-d*x-c)-b/sin(d*x+c))

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Maxima [A]  time = 1.47307, size = 42, normalized size = 1.62 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a + \frac{b}{\sin \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d*x + c + 1/tan(d*x + c))*a + b/sin(d*x + c))/d

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Fricas [A]  time = 0.807459, size = 82, normalized size = 3.15 \begin{align*} -\frac{a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + b}{d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*d*x*sin(d*x + c) + a*cos(d*x + c) + b)/(d*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15339, size = 70, normalized size = 2.69 \begin{align*} -\frac{2 \,{\left (d x + c\right )} a - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{a + b}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(d*x + c)*a - a*tan(1/2*d*x + 1/2*c) + b*tan(1/2*d*x + 1/2*c) + (a + b)/tan(1/2*d*x + 1/2*c))/d