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

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

[Out]

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

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Rubi [A]  time = 0.0239717, 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} \[ -\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) - (Cot[c + d*x]*(a + a*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+a \sec (c+d x)) \, dx &=-\frac{\cot (c+d x) (a+a \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac{\cot (c+d x) (a+a \sec (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 0.0287506, size = 43, normalized size = 1.65 \[ -\frac{a \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}-\frac{a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*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 = 0.043, size = 35, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( a \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -{\frac{a}{\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+a*sec(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.75011, size = 42, normalized size = 1.62 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a + \frac{a}{\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+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.895697, 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 ) + a}{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+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3252, size = 35, normalized size = 1.35 \begin{align*} -\frac{{\left (d x + c\right )} a + \frac{a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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