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(d*x+c)*(a+a*sec(d*x+c))/d

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [C]  time = 0.03, size = 43, normalized size = 1.65 \[ -\frac {a \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}-\frac {a \csc (c+d x)}{d} \]

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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fricas [A]  time = 0.94, size = 33, normalized size = 1.27 \[ -\frac {a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + a}{d \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.23, size = 26, normalized size = 1.00 \[ -\frac {{\left (d x + c\right )} a + \frac {a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} \]

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

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maple [A]  time = 0.48, size = 35, normalized size = 1.35 \[ \frac {a \left (-\cot \left (d x +c \right )-d x -c \right )-\frac {a}{\sin \left (d x +c \right )}}{d} \]

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 = 1.22, size = 31, normalized size = 1.19 \[ -\frac {{\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a + \frac {a}{\sin \left (d x + c\right )}}{d} \]

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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mupad [B]  time = 1.07, size = 19, normalized size = 0.73 \[ -\frac {a\,\left (\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*(cot(c/2 + (d*x)/2) + d*x))/d

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

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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