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

Optimal. Leaf size=35 \[ \frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {a^2 \log (\cos (c+d x))}{d} \]

[Out]

2*a^2*ln(1-cos(d*x+c))/d-a^2*ln(cos(d*x+c))/d

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3879, 72} \[ \frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {a^2 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*Log[1 - Cos[c + d*x]])/d - (a^2*Log[Cos[c + d*x]])/d

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int \cot (c+d x) (a+a \sec (c+d x))^2 \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {a+a x}{x (a-a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {2}{-1+x}+\frac {1}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {a^2 \log (\cos (c+d x))}{d}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 29, normalized size = 0.83 \[ -\frac {a^2 \left (\log (\cos (c+d x))-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*(Log[Cos[c + d*x]] - 4*Log[Sin[(c + d*x)/2]]))/d)

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fricas [A]  time = 0.77, size = 35, normalized size = 1.00 \[ -\frac {a^{2} \log \left (-\cos \left (d x + c\right )\right ) - 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*log(-cos(d*x + c)) - 2*a^2*log(-1/2*cos(d*x + c) + 1/2))/d

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giac [A]  time = 0.26, size = 64, normalized size = 1.83 \[ \frac {2 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a^{2} \log \left ({\left | \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1 \right |}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - a^2*log(abs((cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)
^2 - 1)))/d

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maple [A]  time = 0.53, size = 34, normalized size = 0.97 \[ -\frac {a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {2 a^{2} \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/d*ln(sec(d*x+c))+2*a^2/d*ln(-1+sec(d*x+c))

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maxima [A]  time = 0.39, size = 31, normalized size = 0.89 \[ \frac {2 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - a^{2} \log \left (\cos \left (d x + c\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^2*log(cos(d*x + c) - 1) - a^2*log(cos(d*x + c)))/d

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mupad [B]  time = 1.25, size = 36, normalized size = 1.03 \[ \frac {a^2\,\left (4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*(4*log(tan(c/2 + (d*x)/2)) - log(tan(c/2 + (d*x)/2)^4 - 1)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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