3.42 \(\int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=48 \[ \frac {a^3 \sec (c+d x)}{d}+\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {3 a^3 \log (\cos (c+d x))}{d} \]

[Out]

4*a^3*ln(1-cos(d*x+c))/d-3*a^3*ln(cos(d*x+c))/d+a^3*sec(d*x+c)/d

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Rubi [A]  time = 0.05, antiderivative size = 48, 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, 88} \[ \frac {a^3 \sec (c+d x)}{d}+\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {3 a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]

[Out]

(4*a^3*Log[1 - Cos[c + d*x]])/d - (3*a^3*Log[Cos[c + d*x]])/d + (a^3*Sec[c + d*x])/d

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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))^3 \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {(a+a x)^2}{x^2 (a-a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {4 a}{-1+x}+\frac {a}{x^2}+\frac {3 a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {3 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \sec (c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 36, normalized size = 0.75 \[ \frac {a^3 \left (\sec (c+d x)+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log (\cos (c+d x))\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*(-3*Log[Cos[c + d*x]] + 8*Log[Sin[(c + d*x)/2]] + Sec[c + d*x]))/d

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fricas [A]  time = 0.79, size = 61, normalized size = 1.27 \[ -\frac {3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{3} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a^{3}}{d \cos \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-(3*a^3*cos(d*x + c)*log(-cos(d*x + c)) - 4*a^3*cos(d*x + c)*log(-1/2*cos(d*x + c) + 1/2) - a^3)/(d*cos(d*x +
c))

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giac [B]  time = 0.32, size = 145, normalized size = 3.02 \[ \frac {4 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 3 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {5 \, a^{3} + \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

(4*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
+ 1)) - 3*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (5*a^3 + 3*a^3*(cos(d*x + c) - 1)/(cos(d*
x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/d

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maple [A]  time = 0.51, size = 47, normalized size = 0.98 \[ \frac {a^{3} \sec \left (d x +c \right )}{d}-\frac {a^{3} \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {4 a^{3} \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+a*sec(d*x+c))^3,x)

[Out]

a^3*sec(d*x+c)/d-a^3/d*ln(sec(d*x+c))+4*a^3/d*ln(-1+sec(d*x+c))

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maxima [A]  time = 0.60, size = 43, normalized size = 0.90 \[ \frac {4 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 3 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {a^{3}}{\cos \left (d x + c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

(4*a^3*log(cos(d*x + c) - 1) - 3*a^3*log(cos(d*x + c)) + a^3/cos(d*x + c))/d

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mupad [B]  time = 1.20, size = 86, normalized size = 1.79 \[ \frac {8\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,a^3}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {3\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)*(a + a/cos(c + d*x))^3,x)

[Out]

(8*a^3*log(tan(c/2 + (d*x)/2)))/d - (2*a^3)/(d*(tan(c/2 + (d*x)/2)^2 - 1)) - (3*a^3*log(tan(c/2 + (d*x)/2)^2 -
1))/d - (a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int 3 \cot {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \cot {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+a*sec(d*x+c))**3,x)

[Out]

a**3*(Integral(3*cot(c + d*x)*sec(c + d*x), x) + Integral(3*cot(c + d*x)*sec(c + d*x)**2, x) + Integral(cot(c
+ d*x)*sec(c + d*x)**3, x) + Integral(cot(c + d*x), x))

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