Integrand size = 19, antiderivative size = 48 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {3 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \sec (c+d x)}{d} \] Output:
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
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 \left (-3 \log (\cos (c+d x))+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec (c+d x)\right )}{d} \] Input:
Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]
Output:
(a^3*(-3*Log[Cos[c + d*x]] + 8*Log[Sin[(c + d*x)/2]] + Sec[c + d*x]))/d
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 25, 4367, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot (c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle -\frac {a^2 \int \frac {a (\cos (c+d x)+1)^2 \sec ^2(c+d x)}{1-\cos (c+d x)}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^3 \int \frac {(\cos (c+d x)+1)^2 \sec ^2(c+d x)}{1-\cos (c+d x)}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {a^3 \int \left (\sec ^2(c+d x)+3 \sec (c+d x)-\frac {4}{\cos (c+d x)-1}\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 (-\sec (c+d x)-4 \log (1-\cos (c+d x))+3 \log (\cos (c+d x)))}{d}\) |
Input:
Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*(-4*Log[1 - Cos[c + d*x]] + 3*Log[Cos[c + d*x]] - Sec[c + d*x]))/d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[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] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.65
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(79\) |
default | \(\frac {a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(79\) |
risch | \(-i a^{3} x -\frac {2 i a^{3} c}{d}+\frac {2 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(89\) |
Input:
int(cot(d*x+c)*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(1/cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+3*a^3*ln(tan(d*x+c))+3*a ^3*ln(csc(d*x+c)-cot(d*x+c))+a^3*ln(sin(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.27 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{3} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a^{3}}{d \cos \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-(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))
\[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \cot {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \cot {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cot {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)*(a+a*sec(d*x+c))**3,x)
Output:
a**3*(Integral(3*cot(c + d*x)*sec(c + d*x), x) + Integral(3*cot(c + d*x)*s ec(c + d*x)**2, x) + Integral(cot(c + d*x)*sec(c + d*x)**3, x) + Integral( cot(c + d*x), x))
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {4 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 3 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {a^{3}}{\cos \left (d x + c\right )}}{d} \] Input:
integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
(4*a^3*log(cos(d*x + c) - 1) - 3*a^3*log(cos(d*x + c)) + a^3/cos(d*x + c)) /d
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {4 \, a^{3} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {3 \, a^{3} \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {a^{3}}{d \cos \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
4*a^3*log(abs(cos(d*x + c) - 1))/d - 3*a^3*log(abs(cos(d*x + c)))/d + a^3/ (d*cos(d*x + c))
Time = 12.87 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.79 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {8\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,a^3}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {3\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cot(c + d*x)*(a + a/cos(c + d*x))^3,x)
Output:
(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
Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.19 \[ \int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-3 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+8 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\cos \left (d x +c \right )+1\right )}{\cos \left (d x +c \right ) d} \] Input:
int(cot(d*x+c)*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1) - 3*cos(c + d*x)*log(t an((c + d*x)/2) - 1) - 3*cos(c + d*x)*log(tan((c + d*x)/2) + 1) + 8*cos(c + d*x)*log(tan((c + d*x)/2)) - cos(c + d*x) + 1))/(cos(c + d*x)*d)