Integrand size = 19, antiderivative size = 61 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {a^2 \log (\cos (c+d x))}{d}+\frac {(a+b)^2 \log (1-\sec (c+d x))}{2 d}+\frac {(a-b)^2 \log (1+\sec (c+d x))}{2 d} \] Output:
a^2*ln(cos(d*x+c))/d+1/2*(a+b)^2*ln(1-sec(d*x+c))/d+1/2*(a-b)^2*ln(1+sec(d *x+c))/d
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {-2 a b \text {arctanh}(\cos (c+d x))-b^2 \log (\cos (c+d x))+\left (a^2+b^2\right ) \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]*(a + b*Sec[c + d*x])^2,x]
Output:
(-2*a*b*ArcTanh[Cos[c + d*x]] - b^2*Log[Cos[c + d*x]] + (a^2 + b^2)*Log[Si n[c + d*x]])/d
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 25, 4373, 524, 219, 240}
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.
\(\displaystyle \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^2}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle -\frac {b^2 \int \frac {\cos (c+d x) (a+b \sec (c+d x))^2}{b \left (b^2-b^2 \sec ^2(c+d x)\right )}d(b \sec (c+d x))}{d}\) |
\(\Big \downarrow \) 524 |
\(\displaystyle -\frac {b^2 \left (\frac {\left (a^2+b^2\right ) \int \frac {b \sec (c+d x)}{b^2-b^2 \sec ^2(c+d x)}d(b \sec (c+d x))}{b^2}+2 a \int \frac {1}{b^2-b^2 \sec ^2(c+d x)}d(b \sec (c+d x))+\frac {a^2 \log (b \sec (c+d x))}{b^2}\right )}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {b^2 \left (\frac {\left (a^2+b^2\right ) \int \frac {b \sec (c+d x)}{b^2-b^2 \sec ^2(c+d x)}d(b \sec (c+d x))}{b^2}+\frac {a^2 \log (b \sec (c+d x))}{b^2}+\frac {2 a \text {arctanh}(\sec (c+d x))}{b}\right )}{d}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle -\frac {b^2 \left (-\frac {\left (a^2+b^2\right ) \log \left (b^2-b^2 \sec ^2(c+d x)\right )}{2 b^2}+\frac {a^2 \log (b \sec (c+d x))}{b^2}+\frac {2 a \text {arctanh}(\sec (c+d x))}{b}\right )}{d}\) |
Input:
Int[Cot[c + d*x]*(a + b*Sec[c + d*x])^2,x]
Output:
-((b^2*((2*a*ArcTanh[Sec[c + d*x]])/b + (a^2*Log[b*Sec[c + d*x]])/b^2 - (( a^2 + b^2)*Log[b^2 - b^2*Sec[c + d*x]^2])/(2*b^2)))/d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))^2/((x_)*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[( c^2/a)*Log[x], x] + (-Simp[(b*c^2 - a*d^2)/a Int[x/(a + b*x^2), x], x] + Simp[2*c*d Int[1/(a + b*x^2), x], x]) /; FreeQ[{a, b, c, d}, x]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )+2 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+b^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
default | \(\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )+2 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+b^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(48\) |
risch | \(-i a^{2} x -\frac {2 i a^{2} c}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(153\) |
Input:
int(cot(d*x+c)*(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*ln(sin(d*x+c))+2*a*b*ln(csc(d*x+c)-cot(d*x+c))+b^2*ln(tan(d*x+c)) )
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {2 \, b^{2} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \] Input:
integrate(cot(d*x+c)*(a+b*sec(d*x+c))^2,x, algorithm="fricas")
Output:
-1/2*(2*b^2*log(-cos(d*x + c)) - (a^2 - 2*a*b + b^2)*log(1/2*cos(d*x + c) + 1/2) - (a^2 + 2*a*b + b^2)*log(-1/2*cos(d*x + c) + 1/2))/d
\[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cot {\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)*(a+b*sec(d*x+c))**2,x)
Output:
Integral((a + b*sec(c + d*x))**2*cot(c + d*x), x)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {2 \, b^{2} \log \left (\cos \left (d x + c\right )\right ) - {\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{2 \, d} \] Input:
integrate(cot(d*x+c)*(a+b*sec(d*x+c))^2,x, algorithm="maxima")
Output:
-1/2*(2*b^2*log(cos(d*x + c)) - (a^2 - 2*a*b + b^2)*log(cos(d*x + c) + 1) - (a^2 + 2*a*b + b^2)*log(cos(d*x + c) - 1))/d
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {b^{2} \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{2 \, d} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \] Input:
integrate(cot(d*x+c)*(a+b*sec(d*x+c))^2,x, algorithm="giac")
Output:
-b^2*log(abs(cos(d*x + c)))/d + 1/2*(a^2 - 2*a*b + b^2)*log(abs(cos(d*x + c) + 1))/d + 1/2*(a^2 + 2*a*b + b^2)*log(abs(cos(d*x + c) - 1))/d
Time = 10.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.57 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {b^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {b^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d}+\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \] Input:
int(cot(c + d*x)*(a + b/cos(c + d*x))^2,x)
Output:
(a^2*log(tan(c/2 + (d*x)/2)))/d + (b^2*log(tan(c/2 + (d*x)/2)))/d - (a^2*l og(tan(c/2 + (d*x)/2)^2 + 1))/d - (b^2*log(tan(c/2 + (d*x)/2)^2 - 1))/d + (2*a*b*log(tan(c/2 + (d*x)/2)))/d
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{2}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{2}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d} \] Input:
int(cot(d*x+c)*(a+b*sec(d*x+c))^2,x)
Output:
( - log(tan((c + d*x)/2)**2 + 1)*a**2 - log(tan((c + d*x)/2) - 1)*b**2 - l og(tan((c + d*x)/2) + 1)*b**2 + log(tan((c + d*x)/2))*a**2 + 2*log(tan((c + d*x)/2))*a*b + log(tan((c + d*x)/2))*b**2)/d