Integrand size = 21, antiderivative size = 69 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-a \left (a^2-3 b^2\right ) x-\frac {b^3 \log (\cos (c+d x))}{d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d} \] Output:
-a*(a^2-3*b^2)*x-b^3*ln(cos(d*x+c))/d+3*a^2*b*ln(sin(d*x+c))/d-a^2*cot(d*x +c)*(a+b*tan(d*x+c))/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a^3 \cot (c+d x)-\frac {1}{2} (i a+b)^3 \log (i-\cot (c+d x))+\frac {1}{2} (i a-b)^3 \log (i+\cot (c+d x))-b^3 \log (\tan (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]
Output:
-((a^3*Cot[c + d*x] - ((I*a + b)^3*Log[I - Cot[c + d*x]])/2 + ((I*a - b)^3 *Log[I + Cot[c + d*x]])/2 - b^3*Log[Tan[c + d*x]])/d)
Time = 0.43 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4048, 3042, 4107, 3042, 25, 3956}
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.
\(\displaystyle \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\tan (c+d x)^2}dx\) |
\(\Big \downarrow \) 4048 |
\(\displaystyle \int \cot (c+d x) \left (\tan ^2(c+d x) b^3+3 a^2 b-a \left (a^2-3 b^2\right ) \tan (c+d x)\right )dx-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^2 b^3+3 a^2 b-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan (c+d x)}dx-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 4107 |
\(\displaystyle 3 a^2 b \int \cot (c+d x)dx+b^3 \int \tan (c+d x)dx-a x \left (a^2-3 b^2\right )-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 a^2 b \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+b^3 \int \tan (c+d x)dx-a x \left (a^2-3 b^2\right )-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 a^2 b \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+b^3 \int \tan (c+d x)dx-a x \left (a^2-3 b^2\right )-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -a x \left (a^2-3 b^2\right )+\frac {3 a^2 b \log (-\sin (c+d x))}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}-\frac {b^3 \log (\cos (c+d x))}{d}\) |
Input:
Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]
Output:
-(a*(a^2 - 3*b^2)*x) - (b^3*Log[Cos[c + d*x]])/d + (3*a^2*b*Log[-Sin[c + d *x]])/d - (a^2*Cot[c + d*x]*(a + b*Tan[c + d*x]))/d
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[B*x, x] + (Simp[A Int[1/Tan[ e + f*x], x], x] + Simp[C Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A, B, C}, x] && NeQ[A, C]
Time = 0.91 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(-\frac {-3 a^{2} b \left (2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (\sec \left (d x +c \right )^{2}\right )\right )-\ln \left (\sec \left (d x +c \right )^{2}\right ) b^{3}+2 a^{3} d x -6 a \,b^{2} d x +2 a^{3} \cot \left (d x +c \right )}{2 d}\) | \(74\) |
derivativedivides | \(\frac {-\frac {a^{3}}{\tan \left (d x +c \right )}+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
default | \(\frac {-\frac {a^{3}}{\tan \left (d x +c \right )}+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
norman | \(\frac {\left (-a^{3}+3 a \,b^{2}\right ) x \tan \left (d x +c \right )-\frac {a^{3}}{d}}{\tan \left (d x +c \right )}+\frac {3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(84\) |
risch | \(-3 i a^{2} b x +i b^{3} x -a^{3} x +3 a \,b^{2} x -\frac {6 i b \,a^{2} c}{d}+\frac {2 i b^{3} c}{d}-\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(114\) |
Input:
int(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/2/d*(-3*a^2*b*(2*ln(tan(d*x+c))-ln(sec(d*x+c)^2))-ln(sec(d*x+c)^2)*b^3+ 2*a^3*d*x-6*a*b^2*d*x+2*a^3*cot(d*x+c))
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 \, a^{2} b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) - b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} d x \tan \left (d x + c\right ) - 2 \, a^{3}}{2 \, d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
1/2*(3*a^2*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c) - b^3*l og(1/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 2*(a^3 - 3*a*b^2)*d*x*tan(d*x + c) - 2*a^3)/(d*tan(d*x + c))
Time = 0.46 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.62 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\begin {cases} \tilde {\infty } a^{3} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{3} \cot ^{2}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{3} x & \text {for}\: c = - d x \\- a^{3} x - \frac {a^{3}}{d \tan {\left (c + d x \right )}} - \frac {3 a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 3 a b^{2} x + \frac {b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \] Input:
integrate(cot(d*x+c)**2*(a+b*tan(d*x+c))**3,x)
Output:
Piecewise((zoo*a**3*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**3*cot(c)** 2, Eq(d, 0)), (zoo*a**3*x, Eq(c, -d*x)), (-a**3*x - a**3/(d*tan(c + d*x)) - 3*a**2*b*log(tan(c + d*x)**2 + 1)/(2*d) + 3*a**2*b*log(tan(c + d*x))/d + 3*a*b**2*x + b**3*log(tan(c + d*x)**2 + 1)/(2*d), True))
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {6 \, a^{2} b \log \left (\tan \left (d x + c\right )\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} - {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac {2 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \] Input:
integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(6*a^2*b*log(tan(d*x + c)) - 2*(a^3 - 3*a*b^2)*(d*x + c) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1) - 2*a^3/tan(d*x + c))/d
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 \, a^{2} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{d} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} - \frac {a^{3}}{d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
3*a^2*b*log(abs(tan(d*x + c)))/d - (a^3 - 3*a*b^2)*(d*x + c)/d - 1/2*(3*a^ 2*b - b^3)*log(tan(d*x + c)^2 + 1)/d - a^3/(d*tan(d*x + c))
Time = 1.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3}{2\,d}-\frac {a^3\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {3\,a^2\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \] Input:
int(cot(c + d*x)^2*(a + b*tan(c + d*x))^3,x)
Output:
(log(tan(c + d*x) - 1i)*(a + b*1i)^3*1i)/(2*d) + (log(tan(c + d*x) + 1i)*( a*1i + b)^3)/(2*d) - (a^3*cot(c + d*x))/d + (3*a^2*b*log(tan(c + d*x)))/d
Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.45 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {-\cos \left (d x +c \right ) a^{3}-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a^{2} b +\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) b^{3}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right ) b^{3}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right ) b^{3}+3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a^{2} b -\sin \left (d x +c \right ) a^{3} d x +3 \sin \left (d x +c \right ) a \,b^{2} d x}{\sin \left (d x +c \right ) d} \] Input:
int(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x)
Output:
( - cos(c + d*x)*a**3 - 3*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**2*b + log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*b**3 - log(tan((c + d*x)/2) - 1)*sin(c + d*x)*b**3 - log(tan((c + d*x)/2) + 1)*sin(c + d*x)*b**3 + 3*lo g(tan((c + d*x)/2))*sin(c + d*x)*a**2*b - sin(c + d*x)*a**3*d*x + 3*sin(c + d*x)*a*b**2*d*x)/(sin(c + d*x)*d)