Integrand size = 40, antiderivative size = 103 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {(a B+b C) x}{a^2+b^2}-\frac {B \cot (c+d x)}{a d}-\frac {(b B-a C) \log (\sin (c+d x))}{a^2 d}+\frac {b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right ) d} \]
-(B*a+C*b)*x/(a^2+b^2)-B*cot(d*x+c)/a/d-(B*b-C*a)*ln(sin(d*x+c))/a^2/d+b^2 *(B*b-C*a)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^2/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {-\frac {2 B \cot (c+d x)}{a}+\frac {i (B+i C) \log (i-\tan (c+d x))}{a+i b}+\frac {2 (-b B+a C) \log (\tan (c+d x))}{a^2}-\frac {(i B+C) \log (i+\tan (c+d x))}{a-i b}+\frac {2 b^2 (b B-a C) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )}}{2 d} \]
((-2*B*Cot[c + d*x])/a + (I*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) + ( 2*(-(b*B) + a*C)*Log[Tan[c + d*x]])/a^2 - ((I*B + C)*Log[I + Tan[c + d*x]] )/(a - I*b) + (2*b^2*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2) ))/(2*d)
Time = 0.81 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4115, 3042, 4092, 3042, 4134, 3042, 25, 3956, 4013}
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.
| \(\displaystyle \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B \tan (c+d x)+C \tan (c+d x)^2}{\tan (c+d x)^3 (a+b \tan (c+d x))}dx\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle \int \frac {\cot ^2(c+d x) (B+C \tan (c+d x))}{a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B+C \tan (c+d x)}{\tan (c+d x)^2 (a+b \tan (c+d x))}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) \left (b B \tan ^2(c+d x)+a B \tan (c+d x)+b B-a C\right )}{a+b \tan (c+d x)}dx}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {b B \tan (c+d x)^2+a B \tan (c+d x)+b B-a C}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle -\frac {-\frac {b^2 (b B-a C) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {(b B-a C) \int \cot (c+d x)dx}{a}+\frac {a x (a B+b C)}{a^2+b^2}}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {b^2 (b B-a C) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {(b B-a C) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {a x (a B+b C)}{a^2+b^2}}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {b^2 (b B-a C) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {(b B-a C) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}+\frac {a x (a B+b C)}{a^2+b^2}}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {-\frac {b^2 (b B-a C) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a x (a B+b C)}{a^2+b^2}+\frac {(b B-a C) \log (-\sin (c+d x))}{a d}}{a}-\frac {B \cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -\frac {-\frac {b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}+\frac {a x (a B+b C)}{a^2+b^2}+\frac {(b B-a C) \log (-\sin (c+d x))}{a d}}{a}-\frac {B \cot (c+d x)}{a d}\) |
-((B*Cot[c + d*x])/(a*d)) - ((a*(a*B + b*C)*x)/(a^2 + b^2) + ((b*B - a*C)* Log[-Sin[c + d*x]])/(a*d) - (b^2*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/a
3.1.30.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1) /(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^ 2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ ((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) *(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f* x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.18
| method | result | size |
| parallelrisch | \(\frac {\left (2 B \,b^{3}-2 C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+\left (B \,a^{2} b -C \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 \left (a^{2}+b^{2}\right ) \left (B b -C a \right ) \ln \left (\tan \left (d x +c \right )\right )-2 a \left (B \left (a^{2}+b^{2}\right ) \cot \left (d x +c \right )+a d x \left (B a +C b \right )\right )}{2 a^{2} d \left (a^{2}+b^{2}\right )}\) | \(122\) |
| derivativedivides | \(\frac {-\frac {B}{a \tan \left (d x +c \right )}+\frac {\left (-B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {\frac {\left (B b -C a \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B a -C b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {\left (B b -C a \right ) b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{2}}}{d}\) | \(123\) |
| default | \(\frac {-\frac {B}{a \tan \left (d x +c \right )}+\frac {\left (-B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {\frac {\left (B b -C a \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B a -C b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {\left (B b -C a \right ) b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{2}}}{d}\) | \(123\) |
| norman | \(\frac {-\frac {B \tan \left (d x +c \right )}{a d}-\frac {\left (B a +C b \right ) x \tan \left (d x +c \right )^{2}}{a^{2}+b^{2}}}{\tan \left (d x +c \right )^{2}}+\frac {\left (B b -C a \right ) b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} d \left (a^{2}+b^{2}\right )}-\frac {\left (B b -C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}+\frac {\left (B b -C a \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{2}+b^{2}\right )}\) | \(148\) |
| risch | \(\frac {x B}{i b -a}-\frac {i x C}{i b -a}+\frac {2 i B b x}{a^{2}}+\frac {2 i B b c}{a^{2} d}-\frac {2 i C x}{a}-\frac {2 i C c}{a d}-\frac {2 i b^{3} B x}{a^{2} \left (a^{2}+b^{2}\right )}-\frac {2 i b^{3} B c}{a^{2} d \left (a^{2}+b^{2}\right )}+\frac {2 i b^{2} C x}{a \left (a^{2}+b^{2}\right )}+\frac {2 i b^{2} C c}{a d \left (a^{2}+b^{2}\right )}-\frac {2 i B}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C}{a d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{a^{2} d \left (a^{2}+b^{2}\right )}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{a d \left (a^{2}+b^{2}\right )}\) | \(320\) |
1/2*((2*B*b^3-2*C*a*b^2)*ln(a+b*tan(d*x+c))+(B*a^2*b-C*a^3)*ln(sec(d*x+c)^ 2)-2*(a^2+b^2)*(B*b-C*a)*ln(tan(d*x+c))-2*a*(B*(a^2+b^2)*cot(d*x+c)+a*d*x* (B*a+C*b)))/a^2/d/(a^2+b^2)
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.72 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {2 \, B a^{3} + 2 \, B a b^{2} + 2 \, {\left (B a^{3} + C a^{2} b\right )} d x \tan \left (d x + c\right ) - {\left (C a^{3} - B a^{2} b + C a b^{2} - B b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + {\left (C a b^{2} - B b^{3}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\right )} d \tan \left (d x + c\right )} \]
-1/2*(2*B*a^3 + 2*B*a*b^2 + 2*(B*a^3 + C*a^2*b)*d*x*tan(d*x + c) - (C*a^3 - B*a^2*b + C*a*b^2 - B*b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan( d*x + c) + (C*a*b^2 - B*b^3)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1))*tan(d*x + c))/((a^4 + a^2*b^2)*d*tan(d*x + c) )
Result contains complex when optimal does not.
Time = 3.72 (sec) , antiderivative size = 2067, normalized size of antiderivative = 20.07 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((nan, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((-B*x - B/(d* tan(c + d*x)) - C*log(tan(c + d*x)**2 + 1)/(2*d) + C*log(tan(c + d*x))/d)/ a, Eq(b, 0)), ((B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c + d*x))/d - B/(2*d*tan(c + d*x)**2) - C*x - C/(d*tan(c + d*x)))/b, Eq(a, 0)), (-3*B*d *x*tan(c + d*x)**2/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - 3*I*B* d*x*tan(c + d*x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - I*B*log( tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan( c + d*x)) + B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) + 2*I*B*log(tan(c + d*x))*tan(c + d*x)**2/(2*a*d* tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - 2*B*log(tan(c + d*x))*tan(c + d* x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - 3*B*tan(c + d*x)/(2*a* d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - 2*I*B/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) + I*C*d*x*tan(c + d*x)**2/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - C*d*x*tan(c + d*x)/(2*a*d*tan(c + d*x)**2 + 2*I*a* d*tan(c + d*x)) - C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) - I*C*log(tan(c + d*x)**2 + 1)*tan(c + d *x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) + 2*C*log(tan(c + d*x)) *tan(c + d*x)**2/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)) + 2*I*C*lo g(tan(c + d*x))*tan(c + d*x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x) ) + I*C*tan(c + d*x)/(2*a*d*tan(c + d*x)**2 + 2*I*a*d*tan(c + d*x)), Eq...
Time = 0.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.27 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {\frac {2 \, {\left (B a + C b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b^{2} - B b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + a^{2} b^{2}} + \frac {{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, {\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{2}} + \frac {2 \, B}{a \tan \left (d x + c\right )}}{2 \, d} \]
-1/2*(2*(B*a + C*b)*(d*x + c)/(a^2 + b^2) + 2*(C*a*b^2 - B*b^3)*log(b*tan( d*x + c) + a)/(a^4 + a^2*b^2) + (C*a - B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 2*(C*a - B*b)*log(tan(d*x + c))/a^2 + 2*B/(a*tan(d*x + c)))/d
Time = 1.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {\frac {2 \, {\left (B a + C b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b^{3} - B b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + a^{2} b^{3}} - \frac {2 \, {\left (C a - B b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (C a \tan \left (d x + c\right ) - B b \tan \left (d x + c\right ) + B a\right )}}{a^{2} \tan \left (d x + c\right )}}{2 \, d} \]
-1/2*(2*(B*a + C*b)*(d*x + c)/(a^2 + b^2) + (C*a - B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 2*(C*a*b^3 - B*b^4)*log(abs(b*tan(d*x + c) + a))/(a^4* b + a^2*b^3) - 2*(C*a - B*b)*log(abs(tan(d*x + c)))/a^2 + 2*(C*a*tan(d*x + c) - B*b*tan(d*x + c) + B*a)/(a^2*tan(d*x + c)))/d
Time = 9.88 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.36 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b^3-C\,a\,b^2\right )}{d\,\left (a^4+a^2\,b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b-C\,a\right )}{a^2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {B\,\mathrm {cot}\left (c+d\,x\right )}{a\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \]