Integrand size = 29, antiderivative size = 87 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=(a A-b B) x+\frac {(a A-b B) \cot (c+d x)}{d}-\frac {(A b+a B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}-\frac {(A b+a B) \log (\sin (c+d x))}{d} \]
(A*a-B*b)*x+(A*a-B*b)*cot(d*x+c)/d-1/2*(A*b+B*a)*cot(d*x+c)^2/d-1/3*a*A*co t(d*x+c)^3/d-(A*b+B*a)*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {2 a A \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )+6 b B \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )+3 (A b+a B) \left (\cot ^2(c+d x)+2 (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )}{6 d} \]
-1/6*(2*a*A*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^ 2] + 6*b*B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(A*b + a*B)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]) ))/d
Time = 0.64 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 4074, 3042, 4012, 25, 3042, 4012, 3042, 4014, 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.
| \(\displaystyle \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 4074 |
| \(\displaystyle \int \cot ^3(c+d x) (A b+a B-(a A-b B) \tan (c+d x))dx-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A b+a B-(a A-b B) \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\cot ^2(c+d x) (a A-b B+(A b+a B) \tan (c+d x))dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot ^2(c+d x) (a A-b B+(A b+a B) \tan (c+d x))dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a A-b B+(A b+a B) \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \cot (c+d x) (A b+a B-(a A-b B) \tan (c+d x))dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {A b+a B-(a A-b B) \tan (c+d x)}{\tan (c+d x)}dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -(a B+A b) \int \cot (c+d x)dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}+x (a A-b B)-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -(a B+A b) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}+x (a A-b B)-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle (a B+A b) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}+x (a A-b B)-\frac {a A \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {(a B+A b) \cot ^2(c+d x)}{2 d}+\frac {(a A-b B) \cot (c+d x)}{d}-\frac {(a B+A b) \log (-\sin (c+d x))}{d}+x (a A-b B)-\frac {a A \cot ^3(c+d x)}{3 d}\) |
(a*A - b*B)*x + ((a*A - b*B)*Cot[c + d*x])/d - ((A*b + a*B)*Cot[c + d*x]^2 )/(2*d) - (a*A*Cot[c + d*x]^3)/(3*d) - ((A*b + a*B)*Log[-Sin[c + d*x]])/d
3.3.38.3.1 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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && N eQ[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_)]), x_Symbol] :> Simp[(b *c - a*d)*(A*b - a*B)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2 ))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m , -1] && NeQ[a^2 + b^2, 0]
Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a A -B b \right ) \arctan \left (\tan \left (d x +c \right )\right )+\left (-A b -B a \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A b +B a}{2 \tan \left (d x +c \right )^{2}}-\frac {-a A +B b}{\tan \left (d x +c \right )}-\frac {a A}{3 \tan \left (d x +c \right )^{3}}}{d}\) | \(106\) |
| default | \(\frac {\frac {\left (A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a A -B b \right ) \arctan \left (\tan \left (d x +c \right )\right )+\left (-A b -B a \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A b +B a}{2 \tan \left (d x +c \right )^{2}}-\frac {-a A +B b}{\tan \left (d x +c \right )}-\frac {a A}{3 \tan \left (d x +c \right )^{3}}}{d}\) | \(106\) |
| norman | \(\frac {\frac {\left (a A -B b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\left (a A -B b \right ) x \left (\tan ^{3}\left (d x +c \right )\right )-\frac {a A}{3 d}-\frac {\left (A b +B a \right ) \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}-\frac {\left (A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(117\) |
| parallelrisch | \(\frac {-2 A \left (\cot ^{3}\left (d x +c \right )\right ) a -3 A b \left (\cot ^{2}\left (d x +c \right )\right )+6 A x a d -3 B a \left (\cot ^{2}\left (d x +c \right )\right )-6 B b d x +6 a A \cot \left (d x +c \right )-6 A \ln \left (\tan \left (d x +c \right )\right ) b +3 A \ln \left (\sec ^{2}\left (d x +c \right )\right ) b -6 B b \cot \left (d x +c \right )-6 B \ln \left (\tan \left (d x +c \right )\right ) a +3 B \ln \left (\sec ^{2}\left (d x +c \right )\right ) a}{6 d}\) | \(123\) |
| risch | \(i A b x +i B a x +A a x -B b x +\frac {2 i A b c}{d}+\frac {2 i a B c}{d}-\frac {2 i \left (3 i A b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i B a \,{\mathrm e}^{4 i \left (d x +c \right )}-6 A a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 i A b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i B a \,{\mathrm e}^{2 i \left (d x +c \right )}+6 A a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 a A +3 B b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}\) | \(215\) |
1/d*(1/2*(A*b+B*a)*ln(1+tan(d*x+c)^2)+(A*a-B*b)*arctan(tan(d*x+c))+(-A*b-B *a)*ln(tan(d*x+c))-1/2*(A*b+B*a)/tan(d*x+c)^2-(-A*a+B*b)/tan(d*x+c)-1/3*a* A/tan(d*x+c)^3)
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {3 \, {\left (B a + A b\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} - 3 \, {\left (2 \, {\left (A a - B b\right )} d x - B a - A b\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (A a - B b\right )} \tan \left (d x + c\right )^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \]
-1/6*(3*(B*a + A*b)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^ 3 - 3*(2*(A*a - B*b)*d*x - B*a - A*b)*tan(d*x + c)^3 - 6*(A*a - B*b)*tan(d *x + c)^2 + 2*A*a + 3*(B*a + A*b)*tan(d*x + c))/(d*tan(d*x + c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (75) = 150\).
Time = 0.85 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.05 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right ) \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a x & \text {for}\: c = - d x \\A a x + \frac {A a}{d \tan {\left (c + d x \right )}} - \frac {A a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {A b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a}{2 d \tan ^{2}{\left (c + d x \right )}} - B b x - \frac {B b}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*A*a*x, Eq(c, 0) & Eq(d, 0)), (x*(A + B*tan(c))*(a + b*tan(c ))*cot(c)**4, Eq(d, 0)), (zoo*A*a*x, Eq(c, -d*x)), (A*a*x + A*a/(d*tan(c + d*x)) - A*a/(3*d*tan(c + d*x)**3) + A*b*log(tan(c + d*x)**2 + 1)/(2*d) - A*b*log(tan(c + d*x))/d - A*b/(2*d*tan(c + d*x)**2) + B*a*log(tan(c + d*x) **2 + 1)/(2*d) - B*a*log(tan(c + d*x))/d - B*a/(2*d*tan(c + d*x)**2) - B*b *x - B*b/(d*tan(c + d*x)), True))
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (A a - B b\right )} {\left (d x + c\right )} + 3 \, {\left (B a + A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (B a + A b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, {\left (A a - B b\right )} \tan \left (d x + c\right )^{2} - 2 \, A a - 3 \, {\left (B a + A b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
1/6*(6*(A*a - B*b)*(d*x + c) + 3*(B*a + A*b)*log(tan(d*x + c)^2 + 1) - 6*( B*a + A*b)*log(tan(d*x + c)) + (6*(A*a - B*b)*tan(d*x + c)^2 - 2*A*a - 3*( B*a + A*b)*tan(d*x + c))/tan(d*x + c)^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (83) = 166\).
Time = 0.76 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.72 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (A a - B b\right )} {\left (d x + c\right )} + 24 \, {\left (B a + A b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 44 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(A*a*tan(1/2*d*x + 1/2*c)^3 - 3*B*a*tan(1/2*d*x + 1/2*c)^2 - 3*A*b*ta n(1/2*d*x + 1/2*c)^2 - 15*A*a*tan(1/2*d*x + 1/2*c) + 12*B*b*tan(1/2*d*x + 1/2*c) + 24*(A*a - B*b)*(d*x + c) + 24*(B*a + A*b)*log(tan(1/2*d*x + 1/2*c )^2 + 1) - 24*(B*a + A*b)*log(abs(tan(1/2*d*x + 1/2*c))) + (44*B*a*tan(1/2 *d*x + 1/2*c)^3 + 44*A*b*tan(1/2*d*x + 1/2*c)^3 + 15*A*a*tan(1/2*d*x + 1/2 *c)^2 - 12*B*b*tan(1/2*d*x + 1/2*c)^2 - 3*B*a*tan(1/2*d*x + 1/2*c) - 3*A*b *tan(1/2*d*x + 1/2*c) - A*a)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 7.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.46 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\left (B\,b-A\,a\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {A\,a}{3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,b+B\,a\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )}{2\,d} \]