Integrand size = 21, antiderivative size = 189 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+(a^4-6*a^2*b^2+b^4)*ln(a*cos(d*x+c)+b*sin(d *x+c))/(a^2+b^2)^4/d-1/3*a^2*tan(d*x+c)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-1 /6*a^2*(a^2+7*b^2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-a*(a^2-3*b^2)/(a^2 +b^2)^3/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 6.25 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.05 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac {\frac {a}{3 b d (a+b \tan (c+d x))^3}+\frac {2 b \left (-\frac {a \left (-\frac {i \log (i-\tan (c+d x))}{2 (a+i b)^4}+\frac {i \log (i+\tan (c+d x))}{2 (a-i b)^4}+\frac {4 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac {b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{b}+\frac {-\frac {\log (i-\tan (c+d x))}{2 (i a-b)^3}+\frac {\log (i+\tan (c+d x))}{2 (i a+b)^3}+\frac {b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{b}\right )}{d}}{2 b} \]
-1/2*Tan[c + d*x]/(b*d*(a + b*Tan[c + d*x])^3) - (a/(3*b*d*(a + b*Tan[c + d*x])^3) + (2*b*(-((a*(((-1/2*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((I/ 2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a*(a - b)*b*(a + b)*Log[a + b*T an[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) - ( a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^ 2)^3*(a + b*Tan[c + d*x]))))/b) + (-1/2*Log[I - Tan[c + d*x]]/(I*a - b)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a* b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/b))/d)/(2*b)
Time = 1.09 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4048, 3042, 4111, 27, 3042, 4012, 3042, 4014, 3042, 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.
| \(\displaystyle \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^3}{(a+b \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {a^2-3 b \tan (c+d x) a+\left (a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^3}dx}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2-3 b \tan (c+d x) a+\left (a^2+3 b^2\right ) \tan (c+d x)^2}{(a+b \tan (c+d x))^3}dx}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \left (2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b\right )}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \frac {2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \int \frac {2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {\left (3 a^2-b^2\right ) b^2+a \left (a^2-3 b^2\right ) \tan (c+d x) b}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {\left (3 a^2-b^2\right ) b^2+a \left (a^2-3 b^2\right ) \tan (c+d x) b}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\frac {4 a b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {b \left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\frac {4 a b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {b \left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {-\frac {a^2 \left (a^2+7 b^2\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {3 \left (\frac {a b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {4 a b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {b \left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}}{a^2+b^2}\right )}{a^2+b^2}}{3 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
-1/3*(a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (-1/2*( a^2*(a^2 + 7*b^2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (3*(((4*a*b^ 2*(a^2 - b^2)*x)/(a^2 + b^2) - (b*(a^4 - 6*a^2*b^2 + b^4)*Log[a*Cos[c + d* x] + b*Sin[c + d*x]])/((a^2 + b^2)*d))/(a^2 + b^2) + (a*b*(a^2 - 3*b^2))/( (a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/(a^2 + b^2))/(3*b*(a^2 + b^2))
3.5.89.3.1 Defintions of rubi rules used
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_.)*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[(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[((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_)*((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_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((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[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Time = 0.46 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(206\) |
| default | \(\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(206\) |
| norman | \(\frac {\frac {\left (a^{6}+6 a^{4} b^{2}-3 a^{2} b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 a d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a \left (-3 a^{4} b +a^{2} b^{3}\right )}{3 d b \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (a^{6}+8 a^{4} b^{2}-9 a^{2} b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{6 a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\) | \(540\) |
| risch | \(-\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 i a \left (-3 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a \,b^{4}+9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+9 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+9 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-18 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+26 i a^{3} b^{2}+13 a^{4} b -22 a^{2} b^{3}+9 b^{5}\right )}{3 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} \left (-i a +b \right )^{3} d \left (i a +b \right )^{4}}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {6 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\) | \(790\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(879\) |
1/d*((a^4-6*a^2*b^2+b^4)/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-a*(a^2-3*b^2)/(a^2 +b^2)^3/(a+b*tan(d*x+c))-1/2*a^2*(a^2+3*b^2)/(a^2+b^2)^2/b^2/(a+b*tan(d*x+ c))^2+1/3*a^3/b^2/(a^2+b^2)/(a+b*tan(d*x+c))^3+1/(a^2+b^2)^4*(1/2*(-a^4+6* a^2*b^2-b^4)*ln(1+tan(d*x+c)^2)+(-4*a^3*b+4*a*b^3)*arctan(tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (185) = 370\).
Time = 0.26 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.78 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {3 \, a^{7} - 30 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + {\left (a^{6} b + 18 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x + 3 \, {\left (a^{7} + 16 \, a^{5} b^{2} - 23 \, a^{3} b^{4} + 6 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 3 \, {\left (9 \, a^{6} b - 26 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \]
1/6*(3*a^7 - 30*a^5*b^2 + 11*a^3*b^4 + (a^6*b + 18*a^4*b^3 - 27*a^2*b^5 - 24*(a^3*b^4 - a*b^6)*d*x)*tan(d*x + c)^3 - 24*(a^6*b - a^4*b^3)*d*x + 3*(a ^7 + 16*a^5*b^2 - 23*a^3*b^4 + 6*a*b^6 - 24*(a^4*b^3 - a^2*b^5)*d*x)*tan(d *x + c)^2 + 3*(a^7 - 6*a^5*b^2 + a^3*b^4 + (a^4*b^3 - 6*a^2*b^5 + b^7)*tan (d*x + c)^3 + 3*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + 3*(a^6*b - 6*a^4*b^3 + a^2*b^5)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + 3*(9*a^6*b - 26*a^4*b^3 + 9*a^2*b^5 - 24*(a^5*b^2 - a^3*b^4)*d*x)*tan(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b ^7 + 4*a^2*b^9 + b^11)*d*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b ^6 + 4*a^3*b^8 + a*b^10)*d*tan(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6* b^5 + 4*a^4*b^7 + a^2*b^9)*d*tan(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8)*d)
Exception generated. \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (185) = 370\).
Time = 0.31 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.13 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{7} + 14 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, {\left (a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b + 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \]
-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^ 6 + b^8) - 6*(a^4 - 6*a^2*b^2 + b^4)*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6* b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (a^7 + 14*a ^5*b^2 - 11*a^3*b^4 + 6*(a^3*b^4 - 3*a*b^6)*tan(d*x + c)^2 + 3*(a^6*b + 8* a^4*b^3 - 9*a^2*b^5)*tan(d*x + c))/(a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 + a^3* b^8 + (a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^10)*tan(d*x + c)^2 + 3*(a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*tan(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (185) = 370\).
Time = 0.90 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.12 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, a^{4} b^{5} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{7} \tan \left (d x + c\right )^{3} + 11 \, b^{9} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 15 \, a b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} b \tan \left (d x + c\right ) + 60 \, a^{6} b^{3} \tan \left (d x + c\right ) - 201 \, a^{4} b^{5} \tan \left (d x + c\right ) + 6 \, a^{2} b^{7} \tan \left (d x + c\right ) + a^{9} + 26 \, a^{7} b^{2} - 63 \, a^{5} b^{4}}{{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]
-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^ 6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6* b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4*b - 6*a^2*b^3 + b^5)*log(abs(b *tan(d*x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (1 1*a^4*b^5*tan(d*x + c)^3 - 66*a^2*b^7*tan(d*x + c)^3 + 11*b^9*tan(d*x + c) ^3 + 39*a^5*b^4*tan(d*x + c)^2 - 210*a^3*b^6*tan(d*x + c)^2 + 15*a*b^8*tan (d*x + c)^2 + 3*a^8*b*tan(d*x + c) + 60*a^6*b^3*tan(d*x + c) - 201*a^4*b^5 *tan(d*x + c) + 6*a^2*b^7*tan(d*x + c) + a^9 + 26*a^7*b^2 - 63*a^5*b^4)/(( a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)^3 ))/d
Time = 4.94 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.90 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {a\,\left (a^6+14\,a^4\,b^2-11\,a^2\,b^4\right )}{6\,b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+8\,a^4\,b^2-9\,a^2\,b^4\right )}{2\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]
(log(a + b*tan(c + d*x))*(1/(a^2 + b^2)^2 - (8*b^2)/(a^2 + b^2)^3 + (8*b^4 )/(a^2 + b^2)^4))/d - (log(tan(c + d*x) + 1i)*1i)/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) - log(tan(c + d*x) - 1i)/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) - ((a*(a^6 - 11*a^2*b^4 + 14*a^4*b^2)) /(6*b^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^2*(3*a*b^4 - a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (tan(c + d*x)*(a^6 - 9*a^2 *b^4 + 8*a^4*b^2))/(2*b*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^ 3*tan(c + d*x)^3 + 3*a*b^2*tan(c + d*x)^2 + 3*a^2*b*tan(c + d*x)))