Integrand size = 21, antiderivative size = 239 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
b*(3*a^2-b^2)*x/(a^2+b^2)^3-a*(a^2-3*b^2)*ln(cos(d*x+c))/(a^2+b^2)^3/d-a^3 *(3*a^4+9*a^2*b^2+10*b^4)*ln(a+b*tan(d*x+c))/b^4/(a^2+b^2)^3/d+(3*a^4+6*a^ 2*b^2+b^4)*tan(d*x+c)/b^3/(a^2+b^2)^2/d-1/2*a^2*tan(d*x+c)^3/b/(a^2+b^2)/d /(a+b*tan(d*x+c))^2-1/2*a^2*(3*a^2+7*b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)^2/d/( a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.56 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {\log (i-\tan (c+d x))}{(a+i b)^3}+\frac {\log (i+\tan (c+d x))}{(a-i b)^3}-\frac {2 a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3}+\frac {a^3 \left (3 a^2+2 b^2\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (6 a^4+11 a^2 b^2+3 b^4\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \]
(Log[I - Tan[c + d*x]]/(a + I*b)^3 + Log[I + Tan[c + d*x]]/(a - I*b)^3 - ( 2*a^3*(3*a^4 + 9*a^2*b^2 + 10*b^4)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^ 2)^3) + (a^3*(3*a^2 + 2*b^2))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + ( 2*Tan[c + d*x]^3)/(b*(a + b*Tan[c + d*x])^2) - (2*a^2*(6*a^4 + 11*a^2*b^2 + 3*b^4))/(b^4*(a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*d)
Time = 1.51 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4048, 3042, 4128, 27, 3042, 4130, 25, 3042, 4109, 3042, 3956, 4100, 16}
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.
| \(\displaystyle \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^5}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {\tan ^2(c+d x) \left (3 a^2-2 b \tan (c+d x) a+\left (3 a^2+2 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x)^2 \left (3 a^2-2 b \tan (c+d x) a+\left (3 a^2+2 b^2\right ) \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {\int \frac {2 \tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (3 a^4+6 b^2 a^2+b^4\right ) \tan ^2(c+d x)+a^2 \left (3 a^2+7 b^2\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {\tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (3 a^4+6 b^2 a^2+b^4\right ) \tan ^2(c+d x)+a^2 \left (3 a^2+7 b^2\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \int \frac {\tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (3 a^4+6 b^2 a^2+b^4\right ) \tan (c+d x)^2+a^2 \left (3 a^2+7 b^2\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int -\frac {-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+3 a \left (a^2+b^2\right )^2 \tan ^2(c+d x)+a \left (3 a^4+6 b^2 a^2+b^4\right )}{a+b \tan (c+d x)}dx}{b}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+3 a \left (a^2+b^2\right )^2 \tan ^2(c+d x)+a \left (3 a^4+6 b^2 a^2+b^4\right )}{a+b \tan (c+d x)}dx}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+3 a \left (a^2+b^2\right )^2 \tan (c+d x)^2+a \left (3 a^4+6 b^2 a^2+b^4\right )}{a+b \tan (c+d x)}dx}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {a b^3 \left (a^2-3 b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^4 x \left (3 a^2-b^2\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {a b^3 \left (a^2-3 b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^4 x \left (3 a^2-b^2\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^4 x \left (3 a^2-b^2\right )}{a^2+b^2}+\frac {a b^3 \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {b^4 x \left (3 a^2-b^2\right )}{a^2+b^2}+\frac {a b^3 \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^4 x \left (3 a^2-b^2\right )}{a^2+b^2}+\frac {a b^3 \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
-1/2*(a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (-((a ^2*(3*a^2 + 7*b^2)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (2*(-((-((b^4*(3*a^2 - b^2)*x)/(a^2 + b^2)) + (a*b^3*(a^2 - 3*b^2)*Log[ Cos[c + d*x]])/((a^2 + b^2)*d) + (a^3*(3*a^4 + 9*a^2*b^2 + 10*b^4)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d))/b) + ((3*a^4 + 6*a^2*b^2 + b^4)*Tan[c + d*x])/(b*d)))/(b*(a^2 + b^2)))/(2*b*(a^2 + b^2))
3.5.78.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a *C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[( 1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( a^2 + b^2) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & & NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C , 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[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, 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] && GtQ[m, 0] && LtQ[n, -1]
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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Time = 0.58 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )}{b^{3}}+\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (3 a^{2}+5 b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{5}}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{3} \left (3 a^{4}+9 a^{2} b^{2}+10 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(186\) |
| default | \(\frac {\frac {\tan \left (d x +c \right )}{b^{3}}+\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (3 a^{2}+5 b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{5}}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{3} \left (3 a^{4}+9 a^{2} b^{2}+10 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(186\) |
| norman | \(\frac {\frac {\tan ^{3}\left (d x +c \right )}{b d}+\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {a^{2} \left (9 a^{5}+17 a^{3} b^{2}+4 a \,b^{4}\right )}{2 d \,b^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a \left (6 a^{5}+11 a^{3} b^{2}+3 a \,b^{4}\right ) \tan \left (d x +c \right )}{d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{2} \left (3 a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a^{3} \left (3 a^{4}+9 a^{2} b^{2}+10 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{4}}\) | \(391\) |
| parallelrisch | \(\frac {-26 b^{2} a^{7}-21 b^{4} a^{5}-4 b^{6} a^{3}+2 \left (\tan ^{3}\left (d x +c \right )\right ) b^{9}-9 a^{9}+6 \left (\tan ^{3}\left (d x +c \right )\right ) a^{4} b^{5}+\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{5} b^{4}+2 \left (\tan ^{3}\left (d x +c \right )\right ) a^{6} b^{3}-3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{6}-12 \tan \left (d x +c \right ) a^{8} b -34 \tan \left (d x +c \right ) a^{6} b^{3}-28 \tan \left (d x +c \right ) a^{4} b^{5}-6 \tan \left (d x +c \right ) a^{2} b^{7}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{7} b^{2}-20 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{5} b^{4}+6 x \,a^{4} b^{5} d -2 x \,a^{2} b^{7} d -12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{8} b -36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{6} b^{3}-40 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{5}+6 \left (\tan ^{3}\left (d x +c \right )\right ) a^{2} b^{7}+12 x \tan \left (d x +c \right ) a^{3} b^{6} d -4 x \tan \left (d x +c \right ) a \,b^{8} d +6 x \left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{7} d -2 x \left (\tan ^{2}\left (d x +c \right )\right ) b^{9} d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{6}-3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{8}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{7} b^{2}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{5} b^{4}-20 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{6}+2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{5}-6 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{7}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{9}}{2 \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{4}}\) | \(620\) |
| risch | \(\frac {i x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}-\frac {6 i a x}{b^{4}}-\frac {6 i a c}{b^{4} d}+\frac {6 i a^{7} x}{b^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i a^{7} c}{b^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {18 i a^{5} x}{b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {18 i a^{5} c}{b^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {20 i a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {20 i a^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 i \left (8 i a^{4} b^{3}-6 i a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{7}-2 i b^{7} {\mathrm e}^{2 i \left (d x +c \right )}-6 i a^{6} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 i a^{2} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+8 a^{5} b^{2}+a \,b^{6}+3 i a^{6} b -3 i a^{6} b \,{\mathrm e}^{2 i \left (d x +c \right )}-10 i a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-5 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-3 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{2} b^{5}-i a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+2 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+i b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+i b^{7}+3 a^{3} b^{4}+3 a^{7} {\mathrm e}^{4 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (-i b +a \right )^{3} b^{3} d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{4} d}-\frac {3 a^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {9 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {10 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(866\) |
1/d*(1/b^3*tan(d*x+c)+1/(a^2+b^2)^3*(1/2*(a^3-3*a*b^2)*ln(1+tan(d*x+c)^2)+ (3*a^2*b-b^3)*arctan(tan(d*x+c)))-1/b^4*a^4*(3*a^2+5*b^2)/(a^2+b^2)^2/(a+b *tan(d*x+c))+1/2/b^4*a^5/(a^2+b^2)/(a+b*tan(d*x+c))^2-1/b^4*a^3*(3*a^4+9*a ^2*b^2+10*b^4)/(a^2+b^2)^3*ln(a+b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (235) = 470\).
Time = 0.29 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.30 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 \, a^{7} b^{2} + 9 \, a^{5} b^{4} - 2 \, {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d x - {\left (9 \, a^{7} b^{2} + 23 \, a^{5} b^{4} + 12 \, a^{3} b^{6} + 4 \, a b^{8} + 2 \, {\left (3 \, a^{2} b^{7} - b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{9} + 9 \, a^{7} b^{2} + 10 \, a^{5} b^{4} + {\left (3 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{8} b + 9 \, a^{6} b^{3} + 10 \, a^{4} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, a^{8} b + 6 \, a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7} + 2 \, {\left (3 \, a^{3} b^{6} - a b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} d\right )}} \]
-1/2*(3*a^7*b^2 + 9*a^5*b^4 - 2*(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*ta n(d*x + c)^3 - 2*(3*a^4*b^5 - a^2*b^7)*d*x - (9*a^7*b^2 + 23*a^5*b^4 + 12* a^3*b^6 + 4*a*b^8 + 2*(3*a^2*b^7 - b^9)*d*x)*tan(d*x + c)^2 + (3*a^9 + 9*a ^7*b^2 + 10*a^5*b^4 + (3*a^7*b^2 + 9*a^5*b^4 + 10*a^3*b^6)*tan(d*x + c)^2 + 2*(3*a^8*b + 9*a^6*b^3 + 10*a^4*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*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6 + (a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2 + 2*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c))*log(1/( tan(d*x + c)^2 + 1)) - 2*(3*a^8*b + 6*a^6*b^3 - 2*a^4*b^5 + a^2*b^7 + 2*(3 *a^3*b^6 - a*b^8)*d*x)*tan(d*x + c))/((a^6*b^6 + 3*a^4*b^8 + 3*a^2*b^10 + b^12)*d*tan(d*x + c)^2 + 2*(a^7*b^5 + 3*a^5*b^7 + 3*a^3*b^9 + a*b^11)*d*ta n(d*x + c) + (a^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*d)
Exception generated. \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
Time = 0.41 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.23 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, a^{7} + 9 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, a^{7} + 9 \, a^{5} b^{2} + 2 \, {\left (3 \, a^{6} b + 5 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
1/2*(2*(3*a^2*b - b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*( 3*a^7 + 9*a^5*b^2 + 10*a^3*b^4)*log(b*tan(d*x + c) + a)/(a^6*b^4 + 3*a^4*b ^6 + 3*a^2*b^8 + b^10) + (a^3 - 3*a*b^2)*log(tan(d*x + c)^2 + 1)/(a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6) - (5*a^7 + 9*a^5*b^2 + 2*(3*a^6*b + 5*a^4*b^3)* tan(d*x + c))/(a^6*b^4 + 2*a^4*b^6 + a^2*b^8 + (a^4*b^6 + 2*a^2*b^8 + b^10 )*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*tan(d*x + c)) + 2*tan(d *x + c)/b^3)/d
Time = 1.80 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.36 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, a^{7} + 9 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {9 \, a^{7} b^{2} \tan \left (d x + c\right )^{2} + 27 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} + 30 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 12 \, a^{8} b \tan \left (d x + c\right ) + 38 \, a^{6} b^{3} \tan \left (d x + c\right ) + 50 \, a^{4} b^{5} \tan \left (d x + c\right ) + 4 \, a^{9} + 13 \, a^{7} b^{2} + 21 \, a^{5} b^{4}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac {2 \, \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
1/2*(2*(3*a^2*b - b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (a^ 3 - 3*a*b^2)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(3*a^7 + 9*a^5*b^2 + 10*a^3*b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10) + (9*a^7*b^2*tan(d*x + c)^2 + 27*a^5*b^4*t an(d*x + c)^2 + 30*a^3*b^6*tan(d*x + c)^2 + 12*a^8*b*tan(d*x + c) + 38*a^6 *b^3*tan(d*x + c) + 50*a^4*b^5*tan(d*x + c) + 4*a^9 + 13*a^7*b^2 + 21*a^5* b^4)/((a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)^2) + 2 *tan(d*x + c)/b^3)/d
Time = 4.89 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.10 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^6+5\,a^4\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {5\,a^7+9\,a^5\,b^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^4+9\,a^2\,b^2+10\,b^4\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \]
tan(c + d*x)/(b^3*d) - ((tan(c + d*x)*(3*a^6 + 5*a^4*b^2))/(a^4 + b^4 + 2* a^2*b^2) + (5*a^7 + 9*a^5*b^2)/(2*b*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2*b^3 + b^5*tan(c + d*x)^2 + 2*a*b^4*tan(c + d*x))) - (log(tan(c + d*x) - 1i)*1i )/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) - log(tan(c + d*x) + 1i)/(2*d* (3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (a^3*log(a + b*tan(c + d*x))*(3*a^4 + 10*b^4 + 9*a^2*b^2))/(b^4*d*(a^2 + b^2)^3)