Integrand size = 21, antiderivative size = 155 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \] Output:
(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*ln(cos(d*x+c))/(a^2+b^2)^2/d-2*a^3*(a^2+2*b^ 2)*ln(a+b*tan(d*x+c))/b^3/(a^2+b^2)^2/d+(2*a^2+b^2)*tan(d*x+c)/b^2/(a^2+b^ 2)/d-a^2*tan(d*x+c)^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {4 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2}-\frac {2 a^2 \left (2 a^2+b^2\right )}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 \tan ^2(c+d x)}{b (a+b \tan (c+d x))}}{2 d} \] Input:
Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]
Output:
(((-I)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (4*a^3*(a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2 ) - (2*a^2*(2*a^2 + b^2))/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])) + (2*Tan[ c + d*x]^2)/(b*(a + b*Tan[c + d*x])))/(2*d)
Time = 1.01 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4048, 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.
| \(\displaystyle \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^4}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan (c+d x)^2\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -\frac {\tan (c+d x) b^3+2 a \left (a^2+b^2\right ) \tan ^2(c+d x)+a \left (2 a^2+b^2\right )}{a+b \tan (c+d x)}dx}{b}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {\tan (c+d x) b^3+2 a \left (a^2+b^2\right ) \tan ^2(c+d x)+a \left (2 a^2+b^2\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {\tan (c+d x) b^3+2 a \left (a^2+b^2\right ) \tan (c+d x)^2+a \left (2 a^2+b^2\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a b^3 \int \tan (c+d x)dx}{a^2+b^2}+\frac {2 a^3 \left (a^2+2 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^2 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a b^3 \int \tan (c+d x)dx}{a^2+b^2}+\frac {2 a^3 \left (a^2+2 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^2 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a^3 \left (a^2+2 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {2 a b^3 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a^3 \left (a^2+2 b^2\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {2 a b^3 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^2 x \left (a^2-b^2\right )}{a^2+b^2}-\frac {2 a b^3 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
Input:
Int[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]
Output:
-((a^2*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (-((-((b^ 2*(a^2 - b^2)*x)/(a^2 + b^2)) - (2*a*b^3*Log[Cos[c + d*x]])/((a^2 + b^2)*d ) + (2*a^3*(a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d))/b) + ((2*a^2 + b^2)*Tan[c + d*x])/(b*d))/(b*(a^2 + b^2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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[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 = 0.62 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )}{b^{2}}-\frac {a^{4}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(125\) |
| default | \(\frac {\frac {\tan \left (d x +c \right )}{b^{2}}-\frac {a^{4}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(125\) |
| norman | \(\frac {\frac {\tan \left (d x +c \right )^{2}}{b d}+\frac {\left (a^{2}-b^{2}\right ) a x}{a^{4}+2 b^{2} a^{2}+b^{4}}+\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 b^{2} a^{2}+b^{4}}-\frac {\left (2 a^{3}+a \,b^{2}\right ) a}{d \,b^{3} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}-\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}\) | \(207\) |
| parallelrisch | \(-\frac {-x \tan \left (d x +c \right ) a^{2} b^{4} d +x \tan \left (d x +c \right ) b^{6} d +\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a \,b^{5}+2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b +4 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{3}-x \,a^{3} b^{3} d +a x \,b^{5} d -\tan \left (d x +c \right )^{2} a^{4} b^{2}-2 \tan \left (d x +c \right )^{2} a^{2} b^{4}-\tan \left (d x +c \right )^{2} b^{6}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{2} b^{4}+2 a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )+4 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4} b^{2}+2 a^{6}+3 a^{4} b^{2}+a^{2} b^{4}}{\left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}\) | \(272\) |
| risch | \(-\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {4 i a^{5} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3}}+\frac {4 i a^{5} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}+\frac {8 i a^{3} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b}+\frac {8 i a^{3} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b d}-\frac {4 i a x}{b^{3}}-\frac {4 i a c}{b^{3} d}+\frac {2 i \left (-2 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4}+2 b^{2} a^{2}+b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) b^{2} d}-\frac {2 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b d}+\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\) | \(436\) |
Input:
int(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/b^2*tan(d*x+c)-1/b^3*a^4/(a^2+b^2)/(a+b*tan(d*x+c))-2/b^3*a^3*(a^2+ 2*b^2)/(a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^2*(-a*b*ln(1+tan(d*x+c)^ 2)+(a^2-b^2)*arctan(tan(d*x+c))))
Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.86 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^{4} b^{2} - {\left (a^{3} b^{3} - a b^{5}\right )} d x - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{6} + 2 \, a^{4} b^{2} + {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, a^{5} b + 2 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d} \] Input:
integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-(a^4*b^2 - (a^3*b^3 - a*b^5)*d*x - (a^4*b^2 + 2*a^2*b^4 + b^6)*tan(d*x + c)^2 + (a^6 + 2*a^4*b^2 + (a^5*b + 2*a^3*b^3)*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)) - (a^6 + 2*a^4 *b^2 + a^2*b^4 + (a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - (2*a^5*b + 2*a^3*b^3 + a*b^5 + (a^2*b^4 - b^6)*d*x)*tan(d*x + c))/((a^4*b^4 + 2*a^2*b^6 + b^8)*d*tan(d*x + c) + (a^5*b^3 + 2*a^3*b^5 + a*b^7)*d)
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 2312, normalized size of antiderivative = 14.92 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)**4/(a+b*tan(d*x+c))**2,x)
Output:
Piecewise((zoo*x*tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((x + tan(c + d*x)**3/(3*d) - tan(c + d*x)/d)/a**2, Eq(b, 0)), (-9*d*x*tan(c + d*x)**2/ (4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 18*I*d*x *tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2 *d) + 9*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d ) + 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4 *I*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2 *d*tan(c + d*x) - 4*b**2*d) + 19*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 14*I/(4*b**2*d*tan(c + d*x)**2 - 8*I *b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (-9*d*x*tan(c + d*x)**2/(4 *b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 18*I*d*x*t an(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d ) + 9*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I *log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c ...
Time = 0.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {a^{4}}{a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )} + \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac {\tan \left (d x + c\right )}{b^{2}}}{d} \] Input:
integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-(a^4/(a^3*b^3 + a*b^5 + (a^2*b^4 + b^6)*tan(d*x + c)) + a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^ 2 + b^4) + 2*(a^5 + 2*a^3*b^2)*log(b*tan(d*x + c) + a)/(a^4*b^3 + 2*a^2*b^ 5 + b^7) - tan(d*x + c)/b^2)/d
Time = 0.22 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} - \frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{3} d + 2 \, a^{2} b^{5} d + b^{7} d} + \frac {\tan \left (d x + c\right )}{b^{2} d} - \frac {a^{6} + a^{4} b^{2}}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} b^{3} d} \] Input:
integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
-a*b*log(tan(d*x + c)^2 + 1)/(a^4*d + 2*a^2*b^2*d + b^4*d) + (a^2 - b^2)*( d*x + c)/(a^4*d + 2*a^2*b^2*d + b^4*d) - 2*(a^5 + 2*a^3*b^2)*log(abs(b*tan (d*x + c) + a))/(a^4*b^3*d + 2*a^2*b^5*d + b^7*d) + tan(d*x + c)/(b^2*d) - (a^6 + a^4*b^2)/((a^2 + b^2)^2*(b*tan(d*x + c) + a)*b^3*d)
Time = 1.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )}{b^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^4}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^3+a\,b^2\right )\,\left (a^2+b^2\right )}-\frac {2\,a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+2\,b^2\right )}{b^3\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \] Input:
int(tan(c + d*x)^4/(a + b*tan(c + d*x))^2,x)
Output:
tan(c + d*x)/(b^2*d) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^ 2)) - log(tan(c + d*x) - 1i)/(2*d*(2*a*b - a^2*1i + b^2*1i)) - a^4/(b*d*(a *b^2 + b^3*tan(c + d*x))*(a^2 + b^2)) - (2*a^3*log(a + b*tan(c + d*x))*(a^ 2 + 2*b^2))/(b^3*d*(a^2 + b^2)^2)
Time = 0.15 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.01 \[ \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a \,b^{5}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{2} b^{4}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{5} b -4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{6}-4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b^{2}+\tan \left (d x +c \right )^{2} a^{4} b^{2}+2 \tan \left (d x +c \right )^{2} a^{2} b^{4}+\tan \left (d x +c \right )^{2} b^{6}+2 \tan \left (d x +c \right ) a^{5} b +3 \tan \left (d x +c \right ) a^{3} b^{3}+\tan \left (d x +c \right ) a^{2} b^{4} d x +\tan \left (d x +c \right ) a \,b^{5}-\tan \left (d x +c \right ) b^{6} d x +a^{3} b^{3} d x -a \,b^{5} d x}{b^{3} d \left (\tan \left (d x +c \right ) a^{4} b +2 \tan \left (d x +c \right ) a^{2} b^{3}+\tan \left (d x +c \right ) b^{5}+a^{5}+2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:
int(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x)
Output:
( - log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a*b**5 - log(tan(c + d*x)**2 + 1 )*a**2*b**4 - 2*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**5*b - 4*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**3*b**3 - 2*log(tan(c + d*x)*b + a)*a**6 - 4* log(tan(c + d*x)*b + a)*a**4*b**2 + tan(c + d*x)**2*a**4*b**2 + 2*tan(c + d*x)**2*a**2*b**4 + tan(c + d*x)**2*b**6 + 2*tan(c + d*x)*a**5*b + 3*tan(c + d*x)*a**3*b**3 + tan(c + d*x)*a**2*b**4*d*x + tan(c + d*x)*a*b**5 - tan (c + d*x)*b**6*d*x + a**3*b**3*d*x - a*b**5*d*x)/(b**3*d*(tan(c + d*x)*a** 4*b + 2*tan(c + d*x)*a**2*b**3 + tan(c + d*x)*b**5 + a**5 + 2*a**3*b**2 + a*b**4))