\(\int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [468]

Optimal result

Integrand size = 21, antiderivative size = 197 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^2 d}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

2*a*b*x/(a^2+b^2)^2-(a^2-b^2)*ln(cos(d*x+c))/(a^2+b^2)^2/d+a^4*(3*a^2+5*b^2)*ln(a+b*tan(d*x+c))/b^4/(a^2+b^2)^
2/d-a*(3*a^2+2*b^2)*tan(d*x+c)/b^3/(a^2+b^2)/d+1/2*(3*a^2+b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)/d-a^2*tan(d*x+c)^3/b
/(a^2+b^2)/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3728, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )}-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )}+\frac {a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^2} \]

[In]

Int[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]

[Out]

(2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^4*(3*a^2 + 5*b^2)*Log[a + b*T
an[c + d*x]])/(b^4*(a^2 + b^2)^2*d) - (a*(3*a^2 + 2*b^2)*Tan[c + d*x])/(b^3*(a^2 + b^2)*d) + ((3*a^2 + b^2)*Ta
n[c + d*x]^2)/(2*b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(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] - D
ist[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] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3698

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
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]

Rule 3707

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] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), I
nt[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(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]

Rule 3728

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] + Dist[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] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^2(c+d x) \left (3 a^2-a b \tan (c+d x)+\left (3 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 {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-2 a \left (3 a^2+b^2\right )-2 b^3 \tan (c+d x)-2 a \left (3 a^2+2 b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )} \\ & = -\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {2 a^2 \left (3 a^2+2 b^2\right )+2 a b^3 \tan (c+d x)+2 \left (3 a^2-b^2\right ) \left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )} \\ & = \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2-b^2\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^4 \left (3 a^2+5 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^2} \\ & = \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^4 \left (3 a^2+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^2 d} \\ & = \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^2 d}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {b \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {b \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2}+\frac {6 a^5+4 a^3 b^2}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {3 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))}+\frac {\tan ^3(c+d x)}{a+b \tan (c+d x)}}{2 b d} \]

[In]

Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]

[Out]

((b*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (b*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*a^4*(3*a^2 + 5*b^2)*Log[a
+ b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2) + (6*a^5 + 4*a^3*b^2)/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])) - (3*a*Tan
[c + d*x]^2)/(b*(a + b*Tan[c + d*x])) + Tan[c + d*x]^3/(a + b*Tan[c + d*x]))/(2*b*d)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 a \tan \left (d x +c \right )}{b^{3}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{5}}{b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(142\)
default	\(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 a \tan \left (d x +c \right )}{b^{3}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{5}}{b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(142\)
norman	\(\frac {\frac {\left (3 a^{4}+2 a^{2} b^{2}\right ) a}{d \,b^{4} \left (a^{2}+b^{2}\right )}+\frac {\tan ^{3}\left (d x +c \right )}{2 b d}-\frac {3 a \left (\tan ^{2}\left (d x +c \right )\right )}{2 b^{2} d}+\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d \,b^{4}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(225\)
parallelrisch	\(\frac {-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) b^{7}+6 a^{7}+10 a^{5} b^{2}+\left (\tan ^{3}\left (d x +c \right )\right ) a^{4} b^{3}+2 \left (\tan ^{3}\left (d x +c \right )\right ) a^{2} b^{5}-3 \left (\tan ^{2}\left (d x +c \right )\right ) a^{5} b^{2}-6 \left (\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{4}-3 \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{6}+\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{4}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{6}+10 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{5} b^{2}+4 a^{3} b^{4}+4 a^{2} b^{5} x d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{5}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{6} b +10 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{3}+\left (\tan ^{3}\left (d x +c \right )\right ) b^{7}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{7}+4 b^{6} a \tan \left (d x +c \right ) x d}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d \,b^{4}}\)	\(337\)
risch	\(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {6 i a^{2} x}{b^{4}}+\frac {6 i a^{2} c}{b^{4} d}-\frac {2 i x}{b^{2}}-\frac {2 i c}{b^{2} d}-\frac {6 i a^{6} x}{b^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {6 i a^{6} c}{b^{4} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {10 i a^{4} x}{b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {10 i a^{4} c}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i \left (2 a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+4 a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-4 i a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a^{3} b^{2}+2 a \,b^{4}+6 a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{5}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) b^{3} d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{4} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}+\frac {3 a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{4} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {5 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(582\)

[In]

int(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/b^3*(-1/2*b*tan(d*x+c)^2+2*a*tan(d*x+c))+1/(a^2+b^2)^2*(1/2*(a^2-b^2)*ln(1+tan(d*x+c)^2)+2*a*b*arctan(
tan(d*x+c)))+1/b^4*a^4*(3*a^2+5*b^2)/(a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/b^4*a^5/(a^2+b^2)/(a+b*tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.73 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, a^{2} b^{5} d x + 3 \, a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{7} + 5 \, a^{5} b^{2} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, a^{7} + 5 \, a^{5} b^{2} + a^{3} b^{4} - a b^{6} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (4 \, a b^{6} d x - 6 \, a^{6} b - 7 \, a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} d\right )}} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/2*(4*a^2*b^5*d*x + 3*a^5*b^2 + 2*a^3*b^4 + a*b^6 + (a^4*b^3 + 2*a^2*b^5 + b^7)*tan(d*x + c)^3 - 3*(a^5*b^2 +
2*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + (3*a^7 + 5*a^5*b^2 + (3*a^6*b + 5*a^4*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)) - (3*a^7 + 5*a^5*b^2 + a^3*b^4 - a*b^6 + (3*a^6*b +
5*a^4*b^3 + a^2*b^5 - b^7)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) + (4*a*b^6*d*x - 6*a^6*b - 7*a^4*b^3 - 2*
a^2*b^5 + b^7)*tan(d*x + c))/((a^4*b^5 + 2*a^2*b^7 + b^9)*d*tan(d*x + c) + (a^5*b^4 + 2*a^3*b^6 + a*b^8)*d)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.54 (sec) , antiderivative size = 2837, normalized size of antiderivative = 14.40 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)**5/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x*tan(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**
4/(4*d) - tan(c + d*x)**2/(2*d))/a**2, Eq(b, 0)), (-15*I*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) - 30*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) + 15*I*d*x/(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)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 16*I*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) + 8*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) + 2*tan(c + d*x)**4/(4*b**2*d*tan(c + d*x)**2 -
8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*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) + 29*I*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) + 22/(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)), (15*I*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) - 30*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) - 15*I*d*x/(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)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 16*I*lo
g(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) + 8*log(ta
n(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) + 2*tan(c + d*x)**4/(4*b**2
*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*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) - 29*I*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) + 22/(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)), (x*tan(c)**5/(a + b*
tan(c))**2, Eq(d, 0)), (6*a**7*log(a/b + tan(c + d*x))/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b*
*6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 6*a**7/(2*a**5*b**4*d + 2*a**4*b**5*
d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 6*a**6*b*l
og(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*
d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 10*a**5*b**2*log(a/b + tan(c + d*x))/(2*a**5*b**4*d + 2
*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) -
3*a**5*b**2*tan(c + d*x)**2/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c
+ d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 10*a**5*b**2/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a
**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 10*a**4*b**3*log(a/b + tan(c +
d*x))*tan(c + d*x)/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) +
2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + a**4*b**3*tan(c + d*x)**3/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) +
4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + a**3*b**4*log(tan(c + d*x)
**2 + 1)/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d
+ 2*b**9*d*tan(c + d*x)) - 6*a**3*b**4*tan(c + d*x)**2/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b
**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 4*a**3*b**4/(2*a**5*b**4*d + 2*a**4
*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 4*a*
*2*b**5*d*x/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**
8*d + 2*b**9*d*tan(c + d*x)) + a**2*b**5*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**5*b**4*d + 2*a**4*b**5*d*
tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) + 2*a**2*b**5*
tan(c + d*x)**3/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a
*b**8*d + 2*b**9*d*tan(c + d*x)) + 4*a*b**6*d*x*tan(c + d*x)/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a
**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) - a*b**6*log(tan(c + d*x)**2 + 1
)/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b*
*9*d*tan(c + d*x)) - 3*a*b**6*tan(c + d*x)**2/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*
a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)) - b**7*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2
*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d
*tan(c + d*x)) + b**7*tan(c + d*x)**3/(2*a**5*b**4*d + 2*a**4*b**5*d*tan(c + d*x) + 4*a**3*b**6*d + 4*a**2*b**
7*d*tan(c + d*x) + 2*a*b**8*d + 2*b**9*d*tan(c + d*x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, a^{5}}{a^{3} b^{4} + a b^{6} + {\left (a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )} + \frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, a^{6} + 5 \, a^{4} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b \tan \left (d x + c\right )^{2} - 4 \, a \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(2*a^5/(a^3*b^4 + a*b^6 + (a^2*b^5 + b^7)*tan(d*x + c)) + 4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*a
^6 + 5*a^4*b^2)*log(b*tan(d*x + c) + a)/(a^4*b^4 + 2*a^2*b^6 + b^8) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4
+ 2*a^2*b^2 + b^4) + (b*tan(d*x + c)^2 - 4*a*tan(d*x + c))/b^3)/d

Giac [A] (verification not implemented)

none

Time = 1.61 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.12 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, a^{6} + 5 \, a^{4} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}} - \frac {2 \, {\left (3 \, a^{6} b \tan \left (d x + c\right ) + 5 \, a^{4} b^{3} \tan \left (d x + c\right ) + 2 \, a^{7} + 4 \, a^{5} b^{2}\right )}}{{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} + \frac {b^{2} \tan \left (d x + c\right )^{2} - 4 \, a b \tan \left (d x + c\right )}{b^{4}}}{2 \, d} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2
*(3*a^6 + 5*a^4*b^2)*log(abs(b*tan(d*x + c) + a))/(a^4*b^4 + 2*a^2*b^6 + b^8) - 2*(3*a^6*b*tan(d*x + c) + 5*a^
4*b^3*tan(d*x + c) + 2*a^7 + 4*a^5*b^2)/((a^4*b^4 + 2*a^2*b^6 + b^8)*(b*tan(d*x + c) + a)) + (b^2*tan(d*x + c)
^2 - 4*a*b*tan(d*x + c))/b^4)/d

Mupad [B] (verification not implemented)

Time = 4.87 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^2\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^6+5\,a^4\,b^2\right )}{d\,\left (a^4\,b^4+2\,a^2\,b^6+b^8\right )}-\frac {2\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {a^5}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^4+a\,b^3\right )\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]

[In]

int(tan(c + d*x)^5/(a + b*tan(c + d*x))^2,x)

[Out]

log(tan(c + d*x) - 1i)/(2*d*(a*b*2i + a^2 - b^2)) + (log(tan(c + d*x) + 1i)*1i)/(2*d*(2*a*b + a^2*1i - b^2*1i)
) + tan(c + d*x)^2/(2*b^2*d) + (log(a + b*tan(c + d*x))*(3*a^6 + 5*a^4*b^2))/(d*(b^8 + 2*a^2*b^6 + a^4*b^4)) -
(2*a*tan(c + d*x))/(b^3*d) + a^5/(b*d*(a*b^3 + b^4*tan(c + d*x))*(a^2 + b^2))