\(\int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [487]

Optimal result

Integrand size = 21, antiderivative size = 256 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a^2 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4 d}-\frac {a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+3 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

4*a*b*(a^2-b^2)*x/(a^2+b^2)^4-(a^4-6*a^2*b^2+b^4)*ln(cos(d*x+c))/(a^2+b^2)^4/d+a^2*(a^6+4*a^4*b^2+5*a^2*b^4+10
*b^6)*ln(a+b*tan(d*x+c))/b^4/(a^2+b^2)^4/d-1/3*a^2*tan(d*x+c)^3/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-1/2*a^2*(a^2+
3*b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2+a^3*(a^4+3*a^2*b^2+6*b^4)/b^4/(a^2+b^2)^3/d/(a+b*tan(
d*x+c))

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3646, 3726, 3716, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+3 b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {a^2 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))} \]

[In]

Int[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^4,x]

[Out]

(4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a^2*(a^
6 + 4*a^4*b^2 + 5*a^2*b^4 + 10*b^6)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) - (a^2*Tan[c + d*x]^3)/(3*b
*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 3*b^2)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[
c + d*x])^2) + (a^3*(a^4 + 3*a^2*b^2 + 6*b^4))/(b^4*(a^2 + b^2)^3*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 3716

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*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[c^2 + d^2, 0] && LtQ[n, -1]

Rule 3726

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\tan ^2(c+d x) \left (3 a^2-3 a b \tan (c+d x)+3 \left (a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )} \\ & = -\frac {a^2 \tan ^3(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+3 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))^2}+\frac {\int \frac {\tan (c+d x) \left (6 a^2 \left (a^2+3 b^2\right )-12 a b^3 \tan (c+d x)+6 \left (a^2+b^2\right )^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {a^2 \tan ^3(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+3 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))^2}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {6 a^2 \left (a^4+3 a^2 b^2+6 b^4\right )+6 a b^3 \left (a^2-3 b^2\right ) \tan (c+d x)+6 \left (a^2+b^2\right )^3 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )^3} \\ & = \frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {a^2 \tan ^3(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+3 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))^2}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^4}+\frac {\left (a^2 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\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 )^4} \\ & = \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 (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan ^3(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+3 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))^2}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\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 )^4 d} \\ & = \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 (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a^2 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4 d}-\frac {a^2 \tan ^3(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+3 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))^2}+\frac {a^3 \left (a^4+3 a^2 b^2+6 b^4\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.83 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \log (i-\tan (c+d x))}{(a+i b)^4}+\frac {3 \log (i+\tan (c+d x))}{(a-i b)^4}+\frac {a^2 \left (6 \left (a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\right ) \log (a+b \tan (c+d x))+\frac {a \left (a^2+b^2\right ) \left (11 a^6+34 a^4 b^2+47 a^2 b^4+3 a b \left (9 a^4+28 a^2 b^2+35 b^4\right ) \tan (c+d x)+6 b^2 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3}\right )}{b^4 \left (a^2+b^2\right )^4}}{6 d} \]

[In]

Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^4,x]

[Out]

((3*Log[I - Tan[c + d*x]])/(a + I*b)^4 + (3*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (a^2*(6*(a^6 + 4*a^4*b^2 + 5*
a^2*b^4 + 10*b^6)*Log[a + b*Tan[c + d*x]] + (a*(a^2 + b^2)*(11*a^6 + 34*a^4*b^2 + 47*a^2*b^4 + 3*a*b*(9*a^4 +
28*a^2*b^2 + 35*b^4)*Tan[c + d*x] + 6*b^2*(3*a^4 + 9*a^2*b^2 + 10*b^4)*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3
))/(b^4*(a^2 + b^2)^4))/(6*d)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2} \left (a^{6}+4 a^{4} b^{2}+5 a^{2} b^{4}+10 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} b^{4}}-\frac {a^{4} \left (3 a^{2}+5 b^{2}\right )}{2 b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{5}}{3 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{3} \left (3 a^{4}+9 a^{2} b^{2}+10 b^{4}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(234\)
default	\(\frac {\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2} \left (a^{6}+4 a^{4} b^{2}+5 a^{2} b^{4}+10 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} b^{4}}-\frac {a^{4} \left (3 a^{2}+5 b^{2}\right )}{2 b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{5}}{3 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{3} \left (3 a^{4}+9 a^{2} b^{2}+10 b^{4}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(234\)
norman	\(\frac {\frac {a \left (3 a^{6}+9 a^{4} b^{2}+10 a^{2} b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a^{3} \left (11 a^{6}+34 a^{4} b^{2}+47 a^{2} b^{4}\right )}{6 d \,b^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {4 \left (a^{2}-b^{2}\right ) b \,a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {a^{2} \left (9 a^{6}+28 a^{4} b^{2}+35 a^{2} b^{4}\right ) \tan \left (d x +c \right )}{2 b^{3} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{2} \left (a^{6}+4 a^{4} b^{2}+5 a^{2} b^{4}+10 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\)	\(566\)
parallelrisch	\(\text {Expression too large to display}\)	\(993\)
risch	\(\frac {2 i c}{d \,b^{4}}-\frac {20 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 {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {8 i a^{6} x}{b^{2} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {8 i a^{6} c}{b^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {10 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {2 i \left (8 b^{2} a^{7}+19 b^{4} a^{5}-30 b^{6} a^{3}-9 i a^{8} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a^{9}-45 i a^{4} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a^{4} b^{5}+3 a^{9} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{9} {\mathrm e}^{2 i \left (d x +c \right )}-18 i a^{6} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-9 b^{4} a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+84 b^{4} a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+30 b^{2} a^{7} {\mathrm e}^{2 i \left (d x +c \right )}-30 a^{3} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+60 a^{3} b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-15 i a^{4} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-30 i b^{3} a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 i b \,a^{8}+22 i a^{6} b^{3}-3 i a^{8} b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 \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 )^{3} \left (i b +a \right )^{3} b^{3} d \left (-i b +a \right )^{4}}-\frac {2 i a^{8} c}{b^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {10 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 {20 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 a^{8} x}{b^{4} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 i x}{b^{4}}+\frac {a^{8} \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 )}+\frac {4 a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {5 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 {10 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 )}+1\right )}{d \,b^{4}}\)	\(1045\)

[In]

int(tan(d*x+c)^5/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(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)))+a^2*(a^6+
4*a^4*b^2+5*a^2*b^4+10*b^6)/(a^2+b^2)^4/b^4*ln(a+b*tan(d*x+c))-1/2*a^4/b^4*(3*a^2+5*b^2)/(a^2+b^2)^2/(a+b*tan(
d*x+c))^2+1/3*a^5/b^4/(a^2+b^2)/(a+b*tan(d*x+c))^3+a^3/b^4*(3*a^4+9*a^2*b^2+10*b^4)/(a^2+b^2)^3/(a+b*tan(d*x+c
)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (252) = 504\).

Time = 0.31 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.06 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {3 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 47 \, a^{5} b^{6} - {\left (11 \, a^{8} b^{3} + 42 \, a^{6} b^{5} + 75 \, a^{4} b^{7} - 24 \, {\left (a^{3} b^{8} - a b^{10}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 24 \, {\left (a^{6} b^{5} - a^{4} b^{7}\right )} d x - 3 \, {\left (5 \, a^{9} b^{2} + 18 \, a^{7} b^{4} + 37 \, a^{5} b^{6} - 20 \, a^{3} b^{8} - 24 \, {\left (a^{4} b^{7} - a^{2} b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} + 4 \, a^{9} b^{2} + 5 \, a^{7} b^{4} + 10 \, a^{5} b^{6} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 5 \, a^{4} b^{7} + 10 \, a^{2} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 5 \, a^{5} b^{6} + 10 \, a^{3} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 5 \, a^{6} b^{5} + 10 \, a^{4} b^{7}\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^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \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 )} \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 )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a^{10} b + 5 \, a^{8} b^{3} + 12 \, a^{6} b^{5} - 35 \, a^{4} b^{7} - 24 \, {\left (a^{5} b^{6} - a^{3} b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{7} + 4 \, a^{6} b^{9} + 6 \, a^{4} b^{11} + 4 \, a^{2} b^{13} + b^{15}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{6} + 4 \, a^{7} b^{8} + 6 \, a^{5} b^{10} + 4 \, a^{3} b^{12} + a b^{14}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b^{5} + 4 \, a^{8} b^{7} + 6 \, a^{6} b^{9} + 4 \, a^{4} b^{11} + a^{2} b^{13}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} b^{4} + 4 \, a^{9} b^{6} + 6 \, a^{7} b^{8} + 4 \, a^{5} b^{10} + a^{3} b^{12}\right )} d\right )}} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*a^9*b^2 + 6*a^7*b^4 + 47*a^5*b^6 - (11*a^8*b^3 + 42*a^6*b^5 + 75*a^4*b^7 - 24*(a^3*b^8 - a*b^10)*d*x)*t
an(d*x + c)^3 + 24*(a^6*b^5 - a^4*b^7)*d*x - 3*(5*a^9*b^2 + 18*a^7*b^4 + 37*a^5*b^6 - 20*a^3*b^8 - 24*(a^4*b^7
- a^2*b^9)*d*x)*tan(d*x + c)^2 + 3*(a^11 + 4*a^9*b^2 + 5*a^7*b^4 + 10*a^5*b^6 + (a^8*b^3 + 4*a^6*b^5 + 5*a^4*
b^7 + 10*a^2*b^9)*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 5*a^5*b^6 + 10*a^3*b^8)*tan(d*x + c)^2 + 3*(a^10*b
+ 4*a^8*b^3 + 5*a^6*b^5 + 10*a^4*b^7)*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^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8 + (a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 +
4*a^2*b^9 + b^11)*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)*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)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - 3*(2*a^10*
b + 5*a^8*b^3 + 12*a^6*b^5 - 35*a^4*b^7 - 24*(a^5*b^6 - a^3*b^8)*d*x)*tan(d*x + c))/((a^8*b^7 + 4*a^6*b^9 + 6*
a^4*b^11 + 4*a^2*b^13 + b^15)*d*tan(d*x + c)^3 + 3*(a^9*b^6 + 4*a^7*b^8 + 6*a^5*b^10 + 4*a^3*b^12 + a*b^14)*d*
tan(d*x + c)^2 + 3*(a^10*b^5 + 4*a^8*b^7 + 6*a^6*b^9 + 4*a^4*b^11 + a^2*b^13)*d*tan(d*x + c) + (a^11*b^4 + 4*a
^9*b^6 + 6*a^7*b^8 + 4*a^5*b^10 + a^3*b^12)*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(tan(d*x+c)**5/(a+b*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [A] (verification not implemented)

none

Time = 0.61 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.69 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 10 \, a^{2} b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, a^{9} + 34 \, a^{7} b^{2} + 47 \, a^{5} b^{4} + 6 \, {\left (3 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (9 \, a^{8} b + 28 \, a^{6} b^{3} + 35 \, a^{4} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{4} + 3 \, a^{7} b^{6} + 3 \, a^{5} b^{8} + a^{3} b^{10} + {\left (a^{6} b^{7} + 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} + b^{13}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{6} + 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} + a b^{12}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{5} + 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} + a^{2} b^{11}\right )} \tan \left (d x + c\right )}}{6 \, d} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

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^8 + 4*a^6*b^2 + 5*a^4
*b^4 + 10*a^2*b^6)*log(b*tan(d*x + c) + a)/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) + 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) + (11*a^9 + 34*a^7*b^2
+ 47*a^5*b^4 + 6*(3*a^7*b^2 + 9*a^5*b^4 + 10*a^3*b^6)*tan(d*x + c)^2 + 3*(9*a^8*b + 28*a^6*b^3 + 35*a^4*b^5)*t
an(d*x + c))/(a^9*b^4 + 3*a^7*b^6 + 3*a^5*b^8 + a^3*b^10 + (a^6*b^7 + 3*a^4*b^9 + 3*a^2*b^11 + b^13)*tan(d*x +
c)^3 + 3*(a^7*b^6 + 3*a^5*b^8 + 3*a^3*b^10 + a*b^12)*tan(d*x + c)^2 + 3*(a^8*b^5 + 3*a^6*b^7 + 3*a^4*b^9 + a^
2*b^11)*tan(d*x + c)))/d

Giac [A] (verification not implemented)

none

Time = 2.04 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 10 \, a^{2} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}} - \frac {11 \, a^{8} b^{2} \tan \left (d x + c\right )^{3} + 44 \, a^{6} b^{4} \tan \left (d x + c\right )^{3} + 55 \, a^{4} b^{6} \tan \left (d x + c\right )^{3} + 110 \, a^{2} b^{8} \tan \left (d x + c\right )^{3} + 15 \, a^{9} b \tan \left (d x + c\right )^{2} + 60 \, a^{7} b^{3} \tan \left (d x + c\right )^{2} + 51 \, a^{5} b^{5} \tan \left (d x + c\right )^{2} + 270 \, a^{3} b^{7} \tan \left (d x + c\right )^{2} + 6 \, a^{10} \tan \left (d x + c\right ) + 21 \, a^{8} b^{2} \tan \left (d x + c\right ) - 24 \, a^{6} b^{4} \tan \left (d x + c\right ) + 225 \, a^{4} b^{6} \tan \left (d x + c\right ) - a^{9} b - 26 \, a^{7} b^{3} + 63 \, a^{5} b^{5}}{{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

[In]

integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

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^8 + 4*a^6*b^2 + 5*a^4*b^4 + 10*
a^2*b^6)*log(abs(b*tan(d*x + c) + a))/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) - (11*a^8*b^2*tan(
d*x + c)^3 + 44*a^6*b^4*tan(d*x + c)^3 + 55*a^4*b^6*tan(d*x + c)^3 + 110*a^2*b^8*tan(d*x + c)^3 + 15*a^9*b*tan
(d*x + c)^2 + 60*a^7*b^3*tan(d*x + c)^2 + 51*a^5*b^5*tan(d*x + c)^2 + 270*a^3*b^7*tan(d*x + c)^2 + 6*a^10*tan(
d*x + c) + 21*a^8*b^2*tan(d*x + c) - 24*a^6*b^4*tan(d*x + c) + 225*a^4*b^6*tan(d*x + c) - a^9*b - 26*a^7*b^3 +
63*a^5*b^5)/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*(b*tan(d*x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 5.08 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.46 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {11\,a^9+34\,a^7\,b^2+47\,a^5\,b^4}{6\,b^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (9\,a^8+28\,a^6\,b^2+35\,a^4\,b^4\right )}{2\,b^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^6+9\,a^4\,b^2+10\,a^2\,b^4\right )}{b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^6+4\,a^4\,b^2+5\,a^2\,b^4+10\,b^6\right )}{b^4\,d\,{\left (a^2+b^2\right )}^4}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]

[In]

int(tan(c + d*x)^5/(a + b*tan(c + d*x))^4,x)

[Out]

((11*a^9 + 47*a^5*b^4 + 34*a^7*b^2)/(6*b^4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(9*a^8 + 35*a^
4*b^4 + 28*a^6*b^2))/(2*b^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*tan(c + d*x)^2*(3*a^6 + 10*a^2*b^4 + 9*a
^4*b^2))/(b^2*(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))) + (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^2*log(a + b*tan(c + d*x))*(a^6
+ 10*b^6 + 5*a^2*b^4 + 4*a^4*b^2))/(b^4*d*(a^2 + b^2)^4)