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

Optimal. Leaf size=256 \[ -\frac{a^2 \left (a^2+3 b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 \left (3 a^2 b^2+a^4+6 b^4\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a^2 \left (4 a^4 b^2+5 a^2 b^4+a^6+10 b^6\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

[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]))

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Rubi [A]  time = 0.571474, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3565, 3645, 3635, 3626, 3617, 31, 3475} \[ -\frac{a^2 \left (a^2+3 b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 \left (3 a^2 b^2+a^4+6 b^4\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a^2 \left (4 a^4 b^2+5 a^2 b^4+a^6+10 b^6\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[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 3565

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 3645

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*T
an[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]

Rule 3635

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 3626

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 3617

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 31

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

Rule 3475

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

Rubi steps

\begin{align*} \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 \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 ) \operatorname{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]  time = 6.33977, size = 788, normalized size = 3.08 \[ -\frac{i \left (4 a^6 b^2+5 a^4 b^4+10 a^2 b^6+a^8\right ) \tan ^{-1}(\tan (c+d x)) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^4}-\frac{a^4 \left (3 a^2+13 b^2\right ) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{6 b^2 d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^4}+\frac{\left (10 a^2 b^{16}+10 i a^3 b^{15}+35 a^4 b^{14}+35 i a^5 b^{13}+49 a^6 b^{12}+49 i a^7 b^{11}+38 a^8 b^{10}+38 i a^9 b^9+20 a^{10} b^8+20 i a^{11} b^7+7 a^{12} b^6+7 i a^{13} b^5+a^{14} b^4+i a^{15} b^3\right ) (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^7 d (a-i b)^8 (a+i b)^7 (a+b \tan (c+d x))^4}+\frac{\sec ^4(c+d x) \left (-11 a^4 b^2 \sin (c+d x)-30 a^2 b^4 \sin (c+d x)-3 a^6 \sin (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{3 b^3 d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^4}+\frac{\left (4 a^6 b^2+5 a^4 b^4+10 a^2 b^6+a^8\right ) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^4 d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^4}-\frac{a^4 \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{3 b d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^4}-\frac{\sec ^4(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 d (a+b \tan (c+d x))^4}+\frac{4 a b (a-b) (a+b) (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{d (a-i b)^4 (a+i b)^4 (a+b \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4*(3*a^2 + 13*b^2)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(6*(a - I*b)^3*(a + I*b)^3*b^2*d*(a
+ b*Tan[c + d*x])^4) + (4*a*(a - b)*b*(a + b)*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(
(a - I*b)^4*(a + I*b)^4*d*(a + b*Tan[c + d*x])^4) + ((I*a^15*b^3 + a^14*b^4 + (7*I)*a^13*b^5 + 7*a^12*b^6 + (2
0*I)*a^11*b^7 + 20*a^10*b^8 + (38*I)*a^9*b^9 + 38*a^8*b^10 + (49*I)*a^7*b^11 + 49*a^6*b^12 + (35*I)*a^5*b^13 +
35*a^4*b^14 + (10*I)*a^3*b^15 + 10*a^2*b^16)*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/((
a - I*b)^8*(a + I*b)^7*b^7*d*(a + b*Tan[c + d*x])^4) - (I*(a^8 + 4*a^6*b^2 + 5*a^4*b^4 + 10*a^2*b^6)*ArcTan[Ta
n[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^4*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) -
(Log[Cos[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^4*d*(a + b*Tan[c + d*x])^4) + ((a^8
+ 4*a^6*b^2 + 5*a^4*b^4 + 10*a^2*b^6)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^4*(a*Cos[c + d*x]
+ b*Sin[c + d*x])^4)/(2*b^4*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin
[c + d*x])^3*(-3*a^6*Sin[c + d*x] - 11*a^4*b^2*Sin[c + d*x] - 30*a^2*b^4*Sin[c + d*x]))/(3*(a - I*b)^3*(a + I*
b)^3*b^3*d*(a + b*Tan[c + d*x])^4) - (a^4*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*Tan[c + d*x])/(3*(a
- I*b)^2*(a + I*b)^2*b*d*(a + b*Tan[c + d*x])^4)

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Maple [A]  time = 0.037, size = 448, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{{a}^{8}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{b}^{4}}}+4\,{\frac{{a}^{6}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{b}^{2}}}+5\,{\frac{{a}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+10\,{\frac{{a}^{2}{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{3\,{a}^{6}}{2\,d{b}^{4} \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5\,{a}^{4}}{2\,{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{5}}{3\,d{b}^{4} \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}+3\,{\frac{{a}^{7}}{d{b}^{4} \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+9\,{\frac{{a}^{5}}{{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+10\,{\frac{{a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^4-3/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^2*b^2+1/2/d/(a^2+b^2)^4*ln(1+tan
(d*x+c)^2)*b^4+4/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^3*b-4/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a*b^3+1/d*a^8/(a^2+
b^2)^4/b^4*ln(a+b*tan(d*x+c))+4/d*a^6/(a^2+b^2)^4/b^2*ln(a+b*tan(d*x+c))+5/d*a^4/(a^2+b^2)^4*ln(a+b*tan(d*x+c)
)+10/d*a^2/(a^2+b^2)^4*b^2*ln(a+b*tan(d*x+c))-3/2/d*a^6/b^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^2-5/2/d*a^4/b^2/(a^2+
b^2)^2/(a+b*tan(d*x+c))^2+1/3/d*a^5/b^4/(a^2+b^2)/(a+b*tan(d*x+c))^3+3/d*a^7/b^4/(a^2+b^2)^3/(a+b*tan(d*x+c))+
9/d*a^5/b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))+10/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))

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Maxima [A]  time = 1.62909, size = 585, normalized size = 2.29 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.35836, size = 1698, normalized size = 6.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [A]  time = 3.57605, size = 609, normalized size = 2.38 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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