\(\int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) [282]

Optimal result

Integrand size = 31, antiderivative size = 331 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3686, 3726, 3728, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {a (A b-a B) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3}-\frac {\left (-3 a^4 B+a^3 A b-6 a^2 b^2 B+3 a A b^3-b^4 B\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (-3 a^5 B+a^4 A b-9 a^3 b^2 B+3 a^2 A b^3-10 a b^4 B+6 A b^5\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3} \]

[In]

Int[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]

[Out]

((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[Cos[c
+ d*x]])/((a^2 + b^2)^3*d) + (a^2*(a^4*A*b + 3*a^2*A*b^3 + 6*A*b^5 - 3*a^5*B - 9*a^3*b^2*B - 10*a*b^4*B)*Log[
a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3*d) - ((a^3*A*b + 3*a*A*b^3 - 3*a^4*B - 6*a^2*b^2*B - b^4*B)*Tan[c + d*
x])/(b^3*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(a^
2*A*b + 5*A*b^3 - 3*a^3*B - 7*a*b^2*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^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 3686

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((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)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

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 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]

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 (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a (A b-a B)+2 b (A b-a B) \tan (c+d x)-\left (a A b-3 a^2 B-2 b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )} \\ & = \frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-2 a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)-2 \left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a \left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right )-2 b^3 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 (A b-3 a B) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2} \\ & = \frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\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 )^3} \\ & = \frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\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 )^3 d} \\ & = \frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.22 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.83 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {(A+i B) \log (i-\tan (c+d x))}{(-i a+b)^3}+\frac {(A-i B) \log (i+\tan (c+d x))}{(i a+b)^3}+\frac {2 a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3}+\frac {a^3 \left (-a A b+3 a^2 B+2 b^2 B\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 B \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (-2 a^3 A b-4 a A b^3+6 a^4 B+11 a^2 b^2 B+3 b^4 B\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \]

[In]

Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (d x +c \right ) B}{b^{3}}+\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{2} \left (A \,a^{4} b +3 A \,a^{2} b^{3}+6 A \,b^{5}-3 B \,a^{5}-9 B \,a^{3} b^{2}-10 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (A b -B a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 A \,a^{2} b +4 A \,b^{3}-3 B \,a^{3}-5 B a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(263\)
default	\(\frac {\frac {\tan \left (d x +c \right ) B}{b^{3}}+\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{2} \left (A \,a^{4} b +3 A \,a^{2} b^{3}+6 A \,b^{5}-3 B \,a^{5}-9 B \,a^{3} b^{2}-10 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (A b -B a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 A \,a^{2} b +4 A \,b^{3}-3 B \,a^{3}-5 B a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(263\)
norman	\(\frac {\frac {B \left (\tan ^{3}\left (d x +c \right )\right )}{b d}+\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (2 A \,a^{4} b +4 A \,a^{2} b^{3}-6 B \,a^{5}-11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \tan \left (d x +c \right )}{d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a^{2} \left (3 A \,a^{4} b +7 A \,a^{2} b^{3}-9 B \,a^{5}-17 B \,a^{3} b^{2}-4 B a \,b^{4}\right )}{2 d \,b^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (A \,a^{4} b +3 A \,a^{2} b^{3}+6 A \,b^{5}-3 B \,a^{5}-9 B \,a^{3} b^{2}-10 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{4} d}-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(512\)
parallelrisch	\(\text {Expression too large to display}\)	\(1180\)
risch	\(\text {Expression too large to display}\)	\(1552\)

[In]

int(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (328) = 656\).

Time = 0.38 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.69 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 7 \, A a^{4} b^{5} - 2 \, {\left (B a^{6} b^{3} + 3 \, B a^{4} b^{5} + 3 \, B a^{2} b^{7} + B b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 3 \, A a^{3} b^{6} - B a^{2} b^{7}\right )} d x - {\left (9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 23 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + 12 \, B a^{3} b^{6} + 4 \, B a b^{8} + 2 \, {\left (A a^{3} b^{6} + 3 \, B a^{2} b^{7} - 3 \, A a b^{8} - B b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, B a^{9} - A a^{8} b + 9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 10 \, B a^{5} b^{4} - 6 \, A a^{4} b^{5} + {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 10 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} + 9 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} + 10 \, B a^{4} b^{5} - 6 \, A a^{3} b^{6}\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 \, B a^{9} - A a^{8} b + 9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 3 \, B a^{3} b^{6} - A a^{2} b^{7} + {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 3 \, A a^{2} b^{7} + 3 \, B a b^{8} - A b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} + 9 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} + 9 \, B a^{4} b^{5} - 3 \, A a^{3} b^{6} + 3 \, B a^{2} b^{7} - A a b^{8}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} + 6 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} - 2 \, B a^{4} b^{5} + 4 \, A a^{3} b^{6} + B a^{2} b^{7} + 2 \, {\left (A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 3 \, A a^{2} b^{7} - B a b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} d\right )}} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(3*B*a^7*b^2 - A*a^6*b^3 + 9*B*a^5*b^4 - 7*A*a^4*b^5 - 2*(B*a^6*b^3 + 3*B*a^4*b^5 + 3*B*a^2*b^7 + B*b^9)*
tan(d*x + c)^3 - 2*(A*a^5*b^4 + 3*B*a^4*b^5 - 3*A*a^3*b^6 - B*a^2*b^7)*d*x - (9*B*a^7*b^2 - 3*A*a^6*b^3 + 23*B
*a^5*b^4 - 9*A*a^4*b^5 + 12*B*a^3*b^6 + 4*B*a*b^8 + 2*(A*a^3*b^6 + 3*B*a^2*b^7 - 3*A*a*b^8 - B*b^9)*d*x)*tan(d
*x + c)^2 + (3*B*a^9 - A*a^8*b + 9*B*a^7*b^2 - 3*A*a^6*b^3 + 10*B*a^5*b^4 - 6*A*a^4*b^5 + (3*B*a^7*b^2 - A*a^6
*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 + 10*B*a^3*b^6 - 6*A*a^2*b^7)*tan(d*x + c)^2 + 2*(3*B*a^8*b - A*a^7*b^2 + 9*B
*a^6*b^3 - 3*A*a^5*b^4 + 10*B*a^4*b^5 - 6*A*a^3*b^6)*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*B*a^9 - A*a^8*b + 9*B*a^7*b^2 - 3*A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 +
3*B*a^3*b^6 - A*a^2*b^7 + (3*B*a^7*b^2 - A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 + 9*B*a^3*b^6 - 3*A*a^2*b^7 + 3
*B*a*b^8 - A*b^9)*tan(d*x + c)^2 + 2*(3*B*a^8*b - A*a^7*b^2 + 9*B*a^6*b^3 - 3*A*a^5*b^4 + 9*B*a^4*b^5 - 3*A*a^
3*b^6 + 3*B*a^2*b^7 - A*a*b^8)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - 2*(3*B*a^8*b - A*a^7*b^2 + 6*B*a^6*
b^3 - 3*A*a^5*b^4 - 2*B*a^4*b^5 + 4*A*a^3*b^6 + B*a^2*b^7 + 2*(A*a^4*b^5 + 3*B*a^3*b^6 - 3*A*a^2*b^7 - B*a*b^8
)*d*x)*tan(d*x + c))/((a^6*b^6 + 3*a^4*b^8 + 3*a^2*b^10 + b^12)*d*tan(d*x + c)^2 + 2*(a^7*b^5 + 3*a^5*b^7 + 3*
a^3*b^9 + a*b^11)*d*tan(d*x + c) + (a^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(tan(d*x+c)**4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, B a^{7} - A a^{6} b + 9 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 10 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, B a^{7} - 3 \, A a^{6} b + 9 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3} + 2 \, {\left (3 \, B a^{6} b - 2 \, A a^{5} b^{2} + 5 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, B \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.12 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.53 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, B a^{7} - A a^{6} b + 9 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 10 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {2 \, B \tan \left (d x + c\right )}{b^{3}} + \frac {9 \, B a^{7} b^{2} \tan \left (d x + c\right )^{2} - 3 \, A a^{6} b^{3} \tan \left (d x + c\right )^{2} + 27 \, B a^{5} b^{4} \tan \left (d x + c\right )^{2} - 9 \, A a^{4} b^{5} \tan \left (d x + c\right )^{2} + 30 \, B a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, A a^{2} b^{7} \tan \left (d x + c\right )^{2} + 12 \, B a^{8} b \tan \left (d x + c\right ) - 2 \, A a^{7} b^{2} \tan \left (d x + c\right ) + 38 \, B a^{6} b^{3} \tan \left (d x + c\right ) - 6 \, A a^{5} b^{4} \tan \left (d x + c\right ) + 50 \, B a^{4} b^{5} \tan \left (d x + c\right ) - 28 \, A a^{3} b^{6} \tan \left (d x + c\right ) + 4 \, B a^{9} + 13 \, B a^{7} b^{2} + A a^{6} b^{3} + 21 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(2*(A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (B*a^3 - 3*A*a^
2*b - 3*B*a*b^2 + A*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(3*B*a^7 - A*a^6*b +
9*B*a^5*b^2 - 3*A*a^4*b^3 + 10*B*a^3*b^4 - 6*A*a^2*b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b^4 + 3*a^4*b^6 + 3*
a^2*b^8 + b^10) + 2*B*tan(d*x + c)/b^3 + (9*B*a^7*b^2*tan(d*x + c)^2 - 3*A*a^6*b^3*tan(d*x + c)^2 + 27*B*a^5*b
^4*tan(d*x + c)^2 - 9*A*a^4*b^5*tan(d*x + c)^2 + 30*B*a^3*b^6*tan(d*x + c)^2 - 18*A*a^2*b^7*tan(d*x + c)^2 + 1
2*B*a^8*b*tan(d*x + c) - 2*A*a^7*b^2*tan(d*x + c) + 38*B*a^6*b^3*tan(d*x + c) - 6*A*a^5*b^4*tan(d*x + c) + 50*
B*a^4*b^5*tan(d*x + c) - 28*A*a^3*b^6*tan(d*x + c) + 4*B*a^9 + 13*B*a^7*b^2 + A*a^6*b^3 + 21*B*a^5*b^4 - 11*A*
a^4*b^5)/((a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)^2))/d

Mupad [B] (verification not implemented)

Time = 9.00 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {B\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,B\,a^7-3\,A\,a^6\,b+9\,B\,a^5\,b^2-7\,A\,a^4\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,B\,a^6-2\,A\,a^5\,b+5\,B\,a^4\,b^2-4\,A\,a^3\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^5+A\,a^4\,b-9\,B\,a^3\,b^2+3\,A\,a^2\,b^3-10\,B\,a\,b^4+6\,A\,b^5\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3} \]

[In]

int((tan(c + d*x)^4*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)

[Out]

(log(tan(c + d*x) - 1i)*(A*1i - B))/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - ((5*B*a^7 - 7*A*a^4*b^3 + 9*B*
a^5*b^2 - 3*A*a^6*b)/(2*b*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(3*B*a^6 - 4*A*a^3*b^3 + 5*B*a^4*b^2 - 2*A*
a^5*b))/(a^4 + b^4 + 2*a^2*b^2))/(d*(a^2*b^3 + b^5*tan(c + d*x)^2 + 2*a*b^4*tan(c + d*x))) + (log(tan(c + d*x)
+ 1i)*(A - B*1i))/(2*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) + (B*tan(c + d*x))/(b^3*d) + (a^2*log(a + b*tan(c
+ d*x))*(6*A*b^5 - 3*B*a^5 + 3*A*a^2*b^3 - 9*B*a^3*b^2 + A*a^4*b - 10*B*a*b^4))/(b^4*d*(a^2 + b^2)^3)