3.258 \(\int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=226 \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {b \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \tan (c+d x)}{d}-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \log (\cos (c+d x))}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3592, 3528, 3525, 3475} \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {b \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \tan (c+d x)}{d}-\frac {\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\cos (c+d x))}{d}-x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3475

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

Rule 3525

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3592

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] &&  !LeQ[m, -1]

Rubi steps

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

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Mathematica [C]  time = 3.99, size = 257, normalized size = 1.14 \[ \frac {10 (a A+b B) \left (6 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right )-5 A \left (-60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)+6 b^3 \left (b^2-10 a^2\right ) \tan ^2(c+d x)-20 a b^4 \tan ^3(c+d x)+6 i (a+i b)^5 \log (-\tan (c+d x)+i)-6 (b+i a)^5 \log (\tan (c+d x)+i)-3 b^5 \tan ^4(c+d x)\right )+12 B (a+b \tan (c+d x))^5}{60 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*B*(a + b*Tan[c + d*x])^5 + 10*(a*A + b*B)*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log
[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) - 5*A
*((6*I)*(a + I*b)^5*Log[I - Tan[c + d*x]] - 6*(I*a + b)^5*Log[I + Tan[c + d*x]] - 60*a*b^2*(2*a^2 - b^2)*Tan[c
+ d*x] + 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 - 20*a*b^4*Tan[c + d*x]^3 - 3*b^5*Tan[c + d*x]^4))/(60*b*d)

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fricas [A]  time = 0.55, size = 245, normalized size = 1.08 \[ \frac {12 \, B b^{4} \tan \left (d x + c\right )^{5} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{3} - 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x + 30 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{2} - 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*tan(d*x + c)^5 + 15*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 + 20*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*t
an(d*x + c)^3 - 60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*d*x + 30*(4*B*a^3*b + 6*A*a^2*b^2 - 4
*B*a*b^3 - A*b^4)*tan(d*x + c)^2 - 30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(1/(tan(d*x + c
)^2 + 1)) + 60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c))/d

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giac [B]  time = 10.13, size = 4789, normalized size = 21.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*B*a^4*d*x*tan(d*x)^5*tan(c)^5 + 240*A*a^3*b*d*x*tan(d*x)^5*tan(c)^5 - 360*B*a^2*b^2*d*x*tan(d*x)^5*t
an(c)^5 - 240*A*a*b^3*d*x*tan(d*x)^5*tan(c)^5 + 60*B*b^4*d*x*tan(d*x)^5*tan(c)^5 + 30*A*a^4*log(4*(tan(d*x)^4*
tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan
(d*x)^5*tan(c)^5 - 120*B*a^3*b*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*
x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 180*A*a^2*b^2*log(4*(tan(d*x)^4*tan(c)^2 -
2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^5*ta
n(c)^5 + 120*B*a*b^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*t
an(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 + 30*A*b^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 300*B*
a^4*d*x*tan(d*x)^4*tan(c)^4 - 1200*A*a^3*b*d*x*tan(d*x)^4*tan(c)^4 + 1800*B*a^2*b^2*d*x*tan(d*x)^4*tan(c)^4 +
1200*A*a*b^3*d*x*tan(d*x)^4*tan(c)^4 - 300*B*b^4*d*x*tan(d*x)^4*tan(c)^4 - 120*B*a^3*b*tan(d*x)^5*tan(c)^5 - 1
80*A*a^2*b^2*tan(d*x)^5*tan(c)^5 + 180*B*a*b^3*tan(d*x)^5*tan(c)^5 + 45*A*b^4*tan(d*x)^5*tan(c)^5 - 150*A*a^4*
log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(
tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 600*B*a^3*b*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2
*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 900*A*a^2*b^2*log(4*(tan
(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 +
1))*tan(d*x)^4*tan(c)^4 - 600*B*a*b^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 - 150*A*b^4*log(4*(tan(d*x)^4*tan(c)
^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^
4*tan(c)^4 + 60*B*a^4*tan(d*x)^5*tan(c)^4 + 240*A*a^3*b*tan(d*x)^5*tan(c)^4 - 360*B*a^2*b^2*tan(d*x)^5*tan(c)^
4 - 240*A*a*b^3*tan(d*x)^5*tan(c)^4 + 60*B*b^4*tan(d*x)^5*tan(c)^4 + 60*B*a^4*tan(d*x)^4*tan(c)^5 + 240*A*a^3*
b*tan(d*x)^4*tan(c)^5 - 360*B*a^2*b^2*tan(d*x)^4*tan(c)^5 - 240*A*a*b^3*tan(d*x)^4*tan(c)^5 + 60*B*b^4*tan(d*x
)^4*tan(c)^5 + 600*B*a^4*d*x*tan(d*x)^3*tan(c)^3 + 2400*A*a^3*b*d*x*tan(d*x)^3*tan(c)^3 - 3600*B*a^2*b^2*d*x*t
an(d*x)^3*tan(c)^3 - 2400*A*a*b^3*d*x*tan(d*x)^3*tan(c)^3 + 600*B*b^4*d*x*tan(d*x)^3*tan(c)^3 - 120*B*a^3*b*ta
n(d*x)^5*tan(c)^3 - 180*A*a^2*b^2*tan(d*x)^5*tan(c)^3 + 120*B*a*b^3*tan(d*x)^5*tan(c)^3 + 30*A*b^4*tan(d*x)^5*
tan(c)^3 + 360*B*a^3*b*tan(d*x)^4*tan(c)^4 + 540*A*a^2*b^2*tan(d*x)^4*tan(c)^4 - 660*B*a*b^3*tan(d*x)^4*tan(c)
^4 - 165*A*b^4*tan(d*x)^4*tan(c)^4 - 120*B*a^3*b*tan(d*x)^3*tan(c)^5 - 180*A*a^2*b^2*tan(d*x)^3*tan(c)^5 + 120
*B*a*b^3*tan(d*x)^3*tan(c)^5 + 30*A*b^4*tan(d*x)^3*tan(c)^5 + 120*B*a^2*b^2*tan(d*x)^5*tan(c)^2 + 80*A*a*b^3*t
an(d*x)^5*tan(c)^2 - 20*B*b^4*tan(d*x)^5*tan(c)^2 + 300*A*a^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c)
+ tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 - 1200*B*a^3*
b*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)
/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 - 1800*A*a^2*b^2*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d
*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 1200*B*a*b^3*log(4*
(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)
^2 + 1))*tan(d*x)^3*tan(c)^3 + 300*A*b^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^
2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 - 240*B*a^4*tan(d*x)^4*tan(c)^3 -
960*A*a^3*b*tan(d*x)^4*tan(c)^3 + 1800*B*a^2*b^2*tan(d*x)^4*tan(c)^3 + 1200*A*a*b^3*tan(d*x)^4*tan(c)^3 - 300*
B*b^4*tan(d*x)^4*tan(c)^3 - 240*B*a^4*tan(d*x)^3*tan(c)^4 - 960*A*a^3*b*tan(d*x)^3*tan(c)^4 + 1800*B*a^2*b^2*t
an(d*x)^3*tan(c)^4 + 1200*A*a*b^3*tan(d*x)^3*tan(c)^4 - 300*B*b^4*tan(d*x)^3*tan(c)^4 + 120*B*a^2*b^2*tan(d*x)
^2*tan(c)^5 + 80*A*a*b^3*tan(d*x)^2*tan(c)^5 - 20*B*b^4*tan(d*x)^2*tan(c)^5 - 60*B*a*b^3*tan(d*x)^5*tan(c) - 1
5*A*b^4*tan(d*x)^5*tan(c) - 600*B*a^4*d*x*tan(d*x)^2*tan(c)^2 - 2400*A*a^3*b*d*x*tan(d*x)^2*tan(c)^2 + 3600*B*
a^2*b^2*d*x*tan(d*x)^2*tan(c)^2 + 2400*A*a*b^3*d*x*tan(d*x)^2*tan(c)^2 - 600*B*b^4*d*x*tan(d*x)^2*tan(c)^2 + 3
60*B*a^3*b*tan(d*x)^4*tan(c)^2 + 540*A*a^2*b^2*tan(d*x)^4*tan(c)^2 - 600*B*a*b^3*tan(d*x)^4*tan(c)^2 - 150*A*b
^4*tan(d*x)^4*tan(c)^2 - 480*B*a^3*b*tan(d*x)^3*tan(c)^3 - 720*A*a^2*b^2*tan(d*x)^3*tan(c)^3 + 720*B*a*b^3*tan
(d*x)^3*tan(c)^3 + 180*A*b^4*tan(d*x)^3*tan(c)^3 + 360*B*a^3*b*tan(d*x)^2*tan(c)^4 + 540*A*a^2*b^2*tan(d*x)^2*
tan(c)^4 - 600*B*a*b^3*tan(d*x)^2*tan(c)^4 - 150*A*b^4*tan(d*x)^2*tan(c)^4 - 60*B*a*b^3*tan(d*x)*tan(c)^5 - 15
*A*b^4*tan(d*x)*tan(c)^5 + 12*B*b^4*tan(d*x)^5 - 240*B*a^2*b^2*tan(d*x)^4*tan(c) - 160*A*a*b^3*tan(d*x)^4*tan(
c) + 100*B*b^4*tan(d*x)^4*tan(c) - 300*A*a^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan
(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 + 1200*B*a^3*b*log(4*(tan(d*x)
^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*
tan(d*x)^2*tan(c)^2 + 1800*A*a^2*b^2*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 +
tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 - 1200*B*a*b^3*log(4*(tan(d*x)^4*tan(c
)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)
^2*tan(c)^2 - 300*A*b^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 -
2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 + 360*B*a^4*tan(d*x)^3*tan(c)^2 + 1440*A*a^3*b*tan(
d*x)^3*tan(c)^2 - 2880*B*a^2*b^2*tan(d*x)^3*tan(c)^2 - 1920*A*a*b^3*tan(d*x)^3*tan(c)^2 + 600*B*b^4*tan(d*x)^3
*tan(c)^2 + 360*B*a^4*tan(d*x)^2*tan(c)^3 + 1440*A*a^3*b*tan(d*x)^2*tan(c)^3 - 2880*B*a^2*b^2*tan(d*x)^2*tan(c
)^3 - 1920*A*a*b^3*tan(d*x)^2*tan(c)^3 + 600*B*b^4*tan(d*x)^2*tan(c)^3 - 240*B*a^2*b^2*tan(d*x)*tan(c)^4 - 160
*A*a*b^3*tan(d*x)*tan(c)^4 + 100*B*b^4*tan(d*x)*tan(c)^4 + 12*B*b^4*tan(c)^5 + 60*B*a*b^3*tan(d*x)^4 + 15*A*b^
4*tan(d*x)^4 + 300*B*a^4*d*x*tan(d*x)*tan(c) + 1200*A*a^3*b*d*x*tan(d*x)*tan(c) - 1800*B*a^2*b^2*d*x*tan(d*x)*
tan(c) - 1200*A*a*b^3*d*x*tan(d*x)*tan(c) + 300*B*b^4*d*x*tan(d*x)*tan(c) - 360*B*a^3*b*tan(d*x)^3*tan(c) - 54
0*A*a^2*b^2*tan(d*x)^3*tan(c) + 600*B*a*b^3*tan(d*x)^3*tan(c) + 150*A*b^4*tan(d*x)^3*tan(c) + 480*B*a^3*b*tan(
d*x)^2*tan(c)^2 + 720*A*a^2*b^2*tan(d*x)^2*tan(c)^2 - 720*B*a*b^3*tan(d*x)^2*tan(c)^2 - 180*A*b^4*tan(d*x)^2*t
an(c)^2 - 360*B*a^3*b*tan(d*x)*tan(c)^3 - 540*A*a^2*b^2*tan(d*x)*tan(c)^3 + 600*B*a*b^3*tan(d*x)*tan(c)^3 + 15
0*A*b^4*tan(d*x)*tan(c)^3 + 60*B*a*b^3*tan(c)^4 + 15*A*b^4*tan(c)^4 + 120*B*a^2*b^2*tan(d*x)^3 + 80*A*a*b^3*ta
n(d*x)^3 - 20*B*b^4*tan(d*x)^3 + 150*A*a^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c
)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)*tan(c) - 600*B*a^3*b*log(4*(tan(d*x)^4*tan(
c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x
)*tan(c) - 900*A*a^2*b^2*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 -
2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)*tan(c) + 600*B*a*b^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^
3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)*tan(c) + 150*A*b
^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1
)/(tan(c)^2 + 1))*tan(d*x)*tan(c) - 240*B*a^4*tan(d*x)^2*tan(c) - 960*A*a^3*b*tan(d*x)^2*tan(c) + 1800*B*a^2*b
^2*tan(d*x)^2*tan(c) + 1200*A*a*b^3*tan(d*x)^2*tan(c) - 300*B*b^4*tan(d*x)^2*tan(c) - 240*B*a^4*tan(d*x)*tan(c
)^2 - 960*A*a^3*b*tan(d*x)*tan(c)^2 + 1800*B*a^2*b^2*tan(d*x)*tan(c)^2 + 1200*A*a*b^3*tan(d*x)*tan(c)^2 - 300*
B*b^4*tan(d*x)*tan(c)^2 + 120*B*a^2*b^2*tan(c)^3 + 80*A*a*b^3*tan(c)^3 - 20*B*b^4*tan(c)^3 - 60*B*a^4*d*x - 24
0*A*a^3*b*d*x + 360*B*a^2*b^2*d*x + 240*A*a*b^3*d*x - 60*B*b^4*d*x + 120*B*a^3*b*tan(d*x)^2 + 180*A*a^2*b^2*ta
n(d*x)^2 - 120*B*a*b^3*tan(d*x)^2 - 30*A*b^4*tan(d*x)^2 - 360*B*a^3*b*tan(d*x)*tan(c) - 540*A*a^2*b^2*tan(d*x)
*tan(c) + 660*B*a*b^3*tan(d*x)*tan(c) + 165*A*b^4*tan(d*x)*tan(c) + 120*B*a^3*b*tan(c)^2 + 180*A*a^2*b^2*tan(c
)^2 - 120*B*a*b^3*tan(c)^2 - 30*A*b^4*tan(c)^2 - 30*A*a^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + t
an(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1)) + 120*B*a^3*b*log(4*(tan(d*x)^4*tan(c
)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1)) + 180*A*
a^2*b^2*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c
) + 1)/(tan(c)^2 + 1)) - 120*B*a*b^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 +
tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1)) - 30*A*b^4*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c
) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1)) + 60*B*a^4*tan(d*x) + 240*A*a^3*
b*tan(d*x) - 360*B*a^2*b^2*tan(d*x) - 240*A*a*b^3*tan(d*x) + 60*B*b^4*tan(d*x) + 60*B*a^4*tan(c) + 240*A*a^3*b
*tan(c) - 360*B*a^2*b^2*tan(c) - 240*A*a*b^3*tan(c) + 60*B*b^4*tan(c) + 120*B*a^3*b + 180*A*a^2*b^2 - 180*B*a*
b^3 - 45*A*b^4)/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d*x)^4*tan(c)^4 + 10*d*tan(d*x)^3*tan(c)^3 - 10*d*tan(d*x)^2*
tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)

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maple [B]  time = 0.02, size = 449, normalized size = 1.99 \[ \frac {2 B \left (\tan ^{3}\left (d x +c \right )\right ) a^{2} b^{2}}{d}-\frac {2 B \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}}{d}+\frac {3 A \left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}}{d}+\frac {4 A \left (\tan ^{3}\left (d x +c \right )\right ) a \,b^{3}}{3 d}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{2} b^{2}}{d}+\frac {B \left (\tan ^{5}\left (d x +c \right )\right ) b^{4}}{5 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{4}}{2 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,b^{4}}{2 d}+\frac {A \left (\tan ^{4}\left (d x +c \right )\right ) b^{4}}{4 d}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{4}}{d}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{4}}{d}+\frac {B \,b^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} B \tan \left (d x +c \right )}{d}-\frac {A \left (\tan ^{2}\left (d x +c \right )\right ) b^{4}}{2 d}-\frac {B \left (\tan ^{3}\left (d x +c \right )\right ) b^{4}}{3 d}+\frac {2 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{3} b}{d}-\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B \,a^{3} b}{d}+\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a \,b^{3}}{d}+\frac {6 B \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b^{2}}{d}+\frac {4 A \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{3}}{d}-\frac {6 B \,a^{2} b^{2} \tan \left (d x +c \right )}{d}-\frac {4 A a \,b^{3} \tan \left (d x +c \right )}{d}+\frac {B \left (\tan ^{4}\left (d x +c \right )\right ) a \,b^{3}}{d}-\frac {4 A \arctan \left (\tan \left (d x +c \right )\right ) a^{3} b}{d}+\frac {4 A \,a^{3} b \tan \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.84, size = 246, normalized size = 1.09 \[ \frac {12 \, B b^{4} \tan \left (d x + c\right )^{5} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{3} + 30 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{2} - 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*tan(d*x + c)^5 + 15*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 + 20*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*t
an(d*x + c)^3 + 30*(4*B*a^3*b + 6*A*a^2*b^2 - 4*B*a*b^3 - A*b^4)*tan(d*x + c)^2 - 60*(B*a^4 + 4*A*a^3*b - 6*B*
a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) + 30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(d*x
+ c)^2 + 1) + 60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c))/d

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mupad [B]  time = 6.32, size = 251, normalized size = 1.11 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^4+B\,b^4+4\,A\,a^3\,b-2\,a\,b^2\,\left (2\,A\,b+3\,B\,a\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A\,b^4}{2}+2\,B\,a\,b^3-a^2\,b\,\left (3\,A\,b+2\,B\,a\right )\right )}{d}-x\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3+B\,b^4\right )+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {A\,b^4}{4}+B\,a\,b^3\right )}{d}+\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {A\,a^4}{2}-2\,B\,a^3\,b-3\,A\,a^2\,b^2+2\,B\,a\,b^3+\frac {A\,b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,b^4}{3}-\frac {2\,a\,b^2\,\left (2\,A\,b+3\,B\,a\right )}{3}\right )}{d}+\frac {B\,b^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.08, size = 437, normalized size = 1.93 \[ \begin {cases} \frac {A a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 A a^{3} b x + \frac {4 A a^{3} b \tan {\left (c + d x \right )}}{d} - \frac {3 A a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 A a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 A a b^{3} x + \frac {4 A a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 A a b^{3} \tan {\left (c + d x \right )}}{d} + \frac {A b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {A b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} - B a^{4} x + \frac {B a^{4} \tan {\left (c + d x \right )}}{d} - \frac {2 B a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 B a^{3} b \tan ^{2}{\left (c + d x \right )}}{d} + 6 B a^{2} b^{2} x + \frac {2 B a^{2} b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac {6 B a^{2} b^{2} \tan {\left (c + d x \right )}}{d} + \frac {2 B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {B a b^{3} \tan ^{4}{\left (c + d x \right )}}{d} - \frac {2 B a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} - B b^{4} x + \frac {B b^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {B b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{4} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\relax (c )}\right ) \left (a + b \tan {\relax (c )}\right )^{4} \tan {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*a**4*log(tan(c + d*x)**2 + 1)/(2*d) - 4*A*a**3*b*x + 4*A*a**3*b*tan(c + d*x)/d - 3*A*a**2*b**2*lo
g(tan(c + d*x)**2 + 1)/d + 3*A*a**2*b**2*tan(c + d*x)**2/d + 4*A*a*b**3*x + 4*A*a*b**3*tan(c + d*x)**3/(3*d) -
4*A*a*b**3*tan(c + d*x)/d + A*b**4*log(tan(c + d*x)**2 + 1)/(2*d) + A*b**4*tan(c + d*x)**4/(4*d) - A*b**4*tan
(c + d*x)**2/(2*d) - B*a**4*x + B*a**4*tan(c + d*x)/d - 2*B*a**3*b*log(tan(c + d*x)**2 + 1)/d + 2*B*a**3*b*tan
(c + d*x)**2/d + 6*B*a**2*b**2*x + 2*B*a**2*b**2*tan(c + d*x)**3/d - 6*B*a**2*b**2*tan(c + d*x)/d + 2*B*a*b**3
*log(tan(c + d*x)**2 + 1)/d + B*a*b**3*tan(c + d*x)**4/d - 2*B*a*b**3*tan(c + d*x)**2/d - B*b**4*x + B*b**4*ta
n(c + d*x)**5/(5*d) - B*b**4*tan(c + d*x)**3/(3*d) + B*b**4*tan(c + d*x)/d, Ne(d, 0)), (x*(A + B*tan(c))*(a +
b*tan(c))**4*tan(c), True))

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