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

[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)

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Rubi [A]  time = 0.273254, antiderivative size = 226, normalized size of antiderivative = 1., 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 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]

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 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 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 \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*}

Mathematica [C]  time = 3.52856, size = 257, normalized size = 1.14 \[ \frac{10 (a A+b B) \left (6 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right )-5 A \left (6 b^3 \left (b^2-10 a^2\right ) \tan ^2(c+d x)-60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)-20 a b^4 \tan ^3(c+d x)+6 i (a+i b)^5 \log (-\tan (c+d x)+i)-6 (b+i a)^5 \log (\tan (c+d x)+i)-3 b^5 \tan ^4(c+d x)\right )+12 B (a+b \tan (c+d x))^5}{60 b d} \]

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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Maple [B]  time = 0.014, size = 449, normalized size = 2. \begin{align*}{\frac{B{a}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{A{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{B{a}^{4}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,A \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{3}}{3\,d}}+6\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d}}+4\,{\frac{A\tan \left ( dx+c \right ){a}^{3}b}{d}}-4\,{\frac{Aa{b}^{3}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{3}b}{d}}-2\,{\frac{Ba{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{a}^{3}b}{d}}+2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Ba{b}^{3}}{d}}-4\,{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d}}+4\,{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d}}+{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{4}a{b}^{3}}{d}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{a}^{2}{b}^{2}}{d}}+2\,{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{2}{b}^{2}}{d}}+3\,{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{2}}{d}}-{\frac{A{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d}}+{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{4}{b}^{4}}{4\,d}}+{\frac{B{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{B{b}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{b}^{4}}{2\,d}}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d}}-6\,{\frac{B{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}} \end{align*}

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

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Maxima [A]  time = 1.49841, size = 332, normalized size = 1.47 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.96229, size = 562, normalized size = 2.49 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 1.1846, size = 437, normalized size = 1.93 \begin{align*} \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{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{4} \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

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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Giac [B]  time = 11.4444, size = 6465, normalized size = 28.61 \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)*(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(c)^2 +
1)/(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
(d*x)^5*tan(c)^5 - 120*B*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 180*A*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)^5*ta
n(c)^5 + 120*B*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + t
an(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 + 30*A*b^4*log(4*(tan(c)^2 + 1)/(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(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(c)^2 + 1)/(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(d*x)^4*tan(c)^4 + 600*B*a^3*b*log(4*(tan(c)^2 + 1)/(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(d*x)^4*tan(c)^4 + 900*A*a^2*b^2*log(4*(tan
(c)^2 + 1)/(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(d*x)^4*tan(c)^4 - 600*B*a*b^3*log(4*(tan(c)^2 + 1)/(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(d*x)^4*tan(c)^4 - 150*A*b^4*log(4*(tan(c)^2 + 1)/(ta
n(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(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(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*ta
n(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 1200*B*a^3*
b*log(4*(tan(c)^2 + 1)/(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(d*x)^3*tan(c)^3 - 1800*A*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)^3*tan(c)^3 + 1200*B*a*b^3*log(4*
(tan(c)^2 + 1)/(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(d*x)^3*tan(c)^3 + 300*A*b^4*log(4*(tan(c)^2 + 1)/(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(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(c)^2 + 1)/(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(d*x)^2*tan(c)^2 + 1200*B*a^3*b*log(4*(tan(c)^2
+ 1)/(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(d*x)^2*tan(c)^2 + 1800*A*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)^2*tan(c)^2 - 1200*B*a*b^3*log(4*(tan(c)^2 + 1)/(t
an(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(d*x)
^2*tan(c)^2 - 300*A*b^4*log(4*(tan(c)^2 + 1)/(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(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(c)^2 + 1)/(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(d*x)*tan(c) - 600*B*a^3*b*log(4*(tan(c)^2 + 1)/(
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(d*x
)*tan(c) - 900*A*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)*tan(c) + 600*B*a*b^3*log(4*(tan(c)^2 + 1)/(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(d*x)*tan(c) + 150*A*b
^4*log(4*(tan(c)^2 + 1)/(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(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(c)^2 + 1)/(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)) + 120*B*a^3*b*log(4*(tan(c)^2 + 1)/(t
an(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)) + 180*A*
a^2*b^2*log(4*(tan(c)^2 + 1)/(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)) - 120*B*a*b^3*log(4*(tan(c)^2 + 1)/(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)) - 30*A*b^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*t
an(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 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)