3.479 \(\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=183 \[ -\frac{a^2 \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a^2 \left (3 a^2 b^2+a^4+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.314174, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3565, 3635, 3626, 3617, 31, 3475} \[ -\frac{a^2 \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a^2 \left (3 a^2 b^2+a^4+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a^2*(a^4 + 3*a^2*
b^2 + 6*b^4)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^2)/(2*b*(a^2 + b^2)*d*(a + b*T
an[c + d*x])^2) + (a^3*(a^2 + 3*b^2))/(b^3*(a^2 + b^2)^2*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 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 ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac{a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan (c+d x) \left (2 a^2-2 a b \tan (c+d x)+2 \left (a^2+b^2\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^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{2 a^2 \left (a^2+3 b^2\right )-4 a b^3 \tan (c+d x)+2 \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (b \left (3 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a^2 \left (a^4+3 a^2 b^2+6 b^4\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^3}\\ &=\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\left (a^2 \left (a^4+3 a^2 b^2+6 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^3 d}\\ &=\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}-\frac{a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.99097, size = 351, normalized size = 1.92 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (2 i a^2 \left (3 a^2 b^2+a^4+6 b^4\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2-2 a^2 b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 a b^3 \left (a^2-3 b^2\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2+a^2 \left (3 a^2 b^2+a^4+6 b^4\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )-2 \left (a^2+b^2\right )^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2-2 i a^2 \left (3 a^2 b^2+a^4+6 b^4\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^4 \left (-b^2\right ) \left (a^2+b^2\right )\right )}{2 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-(a^4*b^2*(a^2 + b^2)) - 2*a^2*b*(a^2 + b^2)*(a^2 + 4*b^2)*
Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 2*a*b^3*(a^2 - 3*b^2)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d
*x])^2 + (2*I)*a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - (2*I)*a^2*(a^4 +
3*a^2*b^2 + 6*b^4)*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 2*(a^2 + b^2)^3*Log[Cos[c + d*x]
]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]
*(a*Cos[c + d*x] + b*Sin[c + d*x])^2))/(2*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3)

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Maple [A]  time = 0.03, size = 293, normalized size = 1.6 \begin{align*} -{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) b{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{{a}^{4}}{2\,d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{6}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{{a}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{bd \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+6\,{\frac{b{a}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+2\,{\frac{{a}^{5}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+4\,{\frac{{a}^{3}}{bd \left ({a}^{2}+{b}^{2} \right ) ^{2} \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)^4/(a+b*tan(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.57837, size = 377, normalized size = 2.06 \begin{align*} \frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 6 \, a^{2} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{{\left (3 \, a^{2} b - 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{3 \, a^{6} + 7 \, a^{4} b^{2} + 4 \,{\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} +{\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.24341, size = 1033, normalized size = 5.64 \begin{align*} \frac{a^{6} b^{2} + 7 \, a^{4} b^{4} + 2 \,{\left (a^{5} b^{3} - 3 \, a^{3} b^{5}\right )} d x -{\left (3 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 2 \,{\left (a^{3} b^{5} - 3 \, a b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left (a^{8} + 3 \, a^{6} b^{2} + 6 \, a^{4} b^{4} +{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 6 \, a^{3} b^{5}\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 (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} - 4 \, a^{3} b^{5} - 2 \,{\left (a^{4} b^{4} - 3 \, a^{2} b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) +{\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^6*b^2 + 7*a^4*b^4 + 2*(a^5*b^3 - 3*a^3*b^5)*d*x - (3*a^6*b^2 + 9*a^4*b^4 - 2*(a^3*b^5 - 3*a*b^7)*d*x)*t
an(d*x + c)^2 + (a^8 + 3*a^6*b^2 + 6*a^4*b^4 + (a^6*b^2 + 3*a^4*b^4 + 6*a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 3
*a^5*b^3 + 6*a^3*b^5)*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))
- (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*tan(d*x + c)^2 + 2*(a^7*b +
3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - 2*(a^7*b + 3*a^5*b^3 - 4*a^3*b^5 -
2*(a^4*b^4 - 3*a^2*b^6)*d*x)*tan(d*x + c))/((a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*d*tan(d*x + c)^2 + 2*(a^
7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^10)*d*tan(d*x + c) + (a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*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)**4/(a+b*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 2.41789, size = 410, normalized size = 2.24 \begin{align*} \frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b - 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 (a^{6} + 3 \, a^{4} b^{2} + 6 \, a^{2} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{3 \, a^{6} b \tan \left (d x + c\right )^{2} + 9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 18 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 2 \, a^{7} \tan \left (d x + c\right ) + 6 \, a^{5} b^{2} \tan \left (d x + c\right ) + 28 \, a^{3} b^{4} \tan \left (d x + c\right ) - a^{6} b + 11 \, a^{4} b^{3}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)
/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(a^6 + 3*a^4*b^2 + 6*a^2*b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b^3 +
3*a^4*b^5 + 3*a^2*b^7 + b^9) - (3*a^6*b*tan(d*x + c)^2 + 9*a^4*b^3*tan(d*x + c)^2 + 18*a^2*b^5*tan(d*x + c)^2
+ 2*a^7*tan(d*x + c) + 6*a^5*b^2*tan(d*x + c) + 28*a^3*b^4*tan(d*x + c) - a^6*b + 11*a^4*b^3)/((a^6*b^2 + 3*a
^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan(d*x + c) + a)^2))/d