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

Optimal. Leaf size=172 \[ \frac{a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^2-b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.231723, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3529, 3531, 3530} \[ \frac{a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^2-b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3529

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

Rule 3531

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

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

Mathematica [C]  time = 6.22449, size = 319, normalized size = 1.85 \[ \frac{a \left (\frac{6 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{6 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{24 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac{3 i \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i \log (\tan (c+d x)+i)}{(a-i b)^4}\right )}{6 b d}-\frac{\frac{4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(((3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 - ((3*I)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (24*a*(a - b)*b*(a
+ b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (6*a*b)/((a^2 + b^
2)^2*(a + b*Tan[c + d*x])^2) + (6*b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/(6*b*d) - (Log[I - T
an[c + d*x]]/(I*a - b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 - (2*b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^
2 + b^2)^3 + b/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (4*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*b*d)

________________________________________________________________________________________

Maple [B]  time = 0.032, size = 369, normalized size = 2.2 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{a}{ \left ( 3\,{a}^{2}+3\,{b}^{2} \right ) d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+6\,{\frac{{a}^{2}{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-3\,{\frac{a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.7068, size = 532, normalized size = 3.09 \begin{align*} \frac{\frac{24 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{11 \, a^{5} - 14 \, a^{3} b^{2} - a b^{4} + 6 \,{\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (5 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} +{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4 - 6*a^2*b^2 + b^4)*
log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*
x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (11*a^5 - 14*a^3*b^2 - a*b^4 + 6*(a^3*b^2 - 3*
a*b^4)*tan(d*x + c)^2 + 3*(5*a^4*b - 12*a^2*b^3 - b^5)*tan(d*x + c))/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6 +
(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x +
c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c)))/d

________________________________________________________________________________________

Fricas [B]  time = 2.07402, size = 1135, normalized size = 6.6 \begin{align*} \frac{27 \, a^{5} b^{2} - 18 \, a^{3} b^{4} - a b^{6} -{\left (11 \, a^{4} b^{3} - 30 \, a^{2} b^{5} + 3 \, b^{7} - 24 \,{\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 24 \,{\left (a^{6} b - a^{4} b^{3}\right )} d x - 3 \,{\left (9 \, a^{5} b^{2} - 26 \, a^{3} b^{4} + 9 \, a b^{6} - 24 \,{\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 3 \,{\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} +{\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} 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 ) - 3 \,{\left (6 \, a^{6} b - 23 \, a^{4} b^{3} + 16 \, a^{2} b^{5} + b^{7} - 24 \,{\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \,{\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) +{\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \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,x, algorithm="fricas")

[Out]

1/6*(27*a^5*b^2 - 18*a^3*b^4 - a*b^6 - (11*a^4*b^3 - 30*a^2*b^5 + 3*b^7 - 24*(a^3*b^4 - a*b^6)*d*x)*tan(d*x +
c)^3 + 24*(a^6*b - a^4*b^3)*d*x - 3*(9*a^5*b^2 - 26*a^3*b^4 + 9*a*b^6 - 24*(a^4*b^3 - a^2*b^5)*d*x)*tan(d*x +
c)^2 - 3*(a^7 - 6*a^5*b^2 + a^3*b^4 + (a^4*b^3 - 6*a^2*b^5 + b^7)*tan(d*x + c)^3 + 3*(a^5*b^2 - 6*a^3*b^4 + a*
b^6)*tan(d*x + c)^2 + 3*(a^6*b - 6*a^4*b^3 + a^2*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)) - 3*(6*a^6*b - 23*a^4*b^3 + 16*a^2*b^5 + b^7 - 24*(a^5*b^2 - a^3*b^4)*d*x)*tan
(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6
*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*tan(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*d*t
an(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8)*d)

________________________________________________________________________________________

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

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

Giac [B]  time = 1.30413, size = 541, normalized size = 3.15 \begin{align*} \frac{\frac{24 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{6 \,{\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac{11 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 11 \, b^{7} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{2} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{4} \tan \left (d x + c\right )^{2} + 15 \, a b^{6} \tan \left (d x + c\right )^{2} + 48 \, a^{6} b \tan \left (d x + c\right ) - 219 \, a^{4} b^{3} \tan \left (d x + c\right ) - 6 \, a^{2} b^{5} \tan \left (d x + c\right ) - 3 \, b^{7} \tan \left (d x + c\right ) + 22 \, a^{7} - 69 \, a^{5} b^{2} - 4 \, a^{3} b^{4} - a b^{6}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)*
log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4*b - 6*a^2*b^3 + b^5)*log(abs(
b*tan(d*x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (11*a^4*b^3*tan(d*x + c)^3 - 66*a^2*b
^5*tan(d*x + c)^3 + 11*b^7*tan(d*x + c)^3 + 39*a^5*b^2*tan(d*x + c)^2 - 210*a^3*b^4*tan(d*x + c)^2 + 15*a*b^6*
tan(d*x + c)^2 + 48*a^6*b*tan(d*x + c) - 219*a^4*b^3*tan(d*x + c) - 6*a^2*b^5*tan(d*x + c) - 3*b^7*tan(d*x + c
) + 22*a^7 - 69*a^5*b^2 - 4*a^3*b^4 - a*b^6)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x + c)
+ a)^3))/d