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

Optimal. Leaf size=208 \[ -\frac{a^2 \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \left (7 a^2 b^2+2 a^4+17 b^4\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]

[Out]

((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b
^2)^4*d) - (a^2*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^3*(a^2 + 4*b^2))/(3*b^3*(a^2 +
b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a^2*(2*a^4 + 7*a^2*b^2 + 17*b^4))/(3*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d
*x]))

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Rubi [A]  time = 0.409106, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3565, 3635, 3628, 3531, 3530} \[ -\frac{a^2 \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \left (7 a^2 b^2+2 a^4+17 b^4\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b
^2)^4*d) - (a^2*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^3*(a^2 + 4*b^2))/(3*b^3*(a^2 +
b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a^2*(2*a^4 + 7*a^2*b^2 + 17*b^4))/(3*b^3*(a^2 + b^2)^3*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 3628

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

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 ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac{a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{\tan (c+d x) \left (2 a^2-3 a b \tan (c+d x)+\left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac{a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{2 a^2 \left (a^2+4 b^2\right )-6 a b^3 \tan (c+d x)+\left (a^2+b^2\right ) \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{3 a b^2 \left (a^2-3 b^2\right )-3 b^3 \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{3 b^2 \left (a^2+b^2\right )^3}\\ &=\frac{\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac{a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\left (4 a b \left (a^2-b^2\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^4}\\ &=\frac{\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac{4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 2.11387, size = 236, normalized size = 1.13 \[ -\frac{\frac{2 a^4}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{6 a b^3}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{6 b^3 \left (b^2-3 a^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{24 a b^3 (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac{3 i b^2 \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i b^2 \log (\tan (c+d x)+i)}{(b+i a)^4}+\frac{6 b \tan ^2(c+d x)}{(a+b \tan (c+d x))^3}+\frac{6 a \tan (c+d x)}{(a+b \tan (c+d x))^3}}{6 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 380, normalized size = 1.8 \begin{align*} -2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-6\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{{a}^{4}}{3\,d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{6}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-3\,{\frac{{a}^{4}}{bd \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-6\,{\frac{b{a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+{\frac{{a}^{5}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{3}}{bd \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{b{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-4\,{\frac{{b}^{3}a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}} \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))^4,x)

[Out]

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

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Maxima [B]  time = 1.57973, size = 554, normalized size = 2.66 \begin{align*} \frac{\frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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{12 \,{\left (a^{3} b - a b^{3}\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{6 \,{\left (a^{3} b - a b^{3}\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{a^{8} + 2 \, a^{6} b^{2} + 13 \, a^{4} b^{4} + 3 \,{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 10 \, a^{3} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9} +{\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )}}{3 \, 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))^4,x, algorithm="maxima")

[Out]

1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 12*(a^3*b - a*b^3)*
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) - 6*(a^3*b - a*b^3)*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) - (a^8 + 2*a^6*b^2 + 13*a^4*b^4 + 3*(a^6*b^2 + 3*a^4*b^4
+ 6*a^2*b^6)*tan(d*x + c)^2 + 3*(a^7*b + 3*a^5*b^3 + 10*a^3*b^5)*tan(d*x + c))/(a^9*b^3 + 3*a^7*b^5 + 3*a^5*b
^7 + a^3*b^9 + (a^6*b^6 + 3*a^4*b^8 + 3*a^2*b^10 + b^12)*tan(d*x + c)^3 + 3*(a^7*b^5 + 3*a^5*b^7 + 3*a^3*b^9 +
a*b^11)*tan(d*x + c)^2 + 3*(a^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*tan(d*x + c)))/d

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Fricas [B]  time = 1.80161, size = 1095, normalized size = 5.26 \begin{align*} \frac{9 \, a^{6} b - 13 \, a^{4} b^{3} +{\left (a^{7} + 3 \, a^{5} b^{2} + 24 \, a^{3} b^{4} + 3 \,{\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - 3 \,{\left (a^{6} b - 15 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 3 \,{\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \,{\left (a^{6} b - a^{4} b^{3} +{\left (a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{5} b^{2} - a^{3} b^{4}\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 (a^{7} - 11 \, a^{5} b^{2} + 10 \, a^{3} b^{4} - 3 \,{\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )}{3 \,{\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)^4/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*(9*a^6*b - 13*a^4*b^3 + (a^7 + 3*a^5*b^2 + 24*a^3*b^4 + 3*(a^4*b^3 - 6*a^2*b^5 + b^7)*d*x)*tan(d*x + c)^3
+ 3*(a^7 - 6*a^5*b^2 + a^3*b^4)*d*x - 3*(a^6*b - 15*a^4*b^3 + 6*a^2*b^5 - 3*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*d*x)
*tan(d*x + c)^2 + 6*(a^6*b - a^4*b^3 + (a^3*b^4 - a*b^6)*tan(d*x + c)^3 + 3*(a^4*b^3 - a^2*b^5)*tan(d*x + c)^2
+ 3*(a^5*b^2 - a^3*b^4)*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*(a^7 - 11*a^5*b^2 + 10*a^3*b^4 - 3*(a^6*b - 6*a^4*b^3 + a^2*b^5)*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*tan(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)

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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))**4,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 2.49186, size = 552, normalized size = 2.65 \begin{align*} \frac{\frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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^{3} b - a b^{3}\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{12 \,{\left (a^{3} b^{2} - a b^{4}\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{22 \, a^{3} b^{7} \tan \left (d x + c\right )^{3} - 22 \, a b^{9} \tan \left (d x + c\right )^{3} + 3 \, a^{8} b^{2} \tan \left (d x + c\right )^{2} + 12 \, a^{6} b^{4} \tan \left (d x + c\right )^{2} + 93 \, a^{4} b^{6} \tan \left (d x + c\right )^{2} - 48 \, a^{2} b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{9} b \tan \left (d x + c\right ) + 12 \, a^{7} b^{3} \tan \left (d x + c\right ) + 105 \, a^{5} b^{5} \tan \left (d x + c\right ) - 36 \, a^{3} b^{7} \tan \left (d x + c\right ) + a^{10} + 3 \, a^{8} b^{2} + 37 \, a^{6} b^{4} - 9 \, a^{4} b^{6}}{{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{3 \, 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))^4,x, algorithm="giac")

[Out]

1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^3*b - a*b^3)*l
og(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) + 12*(a^3*b^2 - a*b^4)*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) - (22*a^3*b^7*tan(d*x + c)^3 - 22*a*b^9*tan(d*x
+ c)^3 + 3*a^8*b^2*tan(d*x + c)^2 + 12*a^6*b^4*tan(d*x + c)^2 + 93*a^4*b^6*tan(d*x + c)^2 - 48*a^2*b^8*tan(d*
x + c)^2 + 3*a^9*b*tan(d*x + c) + 12*a^7*b^3*tan(d*x + c) + 105*a^5*b^5*tan(d*x + c) - 36*a^3*b^7*tan(d*x + c)
+ a^10 + 3*a^8*b^2 + 37*a^6*b^4 - 9*a^4*b^6)/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*(b*tan(d*x
+ c) + a)^3))/d