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

Optimal. Leaf size=189 \[ -\frac{a^2 \left (a^2+7 b^2\right )}{6 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

[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^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 7*b^2))/(6*b^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]))

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Rubi [A]  time = 0.324708, antiderivative size = 189, 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, 3628, 3529, 3531, 3530} \[ -\frac{a^2 \left (a^2+7 b^2\right )}{6 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3/(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^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 7*b^2))/(6*b^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 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 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 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 ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac{a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{a^2-3 a b \tan (c+d x)+\left (a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac{a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{-6 a b^2-3 b \left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac{a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (a^2+7 b^2\right )}{6 b^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{-3 b^2 \left (3 a^2-b^2\right )-3 a b \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{3 b \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^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (a^2+7 b^2\right )}{6 b^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^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{a^2 \left (a^2+7 b^2\right )}{6 b^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 = 6.21764, size = 387, normalized size = 2.05 \[ -\frac{\tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{\frac{a}{3 b d (a+b \tan (c+d x))^3}+\frac{2 b \left (\frac{-\frac{2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}}{b}-\frac{a \left (-\frac{b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right )}{b}\right )}{d}}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 376, normalized size = 2. \begin{align*} 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}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{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}^{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}^{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 ) }}-{\frac{{a}^{4}}{2\,{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}}{3\,{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3/(a+b*tan(d*x+c))^4,x)

[Out]

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

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Maxima [B]  time = 1.69413, size = 543, normalized size = 2.87 \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{a^{7} + 14 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \,{\left (a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{6} b + 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} +{\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\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)^3/(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) + (a^7 + 14*a^5*b^2 - 11*a^3*b^4 + 6*(a^3*b^4 -
3*a*b^6)*tan(d*x + c)^2 + 3*(a^6*b + 8*a^4*b^3 - 9*a^2*b^5)*tan(d*x + c))/(a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 +
a^3*b^8 + (a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^1
0)*tan(d*x + c)^2 + 3*(a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*tan(d*x + c)))/d

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Fricas [B]  time = 2.06887, size = 1135, normalized size = 6.01 \begin{align*} \frac{3 \, a^{7} - 30 \, a^{5} b^{2} + 11 \, a^{3} b^{4} +{\left (a^{6} b + 18 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 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 (a^{7} + 16 \, a^{5} b^{2} - 23 \, a^{3} b^{4} + 6 \, 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 (9 \, a^{6} b - 26 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 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)^3/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*a^7 - 30*a^5*b^2 + 11*a^3*b^4 + (a^6*b + 18*a^4*b^3 - 27*a^2*b^5 - 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*(a^7 + 16*a^5*b^2 - 23*a^3*b^4 + 6*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*(9*a^6*b - 26*a^4*b^3 + 9*a^2*b^5 - 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)

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.86303, size = 540, normalized size = 2.86 \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^{5} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{7} \tan \left (d x + c\right )^{3} + 11 \, b^{9} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 15 \, a b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} b \tan \left (d x + c\right ) + 60 \, a^{6} b^{3} \tan \left (d x + c\right ) - 201 \, a^{4} b^{5} \tan \left (d x + c\right ) + 6 \, a^{2} b^{7} \tan \left (d x + c\right ) + a^{9} + 26 \, a^{7} b^{2} - 63 \, a^{5} b^{4}}{{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\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)^3/(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^5*tan(d*x + c)^3 - 66*a^2*
b^7*tan(d*x + c)^3 + 11*b^9*tan(d*x + c)^3 + 39*a^5*b^4*tan(d*x + c)^2 - 210*a^3*b^6*tan(d*x + c)^2 + 15*a*b^8
*tan(d*x + c)^2 + 3*a^8*b*tan(d*x + c) + 60*a^6*b^3*tan(d*x + c) - 201*a^4*b^5*tan(d*x + c) + 6*a^2*b^7*tan(d*
x + c) + a^9 + 26*a^7*b^2 - 63*a^5*b^4)/((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*(b*tan(d*x + c)
+ a)^3))/d