3.5.88 \(\int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [488]

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

[Out]

(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4+4*a*b*(a^2-b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^4/d-1/3*a^2*tan(d*x+
c)^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/3*a^3*(a^2+4*b^2)/b^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-1/3*a^2*(2*a^4+
7*a^2*b^2+17*b^4)/b^3/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.28, antiderivative size = 208, normalized size of antiderivative = 1.00, 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 = {3646, 3716, 3709, 3612, 3611} \begin {gather*} -\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 {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 (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\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} \end {gather*}

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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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 3646

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 3709

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 3716

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.62, size = 236, normalized size = 1.13 \begin {gather*} -\frac {\frac {3 i b^2 \log (i-\tan (c+d x))}{(a+i b)^4}-\frac {3 i b^2 \log (i+\tan (c+d x))}{(i a+b)^4}-\frac {24 a (a-b) b^3 (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 a^4}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {6 a \tan (c+d x)}{(a+b \tan (c+d x))^3}+\frac {6 b \tan ^2(c+d x)}{(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 (-3 a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}}{6 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(((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*Tan[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])))/(b^2*d)

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Maple [A]
time = 0.18, size = 211, normalized size = 1.01

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{4}}{3 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{2} \left (a^{4}+3 a^{2} b^{2}+6 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{3} \left (a^{2}+2 b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(211\)
default	\(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{4}}{3 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{2} \left (a^{4}+3 a^{2} b^{2}+6 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{3} \left (a^{2}+2 b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(211\)
norman	\(\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{3} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {4 b^{3} a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {b^{3} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (a^{7}+3 a^{5} b^{2}+10 a^{3} b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {a \left (a^{5}-3 a^{3} b^{2}\right )}{3 b \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {3 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(529\)
risch	\(-\frac {x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {8 i a^{3} b x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {8 i a \,b^{3} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {8 i a^{3} b c}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) d}+\frac {8 i a \,b^{3} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {4 i a^{2} \left (12 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+9 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-15 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a^{3} b -18 i a \,b^{3}+2 a^{4}-11 a^{2} b^{2}+9 b^{4}\right )}{3 \left (i b +a \right )^{3} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{3} \left (-i b +a \right )^{4} d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(563\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (202) = 404\).
time = 0.51, size = 410, normalized size = 1.97 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (202) = 404\).
time = 1.09, size = 510, normalized size = 2.45 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

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 >> 'NoneType' object has no attribute 'primitive'

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (202) = 404\).
time = 1.32, size = 409, normalized size = 1.97 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 4.64, size = 359, normalized size = 1.73 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a\,b}{{\left (a^2+b^2\right )}^3}-\frac {8\,a\,b^3}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\frac {a^2\,\left (a^6+2\,a^4\,b^2+13\,a^2\,b^4\right )}{3\,b^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^6+3\,a^4\,b^2+6\,a^2\,b^4\right )}{b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+3\,a^4\,b^2+10\,a^2\,b^4\right )}{b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (8*a*b^3)/(a^2 + b^2)^4))/d + log(tan(c + d*x) - 1i)/(2*d*(4
*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)) + (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b^3*4i - a^3*b*4i + a^
4 + b^4 - 6*a^2*b^2)) - ((a^2*(a^6 + 13*a^2*b^4 + 2*a^4*b^2))/(3*b^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (t
an(c + d*x)^2*(a^6 + 6*a^2*b^4 + 3*a^4*b^2))/(b*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*tan(c + d*x)*(a^6 +
10*a^2*b^4 + 3*a^4*b^2))/(b^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^3*tan(c + d*x)^3 + 3*a*b^2*tan
(c + d*x)^2 + 3*a^2*b*tan(c + d*x)))

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