\(\int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [491]

Optimal result

Integrand size = 19, antiderivative size = 172 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612, 3611} \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4} \]

[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 3610

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 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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains complex when optimal does not.

Time = 6.25 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.85 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {a \left (\frac {3 i \log (i-\tan (c+d x))}{(a+i b)^4}-\frac {3 i \log (i+\tan (c+d x))}{(a-i b)^4}-\frac {24 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {6 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{6 b d}-\frac {\frac {\log (i-\tan (c+d x))}{(i a-b)^3}-\frac {\log (i+\tan (c+d x))}{(i a+b)^3}-\frac {2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 b d} \]

[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 [A] (verified)

Time = 0.55 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(191\)
default	\(\frac {\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(191\)
norman	\(\frac {\frac {a \left (a^{2} b^{3}-3 b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}+\frac {a \left (11 a^{4} b^{3}-14 b^{5} a^{2}-b^{7}\right )}{6 b^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {\left (5 a^{4} b^{3}-12 b^{5} a^{2}-b^{7}\right ) \tan \left (d x +c \right )}{2 d \,b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\)	\(541\)
risch	\(\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}+\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 b \left (-9 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}+18 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-13 i a \,b^{4}+6 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-9 i a^{5}+24 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+22 i a^{3} b^{2}+18 a^{4} b -26 a^{2} b^{3}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}-\frac {a^{4} \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 b^{6} a^{2}+b^{8}\right )}+\frac {6 a^{2} b^{2} \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 b^{6} a^{2}+b^{8}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\)	\(792\)
parallelrisch	\(\text {Expression too large to display}\)	\(877\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (168) = 336\).

Time = 0.26 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.07 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {27 \, a^{5} b^{2} - 18 \, a^{3} b^{4} - a b^{6} - {\left (11 \, a^{4} b^{3} - 30 \, a^{2} b^{5} + 3 \, b^{7} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x - 3 \, {\left (9 \, a^{5} b^{2} - 26 \, a^{3} b^{4} + 9 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (6 \, a^{6} b - 23 \, a^{4} b^{3} + 16 \, a^{2} b^{5} + b^{7} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \]

[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)]

Exception generated. \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (168) = 336\).

Time = 0.31 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.29 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, a^{5} - 14 \, a^{3} b^{2} - a b^{4} + 6 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{6 \, d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (168) = 336\).

Time = 0.63 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.33 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 11 \, b^{7} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{2} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{4} \tan \left (d x + c\right )^{2} + 15 \, a b^{6} \tan \left (d x + c\right )^{2} + 48 \, a^{6} b \tan \left (d x + c\right ) - 219 \, a^{4} b^{3} \tan \left (d x + c\right ) - 6 \, a^{2} b^{5} \tan \left (d x + c\right ) - 3 \, b^{7} \tan \left (d x + c\right ) + 22 \, a^{7} - 69 \, a^{5} b^{2} - 4 \, a^{3} b^{4} - a b^{6}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.71 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.06 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {-11\,a^5+14\,a^3\,b^2+a\,b^4}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,a^4\,b+12\,a^2\,b^3+b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

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