\(\int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [274]

Optimal result

Integrand size = 31, antiderivative size = 169 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {(a A+b B) x}{a^2+b^2}+\frac {\left (a^2 A-A b^2+a b B\right ) \cot (c+d x)}{a^3 d}+\frac {(A b-a B) \cot ^2(c+d x)}{2 a^2 d}-\frac {A \cot ^3(c+d x)}{3 a d}+\frac {\left (a^2-b^2\right ) (A b-a B) \log (\sin (c+d x))}{a^4 d}+\frac {b^4 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right ) d} \]

[Out]

(A*a+B*b)*x/(a^2+b^2)+(A*a^2-A*b^2+B*a*b)*cot(d*x+c)/a^3/d+1/2*(A*b-B*a)*cot(d*x+c)^2/a^2/d-1/3*A*cot(d*x+c)^3
/a/d+(a^2-b^2)*(A*b-B*a)*ln(sin(d*x+c))/a^4/d+b^4*(A*b-B*a)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)/d

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {x (a A+b B)}{a^2+b^2}+\frac {(A b-a B) \cot ^2(c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right ) (A b-a B) \log (\sin (c+d x))}{a^4 d}+\frac {b^4 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )}+\frac {\left (a^2 A+a b B-A b^2\right ) \cot (c+d x)}{a^3 d}-\frac {A \cot ^3(c+d x)}{3 a d} \]

[In]

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

[Out]

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{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 3690

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d)
)*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.95 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.15 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {6 \left (a^2 A-A b^2+a b B\right ) \cot (c+d x)}{a^3}+\frac {3 (A b-a B) \cot ^2(c+d x)}{a^2}-\frac {2 A \cot ^3(c+d x)}{a}+\frac {3 (-i A+B) \log (i-\tan (c+d x))}{a+i b}+\frac {6 (a-b) (a+b) (A b-a B) \log (\tan (c+d x))}{a^4}+\frac {3 (i A+B) \log (i+\tan (c+d x))}{a-i b}+\frac {6 b^4 (A b-a B) \log (a+b \tan (c+d x))}{a^4 \left (a^2+b^2\right )}}{6 d} \]

[In]

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

[Out]

((6*(a^2*A - A*b^2 + a*b*B)*Cot[c + d*x])/a^3 + (3*(A*b - a*B)*Cot[c + d*x]^2)/a^2 - (2*A*Cot[c + d*x]^3)/a +
(3*((-I)*A + B)*Log[I - Tan[c + d*x]])/(a + I*b) + (6*(a - b)*(a + b)*(A*b - a*B)*Log[Tan[c + d*x]])/a^4 + (3*
(I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b) + (6*b^4*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)))/(6
*d)

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a A +B b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {-A b +B a}{2 a^{2} \tan \left (d x +c \right )^{2}}-\frac {-A \,a^{2}+A \,b^{2}-B a b}{a^{3} \tan \left (d x +c \right )}+\frac {\left (A \,a^{2} b -A \,b^{3}-B \,a^{3}+B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {A}{3 a \tan \left (d x +c \right )^{3}}+\frac {\left (A b -B a \right ) b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{4}}}{d}\)	\(189\)
default	\(\frac {\frac {\frac {\left (-A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a A +B b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {-A b +B a}{2 a^{2} \tan \left (d x +c \right )^{2}}-\frac {-A \,a^{2}+A \,b^{2}-B a b}{a^{3} \tan \left (d x +c \right )}+\frac {\left (A \,a^{2} b -A \,b^{3}-B \,a^{3}+B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {A}{3 a \tan \left (d x +c \right )^{3}}+\frac {\left (A b -B a \right ) b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{4}}}{d}\)	\(189\)
parallelrisch	\(\frac {\left (6 A \,b^{5}-6 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+\left (-3 A \,a^{4} b +3 B \,a^{5}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (6 A \,a^{4} b -6 A \,b^{5}-6 B \,a^{5}+6 B a \,b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )+6 a \left (-\frac {A \,a^{2} \left (\cot ^{3}\left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )}{3}+\frac {\left (\cot ^{2}\left (d x +c \right )\right ) a \left (a^{2}+b^{2}\right ) \left (A b -B a \right )}{2}+\cot \left (d x +c \right ) \left (a^{2}+b^{2}\right ) \left (A \,a^{2}-A \,b^{2}+B a b \right )+a^{3} d x \left (a A +B b \right )\right )}{6 \left (a^{2}+b^{2}\right ) a^{4} d}\)	\(198\)
norman	\(\frac {\frac {\left (a A +B b \right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {\left (A \,a^{2}-A \,b^{2}+B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{a^{3} d}-\frac {A}{3 a d}+\frac {\left (A b -B a \right ) \tan \left (d x +c \right )}{2 a^{2} d}}{\tan \left (d x +c \right )^{3}}+\frac {\left (A b -B a \right ) \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {b^{4} \left (A b -B a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{4} d}-\frac {\left (A b -B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}\)	\(202\)
risch	\(-\frac {2 i B \,b^{2} x}{a^{3}}-\frac {x A}{i b -a}+\frac {2 i x B}{a}-\frac {2 i B \,b^{2} c}{a^{3} d}-\frac {2 i b^{5} A c}{\left (a^{2}+b^{2}\right ) a^{4} d}+\frac {2 i b^{4} B x}{\left (a^{2}+b^{2}\right ) a^{3}}+\frac {2 i B c}{a d}-\frac {2 i \left (-3 i A a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i A a b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{2}+3 A \,b^{2}-3 B a b \right )}{3 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 i A \,b^{3} c}{a^{4} d}+\frac {2 i b^{4} B c}{\left (a^{2}+b^{2}\right ) a^{3} d}+\frac {2 i A \,b^{3} x}{a^{4}}-\frac {2 i A b x}{a^{2}}-\frac {2 i b^{5} A x}{\left (a^{2}+b^{2}\right ) a^{4}}+\frac {i x B}{i b -a}-\frac {2 i A b c}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{3}}{a^{4} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{a^{3} d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{\left (a^{2}+b^{2}\right ) a^{4} d}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{2}+b^{2}\right ) a^{3} d}\)	\(585\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {2 \, A a^{5} + 2 \, A a^{3} b^{2} + 3 \, {\left (B a^{5} - A a^{4} b - B a b^{4} + A b^{5}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (B a b^{4} - A b^{5}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (B a^{5} - A a^{4} b + B a^{3} b^{2} - A a^{2} b^{3} - 2 \, {\left (A a^{5} + B a^{4} b\right )} d x\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (A a^{5} + B a^{4} b + B a^{2} b^{3} - A a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{5} - A a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{6} + a^{4} b^{2}\right )} d \tan \left (d x + c\right )^{3}} \]

[In]

integrate(cot(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(2*A*a^5 + 2*A*a^3*b^2 + 3*(B*a^5 - A*a^4*b - B*a*b^4 + A*b^5)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*t
an(d*x + c)^3 + 3*(B*a*b^4 - A*b^5)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1))*
tan(d*x + c)^3 + 3*(B*a^5 - A*a^4*b + B*a^3*b^2 - A*a^2*b^3 - 2*(A*a^5 + B*a^4*b)*d*x)*tan(d*x + c)^3 - 6*(A*a
^5 + B*a^4*b + B*a^2*b^3 - A*a*b^4)*tan(d*x + c)^2 + 3*(B*a^5 - A*a^4*b + B*a^3*b^2 - A*a^2*b^3)*tan(d*x + c))
/((a^6 + a^4*b^2)*d*tan(d*x + c)^3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.20 (sec) , antiderivative size = 3016, normalized size of antiderivative = 17.85 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*A*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((A*x + A/(d*tan(c + d*x)) - A/(3*d*tan(c + d*
x)**3) + B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c + d*x))/d - B/(2*d*tan(c + d*x)**2))/a, Eq(b, 0)), ((-
A*log(tan(c + d*x)**2 + 1)/(2*d) + A*log(tan(c + d*x))/d + A/(2*d*tan(c + d*x)**2) - A/(4*d*tan(c + d*x)**4) +
B*x + B/(d*tan(c + d*x)) - B/(3*d*tan(c + d*x)**3))/b, Eq(a, 0)), (15*A*d*x*tan(c + d*x)**4/(6*a*d*tan(c + d*
x)**4 + 6*I*a*d*tan(c + d*x)**3) + 15*I*A*d*x*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3
) + 6*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 6*A*log
(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 12*I*A*log(tan(c + d
*x))*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + 12*A*log(tan(c + d*x))*tan(c + d*x)**
3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + 15*A*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*ta
n(c + d*x)**3) + 9*I*A*tan(c + d*x)**2/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + A*tan(c + d*x)/(6*a
*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 2*I*A/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 9*I*
B*d*x*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + 9*B*d*x*tan(c + d*x)**3/(6*a*d*tan(c
+ d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + 6*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 +
6*I*a*d*tan(c + d*x)**3) + 6*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan
(c + d*x)**3) - 12*B*log(tan(c + d*x))*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 12*
I*B*log(tan(c + d*x))*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) - 9*I*B*tan(c + d*x)**
3/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3) + 3*B*tan(c + d*x)**2/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan
(c + d*x)**3) - 3*I*B*tan(c + d*x)/(6*a*d*tan(c + d*x)**4 + 6*I*a*d*tan(c + d*x)**3), Eq(b, -I*a)), (15*A*d*x*
tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) - 15*I*A*d*x*tan(c + d*x)**3/(6*a*d*tan(c +
d*x)**4 - 6*I*a*d*tan(c + d*x)**3) - 6*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 - 6
*I*a*d*tan(c + d*x)**3) - 6*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c
+ d*x)**3) + 12*I*A*log(tan(c + d*x))*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 12*A
*log(tan(c + d*x))*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 15*A*tan(c + d*x)**3/(6
*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) - 9*I*A*tan(c + d*x)**2/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c
+ d*x)**3) + A*tan(c + d*x)/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 2*I*A/(6*a*d*tan(c + d*x)**4
- 6*I*a*d*tan(c + d*x)**3) + 9*I*B*d*x*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 9*B
*d*x*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 6*B*log(tan(c + d*x)**2 + 1)*tan(c +
d*x)**4/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) - 6*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*
a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) - 12*B*log(tan(c + d*x))*tan(c + d*x)**4/(6*a*d*tan(c + d*x)**4
- 6*I*a*d*tan(c + d*x)**3) + 12*I*B*log(tan(c + d*x))*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c
+ d*x)**3) + 9*I*B*tan(c + d*x)**3/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 3*B*tan(c + d*x)**2/(6*
a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d*x)**3) + 3*I*B*tan(c + d*x)/(6*a*d*tan(c + d*x)**4 - 6*I*a*d*tan(c + d
*x)**3), Eq(b, I*a)), (zoo*A*x/a, Eq(c, -d*x)), (x*(A + B*tan(c))*cot(c)**4/(a + b*tan(c)), Eq(d, 0)), (6*A*a*
*5*d*x*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) + 6*A*a**5*tan(c + d*x)**2/(
6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) - 2*A*a**5/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d
*tan(c + d*x)**3) - 3*A*a**4*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**
2*d*tan(c + d*x)**3) + 6*A*a**4*b*log(tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*
tan(c + d*x)**3) + 3*A*a**4*b*tan(c + d*x)/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) - 2*A*a*
*3*b**2/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) + 3*A*a**2*b**3*tan(c + d*x)/(6*a**6*d*tan(
c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) - 6*A*a*b**4*tan(c + d*x)**2/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b
**2*d*tan(c + d*x)**3) + 6*A*b**5*log(a/b + tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b
**2*d*tan(c + d*x)**3) - 6*A*b**5*log(tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*
tan(c + d*x)**3) + 3*B*a**5*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d
*tan(c + d*x)**3) - 6*B*a**5*log(tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c
+ d*x)**3) - 3*B*a**5*tan(c + d*x)/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) + 6*B*a**4*b*d*
x*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) + 6*B*a**4*b*tan(c + d*x)**2/(6*a
**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c + d*x)**3) - 3*B*a**3*b**2*tan(c + d*x)/(6*a**6*d*tan(c + d*x)**3
+ 6*a**4*b**2*d*tan(c + d*x)**3) + 6*B*a**2*b**3*tan(c + d*x)**2/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan
(c + d*x)**3) - 6*B*a*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*t
an(c + d*x)**3) + 6*B*a*b**4*log(tan(c + d*x))*tan(c + d*x)**3/(6*a**6*d*tan(c + d*x)**3 + 6*a**4*b**2*d*tan(c
+ d*x)**3), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {6 \, {\left (A a + B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {6 \, {\left (B a b^{4} - A b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + a^{4} b^{2}} + \frac {3 \, {\left (B a - A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {6 \, {\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}} - \frac {2 \, A a^{2} - 6 \, {\left (A a^{2} + B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{2} - A a b\right )} \tan \left (d x + c\right )}{a^{3} \tan \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(A*a + B*b)*(d*x + c)/(a^2 + b^2) - 6*(B*a*b^4 - A*b^5)*log(b*tan(d*x + c) + a)/(a^6 + a^4*b^2) + 3*(B*
a - A*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 6*(B*a^3 - A*a^2*b - B*a*b^2 + A*b^3)*log(tan(d*x + c))/a^4 - (
2*A*a^2 - 6*(A*a^2 + B*a*b - A*b^2)*tan(d*x + c)^2 + 3*(B*a^2 - A*a*b)*tan(d*x + c))/(a^3*tan(d*x + c)^3))/d

Giac [A] (verification not implemented)

none

Time = 0.87 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {6 \, {\left (A a + B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {3 \, {\left (B a - A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {6 \, {\left (B a b^{5} - A b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + a^{4} b^{3}} - \frac {6 \, {\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {11 \, B a^{3} \tan \left (d x + c\right )^{3} - 11 \, A a^{2} b \tan \left (d x + c\right )^{3} - 11 \, B a b^{2} \tan \left (d x + c\right )^{3} + 11 \, A b^{3} \tan \left (d x + c\right )^{3} + 6 \, A a^{3} \tan \left (d x + c\right )^{2} + 6 \, B a^{2} b \tan \left (d x + c\right )^{2} - 6 \, A a b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{3} \tan \left (d x + c\right ) + 3 \, A a^{2} b \tan \left (d x + c\right ) - 2 \, A a^{3}}{a^{4} \tan \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(A*a + B*b)*(d*x + c)/(a^2 + b^2) + 3*(B*a - A*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 6*(B*a*b^5 - A*
b^6)*log(abs(b*tan(d*x + c) + a))/(a^6*b + a^4*b^3) - 6*(B*a^3 - A*a^2*b - B*a*b^2 + A*b^3)*log(abs(tan(d*x +
c)))/a^4 + (11*B*a^3*tan(d*x + c)^3 - 11*A*a^2*b*tan(d*x + c)^3 - 11*B*a*b^2*tan(d*x + c)^3 + 11*A*b^3*tan(d*x
+ c)^3 + 6*A*a^3*tan(d*x + c)^2 + 6*B*a^2*b*tan(d*x + c)^2 - 6*A*a*b^2*tan(d*x + c)^2 - 3*B*a^3*tan(d*x + c)
+ 3*A*a^2*b*tan(d*x + c) - 2*A*a^3)/(a^4*tan(d*x + c)^3))/d

Mupad [B] (verification not implemented)

Time = 10.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (A\,a^2+B\,a\,b-A\,b^2\right )}{a^3}-\frac {A}{3\,a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,b-B\,a\right )}{2\,a^2}\right )}{d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,b^5-B\,a\,b^4\right )}{d\,\left (a^6+a^4\,b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^3-A\,a^2\,b-B\,a\,b^2+A\,b^3\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

(cot(c + d*x)^3*((tan(c + d*x)^2*(A*a^2 - A*b^2 + B*a*b))/a^3 - A/(3*a) + (tan(c + d*x)*(A*b - B*a))/(2*a^2)))
/d + (log(a + b*tan(c + d*x))*(A*b^5 - B*a*b^4))/(d*(a^6 + a^4*b^2)) - (log(tan(c + d*x))*(A*b^3 + B*a^3 - A*a
^2*b - B*a*b^2))/(a^4*d) - (log(tan(c + d*x) + 1i)*(A - B*1i))/(2*d*(a*1i + b)) - (log(tan(c + d*x) - 1i)*(A*1
i - B))/(2*d*(a + b*1i))