\(\int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 143 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {1}{32 a^3 d (1-\cos (c+d x))}+\frac {1}{16 a^3 d (1+\cos (c+d x))^4}-\frac {5}{12 a^3 d (1+\cos (c+d x))^3}+\frac {39}{32 a^3 d (1+\cos (c+d x))^2}-\frac {9}{4 a^3 d (1+\cos (c+d x))}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (1+\cos (c+d x))}{64 a^3 d} \]

[Out]

-1/32/a^3/d/(1-cos(d*x+c))+1/16/a^3/d/(1+cos(d*x+c))^4-5/12/a^3/d/(1+cos(d*x+c))^3+39/32/a^3/d/(1+cos(d*x+c))^
2-9/4/a^3/d/(1+cos(d*x+c))-7/64*ln(1-cos(d*x+c))/a^3/d-57/64*ln(1+cos(d*x+c))/a^3/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 90} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {1}{32 a^3 d (1-\cos (c+d x))}-\frac {9}{4 a^3 d (\cos (c+d x)+1)}+\frac {39}{32 a^3 d (\cos (c+d x)+1)^2}-\frac {5}{12 a^3 d (\cos (c+d x)+1)^3}+\frac {1}{16 a^3 d (\cos (c+d x)+1)^4}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (\cos (c+d x)+1)}{64 a^3 d} \]

[In]

Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/32*1/(a^3*d*(1 - Cos[c + d*x])) + 1/(16*a^3*d*(1 + Cos[c + d*x])^4) - 5/(12*a^3*d*(1 + Cos[c + d*x])^3) + 3
9/(32*a^3*d*(1 + Cos[c + d*x])^2) - 9/(4*a^3*d*(1 + Cos[c + d*x])) - (7*Log[1 - Cos[c + d*x]])/(64*a^3*d) - (5
7*Log[1 + Cos[c + d*x]])/(64*a^3*d)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 3964

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^4 \text {Subst}\left (\int \frac {x^6}{(a-a x)^2 (a+a x)^5} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {Subst}\left (\int \left (\frac {1}{32 a^7 (-1+x)^2}+\frac {7}{64 a^7 (-1+x)}+\frac {1}{4 a^7 (1+x)^5}-\frac {5}{4 a^7 (1+x)^4}+\frac {39}{16 a^7 (1+x)^3}-\frac {9}{4 a^7 (1+x)^2}+\frac {57}{64 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {1}{32 a^3 d (1-\cos (c+d x))}+\frac {1}{16 a^3 d (1+\cos (c+d x))^4}-\frac {5}{12 a^3 d (1+\cos (c+d x))^3}+\frac {39}{32 a^3 d (1+\cos (c+d x))^2}-\frac {9}{4 a^3 d (1+\cos (c+d x))}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (1+\cos (c+d x))}{64 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (864+12 \cot ^2\left (\frac {1}{2} (c+d x)\right )+24 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (57 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+7 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-234 \sec ^2\left (\frac {1}{2} (c+d x)\right )+40 \sec ^4\left (\frac {1}{2} (c+d x)\right )-3 \sec ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^3(c+d x)}{96 a^3 d (1+\sec (c+d x))^3} \]

[In]

Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/96*(Cos[(c + d*x)/2]^4*(864 + 12*Cot[(c + d*x)/2]^2 + 24*Cos[(c + d*x)/2]^2*(57*Log[Cos[(c + d*x)/2]] + 7*L
og[Sin[(c + d*x)/2]]) - 234*Sec[(c + d*x)/2]^2 + 40*Sec[(c + d*x)/2]^4 - 3*Sec[(c + d*x)/2]^6)*Sec[c + d*x]^3)
/(a^3*d*(1 + Sec[c + d*x])^3)

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(\frac {\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {5}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {39}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {9}{4 \left (\cos \left (d x +c \right )+1\right )}-\frac {57 \ln \left (\cos \left (d x +c \right )+1\right )}{64}+\frac {1}{32 \cos \left (d x +c \right )-32}-\frac {7 \ln \left (\cos \left (d x +c \right )-1\right )}{64}}{d \,a^{3}}\)	\(91\)
default	\(\frac {\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {5}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {39}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {9}{4 \left (\cos \left (d x +c \right )+1\right )}-\frac {57 \ln \left (\cos \left (d x +c \right )+1\right )}{64}+\frac {1}{32 \cos \left (d x +c \right )-32}-\frac {7 \ln \left (\cos \left (d x +c \right )-1\right )}{64}}{d \,a^{3}}\)	\(91\)
risch	\(\frac {i x}{a^{3}}+\frac {2 i c}{a^{3} d}-\frac {213 \,{\mathrm e}^{9 i \left (d x +c \right )}+606 \,{\mathrm e}^{8 i \left (d x +c \right )}+472 \,{\mathrm e}^{7 i \left (d x +c \right )}-846 \,{\mathrm e}^{6 i \left (d x +c \right )}-1658 \,{\mathrm e}^{5 i \left (d x +c \right )}-846 \,{\mathrm e}^{4 i \left (d x +c \right )}+472 \,{\mathrm e}^{3 i \left (d x +c \right )}+606 \,{\mathrm e}^{2 i \left (d x +c \right )}+213 \,{\mathrm e}^{i \left (d x +c \right )}}{48 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 a^{3} d}-\frac {57 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 a^{3} d}\)	\(193\)

[In]

int(cot(d*x+c)^3/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d/a^3*(1/16/(cos(d*x+c)+1)^4-5/12/(cos(d*x+c)+1)^3+39/32/(cos(d*x+c)+1)^2-9/4/(cos(d*x+c)+1)-57/64*ln(cos(d*
x+c)+1)+1/32/(cos(d*x+c)-1)-7/64*ln(cos(d*x+c)-1))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {426 \, \cos \left (d x + c\right )^{4} + 606 \, \cos \left (d x + c\right )^{3} - 190 \, \cos \left (d x + c\right )^{2} + 171 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 666 \, \cos \left (d x + c\right ) - 272}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}} \]

[In]

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/192*(426*cos(d*x + c)^4 + 606*cos(d*x + c)^3 - 190*cos(d*x + c)^2 + 171*(cos(d*x + c)^5 + 3*cos(d*x + c)^4
+ 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) + 21*(cos(d*x + c)^5 +
3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) - 6
66*cos(d*x + c) - 272)/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d
*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)

Sympy [F]

\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

[In]

integrate(cot(d*x+c)**3/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**3/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (213 \, \cos \left (d x + c\right )^{4} + 303 \, \cos \left (d x + c\right )^{3} - 95 \, \cos \left (d x + c\right )^{2} - 333 \, \cos \left (d x + c\right ) - 136\right )}}{a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} - 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}} + \frac {171 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {21 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{192 \, d} \]

[In]

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/192*(2*(213*cos(d*x + c)^4 + 303*cos(d*x + c)^3 - 95*cos(d*x + c)^2 - 333*cos(d*x + c) - 136)/(a^3*cos(d*x
+ c)^5 + 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 - 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(d*x + c) - a^3) + 171*
log(cos(d*x + c) + 1)/a^3 + 21*log(cos(d*x + c) - 1)/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {12 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {84 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {768 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {504 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{12}}}{768 \, d} \]

[In]

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/768*(12*(7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)*(cos(d*x + c) + 1)/(a^3*(cos(d*x + c) - 1)) - 84*log(a
bs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 + 768*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^
3 + (504*a^9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 132*a^9*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 28*a^
9*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 3*a^9*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/a^12)/d

Mupad [B] (verification not implemented)

Time = 14.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {7\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{32}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{256}}{a^3\,d} \]

[In]

int(cot(c + d*x)^3/(a + a/cos(c + d*x))^3,x)

[Out]

-((7*log(tan(c/2 + (d*x)/2)))/32 - log(tan(c/2 + (d*x)/2)^2 + 1) + cot(c/2 + (d*x)/2)^2/64 + (21*tan(c/2 + (d*
x)/2)^2)/32 - (11*tan(c/2 + (d*x)/2)^4)/64 + (7*tan(c/2 + (d*x)/2)^6)/192 - tan(c/2 + (d*x)/2)^8/256)/(a^3*d)