3.306 \(\int \frac{\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=278 \[ \frac{b^6}{a d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^4}+\frac{\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac{\left (4 a^2-13 a b+12 b^2\right ) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac{5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b)^2 (\sec (c+d x)+1)^2} \]

[Out]

Log[Cos[c + d*x]]/(a^2*d) + ((4*a^2 + 13*a*b + 12*b^2)*Log[1 - Sec[c + d*x]])/(8*(a + b)^4*d) + ((4*a^2 - 13*a
*b + 12*b^2)*Log[1 + Sec[c + d*x]])/(8*(a - b)^4*d) - (b^6*(7*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 -
b^2)^4*d) - 1/(16*(a + b)^2*d*(1 - Sec[c + d*x])^2) - (5*a + 9*b)/(16*(a + b)^3*d*(1 - Sec[c + d*x])) - 1/(16*
(a - b)^2*d*(1 + Sec[c + d*x])^2) - (5*a - 9*b)/(16*(a - b)^3*d*(1 + Sec[c + d*x])) + b^6/(a*(a^2 - b^2)^3*d*(
a + b*Sec[c + d*x]))

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Rubi [A]  time = 0.371301, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3885, 894} \[ \frac{b^6}{a d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^4}+\frac{\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac{\left (4 a^2-13 a b+12 b^2\right ) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac{5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b)^2 (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

Log[Cos[c + d*x]]/(a^2*d) + ((4*a^2 + 13*a*b + 12*b^2)*Log[1 - Sec[c + d*x]])/(8*(a + b)^4*d) + ((4*a^2 - 13*a
*b + 12*b^2)*Log[1 + Sec[c + d*x]])/(8*(a - b)^4*d) - (b^6*(7*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 -
b^2)^4*d) - 1/(16*(a + b)^2*d*(1 - Sec[c + d*x])^2) - (5*a + 9*b)/(16*(a + b)^3*d*(1 - Sec[c + d*x])) - 1/(16*
(a - b)^2*d*(1 + Sec[c + d*x])^2) - (5*a - 9*b)/(16*(a - b)^3*d*(1 + Sec[c + d*x])) + b^6/(a*(a^2 - b^2)^3*d*(
a + b*Sec[c + d*x]))

Rule 3885

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

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=-\frac{b^6 \operatorname{Subst}\left (\int \frac{1}{x (a+x)^2 \left (b^2-x^2\right )^3} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{b^6 \operatorname{Subst}\left (\int \left (\frac{1}{8 b^4 (a+b)^2 (b-x)^3}+\frac{5 a+9 b}{16 b^5 (a+b)^3 (b-x)^2}+\frac{4 a^2+13 a b+12 b^2}{8 b^6 (a+b)^4 (b-x)}+\frac{1}{a^2 b^6 x}+\frac{1}{a (a-b)^3 (a+b)^3 (a+x)^2}+\frac{7 a^2-b^2}{a^2 (a-b)^4 (a+b)^4 (a+x)}-\frac{1}{8 (a-b)^2 b^4 (b+x)^3}+\frac{-5 a+9 b}{16 (a-b)^3 b^5 (b+x)^2}+\frac{-4 a^2+13 a b-12 b^2}{8 (a-b)^4 b^6 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 (a+b)^4 d}+\frac{\left (4 a^2-13 a b+12 b^2\right ) \log (1+\sec (c+d x))}{8 (a-b)^4 d}-\frac{b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^4 d}-\frac{1}{16 (a+b)^2 d (1-\sec (c+d x))^2}-\frac{5 a+9 b}{16 (a+b)^3 d (1-\sec (c+d x))}-\frac{1}{16 (a-b)^2 d (1+\sec (c+d x))^2}-\frac{5 a-9 b}{16 (a-b)^3 d (1+\sec (c+d x))}+\frac{b^6}{a \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.03477, size = 473, normalized size = 1.7 \[ \frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left (\frac{128 i \left (-4 a^4 b^2+6 a^2 b^4+a^6+3 b^6\right ) (c+d x) (a \cos (c+d x)+b)}{(a-b)^4 (a+b)^4}+\frac{8 \left (4 a^2-13 a b+12 b^2\right ) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a-b)^4}+\frac{64 \left (b^8-7 a^2 b^6\right ) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a^2 \left (a^2-b^2\right )^4}-\frac{16 i \left (4 a^2-13 a b+12 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a-b)^4}-\frac{16 i \left (4 a^2+13 a b+12 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{8 \left (4 a^2+13 a b+12 b^2\right ) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{64 b^7}{a^2 (b-a)^3 (a+b)^3}-\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{2 (7 a+11 b) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^3}-\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}+\frac{2 (7 a-11 b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^3}\right )}{64 d (a+b \sec (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

((b + a*Cos[c + d*x])*((64*b^7)/(a^2*(-a + b)^3*(a + b)^3) + ((128*I)*(a^6 - 4*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*(c
+ d*x)*(b + a*Cos[c + d*x]))/((a - b)^4*(a + b)^4) - ((16*I)*(4*a^2 - 13*a*b + 12*b^2)*ArcTan[Tan[c + d*x]]*(
b + a*Cos[c + d*x]))/(a - b)^4 - ((16*I)*(4*a^2 + 13*a*b + 12*b^2)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/
(a + b)^4 + (2*(7*a + 11*b)*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c
+ d*x)/2]^4)/(a + b)^2 + (8*(4*a^2 - 13*a*b + 12*b^2)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]^2])/(a - b)^4
+ (64*(-7*a^2*b^6 + b^8)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)^4) + (8*(4*a^2 + 13*a*
b + 12*b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]^2])/(a + b)^4 + (2*(7*a - 11*b)*(b + a*Cos[c + d*x])*Sec
[(c + d*x)/2]^2)/(a - b)^3 - ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/(a - b)^2)*Sec[c + d*x]^2)/(64*d*(a + b
*Sec[c + d*x])^2)

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Maple [A]  time = 0.09, size = 367, normalized size = 1.3 \begin{align*} -{\frac{{b}^{7}}{d{a}^{2} \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-7\,{\frac{{b}^{6}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{{b}^{8}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}{a}^{2}}}-{\frac{1}{16\,d \left ( a-b \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{7\,a}{16\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{11\,b}{16\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ){a}^{2}}{2\,d \left ( a-b \right ) ^{4}}}-{\frac{13\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) ab}{8\,d \left ( a-b \right ) ^{4}}}+{\frac{3\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ){b}^{2}}{2\,d \left ( a-b \right ) ^{4}}}-{\frac{1}{16\,d \left ( a+b \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,a}{16\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{11\,b}{16\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ){a}^{2}}{2\,d \left ( a+b \right ) ^{4}}}+{\frac{13\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) ab}{8\,d \left ( a+b \right ) ^{4}}}+{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ){b}^{2}}{2\,d \left ( a+b \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x)

[Out]

-1/d*b^7/a^2/(a+b)^3/(a-b)^3/(b+a*cos(d*x+c))-7/d*b^6/(a+b)^4/(a-b)^4*ln(b+a*cos(d*x+c))+1/d*b^8/(a+b)^4/(a-b)
^4/a^2*ln(b+a*cos(d*x+c))-1/16/d/(a-b)^2/(cos(d*x+c)+1)^2+7/16/d/(a-b)^3/(cos(d*x+c)+1)*a-11/16/d/(a-b)^3/(cos
(d*x+c)+1)*b+1/2/d/(a-b)^4*ln(cos(d*x+c)+1)*a^2-13/8/d/(a-b)^4*ln(cos(d*x+c)+1)*a*b+3/2/d/(a-b)^4*ln(cos(d*x+c
)+1)*b^2-1/16/d/(a+b)^2/(-1+cos(d*x+c))^2-7/16/d/(a+b)^3/(-1+cos(d*x+c))*a-11/16/d/(a+b)^3/(-1+cos(d*x+c))*b+1
/2/d/(a+b)^4*ln(-1+cos(d*x+c))*a^2+13/8/d/(a+b)^4*ln(-1+cos(d*x+c))*a*b+3/2/d/(a+b)^4*ln(-1+cos(d*x+c))*b^2

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Maxima [B]  time = 1.11149, size = 753, normalized size = 2.71 \begin{align*} -\frac{\frac{8 \,{\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}} - \frac{{\left (4 \, a^{2} - 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac{{\left (4 \, a^{2} + 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac{2 \,{\left (3 \, a^{6} b - 6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, b^{7} +{\left (5 \, a^{6} b - 13 \, a^{4} b^{3} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} -{\left (4 \, a^{7} - 11 \, a^{5} b^{2} + 7 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} -{\left (7 \, a^{6} b - 17 \, a^{4} b^{3} - 6 \, a^{2} b^{5} - 8 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (a^{7} - 3 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7} +{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{5} +{\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/8*(8*(7*a^2*b^6 - b^8)*log(a*cos(d*x + c) + b)/(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8) - (4*a^
2 - 13*a*b + 12*b^2)*log(cos(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (4*a^2 + 13*a*b + 12*
b^2)*log(cos(d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 2*(3*a^6*b - 6*a^4*b^3 - 5*a^2*b^5 -
4*b^7 + (5*a^6*b - 13*a^4*b^3 - 4*b^7)*cos(d*x + c)^4 - (4*a^7 - 11*a^5*b^2 + 7*a^3*b^4)*cos(d*x + c)^3 - (7*a
^6*b - 17*a^4*b^3 - 6*a^2*b^5 - 8*b^7)*cos(d*x + c)^2 + 3*(a^7 - 3*a^5*b^2 + 2*a^3*b^4)*cos(d*x + c))/(a^8*b -
3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7 + (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)^5 + (a^8*b - 3*a^6*b^3
+ 3*a^4*b^5 - a^2*b^7)*cos(d*x + c)^4 - 2*(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)^3 - 2*(a^8*b -
3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*cos(d*x + c)^2 + (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)))/d

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Fricas [B]  time = 2.64518, size = 3028, normalized size = 10.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/8*(6*a^8*b - 18*a^6*b^3 + 2*a^4*b^5 + 2*a^2*b^7 + 8*b^9 + 2*(5*a^8*b - 18*a^6*b^3 + 13*a^4*b^5 - 4*a^2*b^7 +
4*b^9)*cos(d*x + c)^4 - 2*(4*a^9 - 15*a^7*b^2 + 18*a^5*b^4 - 7*a^3*b^6)*cos(d*x + c)^3 - 2*(7*a^8*b - 24*a^6*
b^3 + 11*a^4*b^5 - 2*a^2*b^7 + 8*b^9)*cos(d*x + c)^2 + 6*(a^9 - 4*a^7*b^2 + 5*a^5*b^4 - 2*a^3*b^6)*cos(d*x + c
) - 8*(7*a^2*b^7 - b^9 + (7*a^3*b^6 - a*b^8)*cos(d*x + c)^5 + (7*a^2*b^7 - b^9)*cos(d*x + c)^4 - 2*(7*a^3*b^6
- a*b^8)*cos(d*x + c)^3 - 2*(7*a^2*b^7 - b^9)*cos(d*x + c)^2 + (7*a^3*b^6 - a*b^8)*cos(d*x + c))*log(a*cos(d*x
+ c) + b) + (4*a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7 + (4*a^9 +
3*a^8*b - 16*a^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^5 + (4*a^8*b + 3*a^7*b^
2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(4*a^9 + 3*a^8*b - 16*a
^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^3 - 2*(4*a^8*b + 3*a^7*b^2 - 16*a^6*b
^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^2 + (4*a^9 + 3*a^8*b - 16*a^7*b^2 - 14*a^
6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (4*a^8*b - 3*a^7*b^2
- 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7 + (4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3
+ 24*a^5*b^4 - 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^5 + (4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*
a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^4
- 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^3 - 2*(4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35
*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^2 + (4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^4 - 35*a^4*b^5 +
12*a^3*b^6)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)
*d*cos(d*x + c)^5 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^4 - 2*(a^11 - 4*a^9*
b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^3 - 2*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*
b^9)*d*cos(d*x + c)^2 + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) + (a^10*b - 4*a^8*
b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{5}{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5/(a+b*sec(d*x+c))**2,x)

[Out]

Integral(cot(c + d*x)**5/(a + b*sec(c + d*x))**2, x)

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Giac [B]  time = 1.49443, size = 1073, normalized size = 3.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/64*(8*(4*a^2 + 13*a*b + 12*b^2)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2
+ 4*a*b^3 + b^4) - 64*(7*a^2*b^6 - b^8)*log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x
+ c) - 1)/(cos(d*x + c) + 1)))/(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8) - (12*a^2*(cos(d*x + c) -
1)/(cos(d*x + c) + 1) - 32*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20*b^2*(cos(d*x + c) - 1)/(cos(d*x + c
) + 1) + a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 2*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + b^2
*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (a^2 + 2*a*b + b^2 +
12*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 32*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20*b^2*(cos(d*x
+ c) - 1)/(cos(d*x + c) + 1) + 48*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 156*a*b*(cos(d*x + c) - 1)^
2/(cos(d*x + c) + 1)^2 + 144*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)^2/((a^4 + 4*a^3
*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(cos(d*x + c) - 1)^2) + 64*(7*a^3*b^6 + 5*a^2*b^7 - 3*a*b^8 - b^9 + 7*a^3*b^6*
(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 7*a^2*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a*b^8*(cos(d*x + c)
- 1)/(cos(d*x + c) + 1) + b^9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^
6 + a^2*b^8)*(a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))) - 64
*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^2)/d