3.305 \(\int \frac{\cot ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=197 \[ \frac{b^4}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac{(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3} \]

[Out]

-(Log[Cos[c + d*x]]/(a^2*d)) - ((a + 2*b)*Log[1 - Sec[c + d*x]])/(2*(a + b)^3*d) - ((a - 2*b)*Log[1 + Sec[c +
d*x]])/(2*(a - b)^3*d) - (b^4*(5*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*
(1 - Sec[c + d*x])) + 1/(4*(a - b)^2*d*(1 + Sec[c + d*x])) + b^4/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

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Rubi [A]  time = 0.228771, antiderivative size = 197, 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} \[ \frac{b^4}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac{(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]

[Out]

-(Log[Cos[c + d*x]]/(a^2*d)) - ((a + 2*b)*Log[1 - Sec[c + d*x]])/(2*(a + b)^3*d) - ((a - 2*b)*Log[1 + Sec[c +
d*x]])/(2*(a - b)^3*d) - (b^4*(5*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*
(1 - Sec[c + d*x])) + 1/(4*(a - b)^2*d*(1 + Sec[c + d*x])) + b^4/(a*(a^2 - b^2)^2*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 ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{x (a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \left (\frac{1}{4 b^3 (a+b)^2 (b-x)^2}+\frac{a+2 b}{2 b^4 (a+b)^3 (b-x)}+\frac{1}{a^2 b^4 x}-\frac{1}{a (a-b)^2 (a+b)^2 (a+x)^2}+\frac{-5 a^2+b^2}{a^2 (a-b)^3 (a+b)^3 (a+x)}-\frac{1}{4 (a-b)^2 b^3 (b+x)^2}+\frac{-a+2 b}{2 (a-b)^3 b^4 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 (a+b)^3 d}-\frac{(a-2 b) \log (1+\sec (c+d x))}{2 (a-b)^3 d}-\frac{b^4 \left (5 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac{1}{4 (a+b)^2 d (1-\sec (c+d x))}+\frac{1}{4 (a-b)^2 d (1+\sec (c+d x))}+\frac{b^4}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [C]  time = 2.05403, size = 351, normalized size = 1.78 \[ \frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left (-\frac{16 i \left (-3 a^2 b^2+a^4-2 b^4\right ) (c+d x) (a \cos (c+d x)+b)}{(a-b)^3 (a+b)^3}+\frac{8 b^4 \left (b^2-5 a^2\right ) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a^2 \left (a^2-b^2\right )^3}-\frac{8 b^5}{a^2 (a-b)^2 (a+b)^2}+\frac{4 (a-2 b) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(b-a)^3}+\frac{8 i (a+2 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a+b)^3}+\frac{8 i (a-2 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a-b)^3}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}-\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}-\frac{4 (a+2 b) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a+b)^3}\right )}{8 d (a+b \sec (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]

[Out]

((b + a*Cos[c + d*x])*((-8*b^5)/(a^2*(a - b)^2*(a + b)^2) - ((16*I)*(a^4 - 3*a^2*b^2 - 2*b^4)*(c + d*x)*(b + a
*Cos[c + d*x]))/((a - b)^3*(a + b)^3) + ((8*I)*(a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a - b)^3
+ ((8*I)*(a + 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/
2]^2)/(a + b)^2 + (4*(a - 2*b)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]^2])/(-a + b)^3 + (8*b^4*(-5*a^2 + b^2
)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)^3) - (4*(a + 2*b)*(b + a*Cos[c + d*x])*Log[Si
n[(c + d*x)/2]^2])/(a + b)^3 - ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^2)*Sec[c + d*x]^2)/(8*d*(a +
b*Sec[c + d*x])^2)

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Maple [A]  time = 0.084, size = 226, normalized size = 1.2 \begin{align*} -{\frac{{b}^{5}}{d{a}^{2} \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-5\,{\frac{{b}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}+{\frac{{b}^{6}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}{a}^{2}}}-{\frac{1}{4\,d \left ( a-b \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) a}{2\,d \left ( a-b \right ) ^{3}}}+{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) b}{d \left ( a-b \right ) ^{3}}}+{\frac{1}{4\,d \left ( a+b \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{2\,d \left ( a+b \right ) ^{3}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{d \left ( a+b \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3/(a+b*sec(d*x+c))^2,x)

[Out]

-1/d*b^5/a^2/(a+b)^2/(a-b)^2/(b+a*cos(d*x+c))-5/d*b^4/(a+b)^3/(a-b)^3*ln(b+a*cos(d*x+c))+1/d*b^6/(a+b)^3/(a-b)
^3/a^2*ln(b+a*cos(d*x+c))-1/4/d/(a-b)^2/(cos(d*x+c)+1)-1/2/d/(a-b)^3*ln(cos(d*x+c)+1)*a+1/d/(a-b)^3*ln(cos(d*x
+c)+1)*b+1/4/d/(a+b)^2/(-1+cos(d*x+c))-1/2/d/(a+b)^3*ln(-1+cos(d*x+c))*a-1/d/(a+b)^3*ln(-1+cos(d*x+c))*b

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Maxima [A]  time = 1.0238, size = 409, normalized size = 2.08 \begin{align*} -\frac{\frac{2 \,{\left (5 \, a^{2} b^{4} - b^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} + \frac{{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{a^{4} b + a^{2} b^{3} + 2 \, b^{5} - 2 \,{\left (a^{4} b + b^{5}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} -{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/2*(2*(5*a^2*b^4 - b^6)*log(a*cos(d*x + c) + b)/(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6) + (a - 2*b)*log(cos(
d*x + c) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (a + 2*b)*log(cos(d*x + c) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3
) + (a^4*b + a^2*b^3 + 2*b^5 - 2*(a^4*b + b^5)*cos(d*x + c)^2 + (a^5 - a^3*b^2)*cos(d*x + c))/(a^6*b - 2*a^4*b
^3 + a^2*b^5 - (a^7 - 2*a^5*b^2 + a^3*b^4)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 + (a^
7 - 2*a^5*b^2 + a^3*b^4)*cos(d*x + c)))/d

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Fricas [B]  time = 1.51635, size = 1455, normalized size = 7.39 \begin{align*} \frac{a^{6} b + a^{2} b^{5} - 2 \, b^{7} - 2 \,{\left (a^{6} b - a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right ) + 2 \,{\left (5 \, a^{2} b^{5} - b^{7} -{\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} -{\left (5 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} +{\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) +{\left (a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} -{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} -{\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{3} +{\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d \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 )} d \cos \left (d x + c\right ) -{\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\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)**3/(a+b*sec(d*x+c))**2,x)

[Out]

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

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Giac [B]  time = 1.39264, size = 886, normalized size = 4.5 \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)^3/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-1/8*(4*(a + 2*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 8*(5*a^2
*b^4 - b^6)*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^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6) - (a^5 - a^4*b - a^3*b^2 + a^2*b^3 + 3*a^5*(cos(d*x + c) - 1)/(cos(
d*x + c) + 1) - 3*a^4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a^3*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1
) + 3*a^2*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20*a*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b^5*(
cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*a^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 4*a^4*b*(cos(d*x + c)
- 1)^2/(cos(d*x + c) + 1)^2 - 2*a^3*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 12*a^2*b^3*(cos(d*x + c)
- 1)^2/(cos(d*x + c) + 1)^2 + 4*a*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 4*b^5*(cos(d*x + c) - 1)^2/(
cos(d*x + c) + 1)^2)/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) -
1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - b*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2)) - (cos(d*x + c) - 1)/((a^2 - 2*a*b + b^2)*(cos(d*x + c) + 1)) - 8*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 1))/a^2)/d