3.263 \(\int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=130 \[ -\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \]

[Out]

-((16*a + 5*b)*Log[1 - Cos[c + d*x]])/(32*d) - ((16*a - 5*b)*Log[1 + Cos[c + d*x]])/(32*d) - (Cot[c + d*x]^6*(
a + b*Sec[c + d*x]))/(6*d) + (Cot[c + d*x]^4*(6*a + 5*b*Sec[c + d*x]))/(24*d) - (Cot[c + d*x]^2*(8*a + 5*b*Sec
[c + d*x]))/(16*d)

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Rubi [A]  time = 0.175971, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3882, 3883, 2668, 633, 31} \[ -\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]

[Out]

-((16*a + 5*b)*Log[1 - Cos[c + d*x]])/(32*d) - ((16*a - 5*b)*Log[1 + Cos[c + d*x]])/(32*d) - (Cot[c + d*x]^6*(
a + b*Sec[c + d*x]))/(6*d) + (Cot[c + d*x]^4*(6*a + 5*b*Sec[c + d*x]))/(24*d) - (Cot[c + d*x]^2*(8*a + 5*b*Sec
[c + d*x]))/(16*d)

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 3883

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))/cot[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[(b + a*Sin[c + d*x])/Cos[
c + d*x], x] /; FreeQ[{a, b, c, d}, x]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{1}{6} \int \cot ^5(c+d x) (-6 a-5 b \sec (c+d x)) \, dx\\ &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}+\frac{1}{24} \int \cot ^3(c+d x) (24 a+15 b \sec (c+d x)) \, dx\\ &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac{1}{48} \int \cot (c+d x) (-48 a-15 b \sec (c+d x)) \, dx\\ &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac{1}{48} \int (-15 b-48 a \cos (c+d x)) \csc (c+d x) \, dx\\ &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac{a \operatorname{Subst}\left (\int \frac{-15 b+x}{2304 a^2-x^2} \, dx,x,-48 a \cos (c+d x)\right )}{d}\\ &=-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac{(16 a-5 b) \operatorname{Subst}\left (\int \frac{1}{48 a-x} \, dx,x,-48 a \cos (c+d x)\right )}{32 d}+\frac{(16 a+5 b) \operatorname{Subst}\left (\int \frac{1}{-48 a-x} \, dx,x,-48 a \cos (c+d x)\right )}{32 d}\\ &=-\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (1+\cos (c+d x))}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.580025, size = 216, normalized size = 1.66 \[ -\frac{a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d}-\frac{b \csc ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{b \csc ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{11 b \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{b \sec ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{b \sec ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{11 b \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}+\frac{5 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]

[Out]

(-11*b*Csc[(c + d*x)/2]^2)/(64*d) + (b*Csc[(c + d*x)/2]^4)/(32*d) - (b*Csc[(c + d*x)/2]^6)/(384*d) + (5*b*Log[
Cos[(c + d*x)/2]])/(16*d) - (5*b*Log[Sin[(c + d*x)/2]])/(16*d) - (a*(6*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*C
ot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d) + (11*b*Sec[(c + d*x)/2]^2)/(64*d) - (b*S
ec[(c + d*x)/2]^4)/(32*d) + (b*Sec[(c + d*x)/2]^6)/(384*d)

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Maple [A]  time = 0.046, size = 185, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{2}a}{2\,d}}-{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}-{\frac{5\,b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}-{\frac{5\,b\cos \left ( dx+c \right ) }{16\,d}}-{\frac{5\,b\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7*(a+b*sec(d*x+c)),x)

[Out]

-1/6/d*a*cot(d*x+c)^6+1/4/d*a*cot(d*x+c)^4-1/2/d*a*cot(d*x+c)^2-1/d*a*ln(sin(d*x+c))-1/6/d*b/sin(d*x+c)^6*cos(
d*x+c)^7+1/24/d*b/sin(d*x+c)^4*cos(d*x+c)^7-1/16/d*b/sin(d*x+c)^2*cos(d*x+c)^7-1/16/d*b*cos(d*x+c)^5-5/48/d*b*
cos(d*x+c)^3-5/16/d*b*cos(d*x+c)-5/16/d*b*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 0.994773, size = 180, normalized size = 1.38 \begin{align*} -\frac{3 \,{\left (16 \, a - 5 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \,{\left (16 \, a + 5 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (33 \, b \cos \left (d x + c\right )^{5} + 72 \, a \cos \left (d x + c\right )^{4} - 40 \, b \cos \left (d x + c\right )^{3} - 108 \, a \cos \left (d x + c\right )^{2} + 15 \, b \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/96*(3*(16*a - 5*b)*log(cos(d*x + c) + 1) + 3*(16*a + 5*b)*log(cos(d*x + c) - 1) - 2*(33*b*cos(d*x + c)^5 +
72*a*cos(d*x + c)^4 - 40*b*cos(d*x + c)^3 - 108*a*cos(d*x + c)^2 + 15*b*cos(d*x + c) + 44*a)/(cos(d*x + c)^6 -
3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1))/d

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Fricas [A]  time = 0.823228, size = 630, normalized size = 4.85 \begin{align*} \frac{66 \, b \cos \left (d x + c\right )^{5} + 144 \, a \cos \left (d x + c\right )^{4} - 80 \, b \cos \left (d x + c\right )^{3} - 216 \, a \cos \left (d x + c\right )^{2} + 30 \, b \cos \left (d x + c\right ) - 3 \,{\left ({\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a + 5 \, b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \,{\left ({\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a - 5 \, b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 88 \, a}{96 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/96*(66*b*cos(d*x + c)^5 + 144*a*cos(d*x + c)^4 - 80*b*cos(d*x + c)^3 - 216*a*cos(d*x + c)^2 + 30*b*cos(d*x +
c) - 3*((16*a - 5*b)*cos(d*x + c)^6 - 3*(16*a - 5*b)*cos(d*x + c)^4 + 3*(16*a - 5*b)*cos(d*x + c)^2 - 16*a +
5*b)*log(1/2*cos(d*x + c) + 1/2) - 3*((16*a + 5*b)*cos(d*x + c)^6 - 3*(16*a + 5*b)*cos(d*x + c)^4 + 3*(16*a +
5*b)*cos(d*x + c)^2 - 16*a - 5*b)*log(-1/2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4
+ 3*d*cos(d*x + c)^2 - d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7*(a+b*sec(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.33109, size = 483, normalized size = 3.72 \begin{align*} -\frac{12 \,{\left (16 \, a + 5 \, b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a + b + \frac{12 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{9 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{87 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{45 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{352 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{110 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac{87 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{45 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{12 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{9 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/384*(12*(16*a + 5*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 384*a*log(abs(-(cos(d*x + c) - 1)/
(cos(d*x + c) + 1) + 1)) - (a + b + 12*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*b*(cos(d*x + c) - 1)/(cos(d
*x + c) + 1) + 87*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 45*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
+ 352*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 110*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x
+ c) + 1)^3/(cos(d*x + c) - 1)^3 - 87*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 45*b*(cos(d*x + c) - 1)/(cos(
d*x + c) + 1) - 12*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 9*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
- a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/d