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]

-1/32*(16*a+5*b)*ln(1-cos(d*x+c))/d-1/32*(16*a-5*b)*ln(1+cos(d*x+c))/d-1/6*cot(d*x+c)^6*(a+b*sec(d*x+c))/d+1/2
4*cot(d*x+c)^4*(6*a+5*b*sec(d*x+c))/d-1/16*cot(d*x+c)^2*(8*a+5*b*sec(d*x+c))/d

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Rubi [A]  time = 0.18, antiderivative size = 130, normalized size of antiderivative = 1.00, 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 31

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

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 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 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]

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*}

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Mathematica [A]  time = 0.62, 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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fricas [A]  time = 0.53, size = 237, normalized size = 1.82 \[ \frac {66 \, b \cos \left (d x + c\right )^{5} + 144 \, a \cos \left (d x + c\right )^{4} - 80 \, b \cos \left (d x + c\right )^{3} - 216 \, a \cos \left (d x + c\right )^{2} + 30 \, b \cos \left (d x + c\right ) - 3 \, {\left ({\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a + 5 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a - 5 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

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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giac [B]  time = 0.57, size = 358, normalized size = 2.75 \[ -\frac {12 \, {\left (16 \, a + 5 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a + b + \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {87 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {352 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {110 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac {87 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \]

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

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maple [A]  time = 0.62, size = 185, normalized size = 1.42 \[ -\frac {a \left (\cot ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {b \left (\cos ^{7}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}+\frac {b \left (\cos ^{7}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}-\frac {b \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}-\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{16 d}-\frac {5 b \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}-\frac {5 b \cos \left (d x +c \right )}{16 d}-\frac {5 b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d} \]

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

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maxima [A]  time = 0.37, size = 133, normalized size = 1.02 \[ -\frac {3 \, {\left (16 \, a - 5 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, {\left (16 \, a + 5 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (33 \, b \cos \left (d x + c\right )^{5} + 72 \, a \cos \left (d x + c\right )^{4} - 40 \, b \cos \left (d x + c\right )^{3} - 108 \, a \cos \left (d x + c\right )^{2} + 15 \, b \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \]

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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mupad [B]  time = 1.50, size = 170, normalized size = 1.31 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{32}-\frac {3\,b}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\left (\frac {29\,a}{2}+\frac {15\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-\frac {3\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{6}+\frac {b}{6}\right )}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {29\,a}{128}-\frac {15\,b}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a}{384}-\frac {b}{384}\right )}{d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a+\frac {5\,b}{16}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)^4*(a/32 - (3*b)/128))/d - (cot(c/2 + (d*x)/2)^6*(a/6 + b/6 - tan(c/2 + (d*x)/2)^2*(2*a + (
3*b)/2) + tan(c/2 + (d*x)/2)^4*((29*a)/2 + (15*b)/2)))/(64*d) - (tan(c/2 + (d*x)/2)^2*((29*a)/128 - (15*b)/128
))/d - (tan(c/2 + (d*x)/2)^6*(a/384 - b/384))/d + (a*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (log(tan(c/2 + (d*x)/2
))*(a + (5*b)/16))/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{7}{\left (c + d x \right )}\, dx \]

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]

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

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