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

Optimal. Leaf size=133 \[ -\frac {15 a}{16 d (1-\cos (c+d x))}-\frac {a}{4 d (\cos (c+d x)+1)}+\frac {9 a}{32 d (1-\cos (c+d x))^2}+\frac {a}{32 d (\cos (c+d x)+1)^2}-\frac {a}{24 d (1-\cos (c+d x))^3}-\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {11 a \log (\cos (c+d x)+1)}{32 d} \]

[Out]

-1/24*a/d/(1-cos(d*x+c))^3+9/32*a/d/(1-cos(d*x+c))^2-15/16*a/d/(1-cos(d*x+c))+1/32*a/d/(1+cos(d*x+c))^2-1/4*a/
d/(1+cos(d*x+c))-21/32*a*ln(1-cos(d*x+c))/d-11/32*a*ln(1+cos(d*x+c))/d

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Rubi [A]  time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3879, 88} \[ -\frac {15 a}{16 d (1-\cos (c+d x))}-\frac {a}{4 d (\cos (c+d x)+1)}+\frac {9 a}{32 d (1-\cos (c+d x))^2}+\frac {a}{32 d (\cos (c+d x)+1)^2}-\frac {a}{24 d (1-\cos (c+d x))^3}-\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {11 a \log (\cos (c+d x)+1)}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(24*d*(1 - Cos[c + d*x])^3) + (9*a)/(32*d*(1 - Cos[c + d*x])^2) - (15*a)/(16*d*(1 - Cos[c + d*x])) + a/(32*
d*(1 + Cos[c + d*x])^2) - a/(4*d*(1 + Cos[c + d*x])) - (21*a*Log[1 - Cos[c + d*x]])/(32*d) - (11*a*Log[1 + Cos
[c + d*x]])/(32*d)

Rule 88

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 3879

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*} \int \cot ^7(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {a^8 \operatorname {Subst}\left (\int \frac {x^6}{(a-a x)^4 (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^8 \operatorname {Subst}\left (\int \left (\frac {1}{8 a^7 (-1+x)^4}+\frac {9}{16 a^7 (-1+x)^3}+\frac {15}{16 a^7 (-1+x)^2}+\frac {21}{32 a^7 (-1+x)}+\frac {1}{16 a^7 (1+x)^3}-\frac {1}{4 a^7 (1+x)^2}+\frac {11}{32 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a}{24 d (1-\cos (c+d x))^3}+\frac {9 a}{32 d (1-\cos (c+d x))^2}-\frac {15 a}{16 d (1-\cos (c+d x))}+\frac {a}{32 d (1+\cos (c+d x))^2}-\frac {a}{4 d (1+\cos (c+d x))}-\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {11 a \log (1+\cos (c+d x))}{32 d}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 165, normalized size = 1.24 \[ -\frac {a \left (64 \cot ^6(c+d x)-96 \cot ^4(c+d x)+192 \cot ^2(c+d x)+\csc ^6\left (\frac {1}{2} (c+d x)\right )-12 \csc ^4\left (\frac {1}{2} (c+d x)\right )+66 \csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^6\left (\frac {1}{2} (c+d x)\right )+12 \sec ^4\left (\frac {1}{2} (c+d x)\right )-66 \sec ^2\left (\frac {1}{2} (c+d x)\right )+120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+384 \log (\tan (c+d x))-120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+384 \log (\cos (c+d x))\right )}{384 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/384*(a*(192*Cot[c + d*x]^2 - 96*Cot[c + d*x]^4 + 64*Cot[c + d*x]^6 + 66*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*
x)/2]^4 + Csc[(c + d*x)/2]^6 - 120*Log[Cos[(c + d*x)/2]] + 384*Log[Cos[c + d*x]] + 120*Log[Sin[(c + d*x)/2]] +
384*Log[Tan[c + d*x]] - 66*Sec[(c + d*x)/2]^2 + 12*Sec[(c + d*x)/2]^4 - Sec[(c + d*x)/2]^6))/d

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fricas [B]  time = 0.62, size = 241, normalized size = 1.81 \[ \frac {66 \, a \cos \left (d x + c\right )^{4} + 78 \, a \cos \left (d x + c\right )^{3} - 158 \, a \cos \left (d x + c\right )^{2} - 58 \, a \cos \left (d x + c\right ) - 33 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 63 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(66*a*cos(d*x + c)^4 + 78*a*cos(d*x + c)^3 - 158*a*cos(d*x + c)^2 - 58*a*cos(d*x + c) - 33*(a*cos(d*x + c
)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(1/2*cos(d*x + c) +
1/2) - 63*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)
*log(-1/2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x
+ c)^2 + d*cos(d*x + c) - d)

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giac [A]  time = 0.42, size = 197, normalized size = 1.48 \[ -\frac {252 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (2 \, a + \frac {21 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {462 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac {42 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(252*a*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)) - (2*a + 21*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 132*a*(cos(d*x + c) - 1)^2/(cos(d*x + c)
+ 1)^2 + 462*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3 - 42*a*(co
s(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/d

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maple [A]  time = 0.61, size = 124, normalized size = 0.93 \[ \frac {a \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a}{24 d \left (-1+\sec \left (d x +c \right )\right )^{3}}+\frac {5 a}{32 d \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {a}{2 d \left (-1+\sec \left (d x +c \right )\right )}-\frac {21 a \ln \left (-1+\sec \left (d x +c \right )\right )}{32 d}+\frac {a}{32 d \left (1+\sec \left (d x +c \right )\right )^{2}}+\frac {3 a}{16 d \left (1+\sec \left (d x +c \right )\right )}-\frac {11 a \ln \left (1+\sec \left (d x +c \right )\right )}{32 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/d*ln(sec(d*x+c))-1/24*a/d/(-1+sec(d*x+c))^3+5/32*a/d/(-1+sec(d*x+c))^2-1/2*a/d/(-1+sec(d*x+c))-21/32*a/d*ln(
-1+sec(d*x+c))+1/32*a/d/(1+sec(d*x+c))^2+3/16*a/d/(1+sec(d*x+c))-11/32*a/d*ln(1+sec(d*x+c))

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maxima [A]  time = 0.57, size = 126, normalized size = 0.95 \[ -\frac {33 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 63 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (33 \, a \cos \left (d x + c\right )^{4} + 39 \, a \cos \left (d x + c\right )^{3} - 79 \, a \cos \left (d x + c\right )^{2} - 29 \, a \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1}}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(33*a*log(cos(d*x + c) + 1) + 63*a*log(cos(d*x + c) - 1) - 2*(33*a*cos(d*x + c)^4 + 39*a*cos(d*x + c)^3
- 79*a*cos(d*x + c)^2 - 29*a*cos(d*x + c) + 44*a)/(cos(d*x + c)^5 - cos(d*x + c)^4 - 2*cos(d*x + c)^3 + 2*cos(
d*x + c)^2 + cos(d*x + c) - 1))/d

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mupad [B]  time = 1.23, size = 118, normalized size = 0.89 \[ \frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {a}{6}\right )}{32\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {21\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (cot(c/2 + (d*x)/2)^6*(a/6 - (7*a*tan(c/2 + (d*x)/2)^2)/4 + 11*a*tan(c/2
+ (d*x)/2)^4))/(32*d) - (7*a*tan(c/2 + (d*x)/2)^2)/(64*d) + (a*tan(c/2 + (d*x)/2)^4)/(128*d) - (21*a*log(tan(
c/2 + (d*x)/2)))/(16*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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