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

Optimal. Leaf size=163 \[ -\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^3}{32 d (a \cos (c+d x)+a)^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{3 a^2}{16 d (a \cos (c+d x)+a)}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (\cos (c+d x)+1)}{32 d} \]

[Out]

-a^4/(24*d*(a - a*Cos[c + d*x])^3) - (5*a^3)/(32*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) -
a^3/(32*d*(a + a*Cos[c + d*x])^2) - (3*a^2)/(16*d*(a + a*Cos[c + d*x])) + (21*a*Log[1 - Cos[c + d*x]])/(32*d)
- (a*Log[Cos[c + d*x]])/d + (11*a*Log[1 + Cos[c + d*x]])/(32*d)

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Rubi [A]  time = 0.149683, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2836, 12, 88} \[ -\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^3}{32 d (a \cos (c+d x)+a)^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{3 a^2}{16 d (a \cos (c+d x)+a)}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (\cos (c+d x)+1)}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^4/(24*d*(a - a*Cos[c + d*x])^3) - (5*a^3)/(32*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) -
a^3/(32*d*(a + a*Cos[c + d*x])^2) - (3*a^2)/(16*d*(a + a*Cos[c + d*x])) + (21*a*Log[1 - Cos[c + d*x]])/(32*d)
- (a*Log[Cos[c + d*x]])/d + (11*a*Log[1 + Cos[c + d*x]])/(32*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]))

Rubi steps

\begin{align*} \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=\frac{a^7 \operatorname{Subst}\left (\int \frac{a}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \frac{1}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \left (-\frac{1}{16 a^5 (a-x)^3}-\frac{3}{16 a^6 (a-x)^2}-\frac{11}{32 a^7 (a-x)}-\frac{1}{a^7 x}+\frac{1}{8 a^4 (a+x)^4}+\frac{5}{16 a^5 (a+x)^3}+\frac{1}{2 a^6 (a+x)^2}+\frac{21}{32 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^3}{32 d (a+a \cos (c+d x))^2}-\frac{3 a^2}{16 d (a+a \cos (c+d x))}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (1+\cos (c+d x))}{32 d}\\ \end{align*}

Mathematica [A]  time = 0.398065, size = 165, normalized size = 1.01 \[ -\frac{a \left (\csc ^6\left (\frac{1}{2} (c+d x)\right )+6 \csc ^4\left (\frac{1}{2} (c+d x)\right )+30 \csc ^2\left (\frac{1}{2} (c+d x)\right )+64 \csc ^6(c+d x)+96 \csc ^4(c+d x)+192 \csc ^2(c+d x)-\sec ^6\left (\frac{1}{2} (c+d x)\right )-6 \sec ^4\left (\frac{1}{2} (c+d x)\right )-30 \sec ^2\left (\frac{1}{2} (c+d x)\right )-120 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-384 \log (\sin (c+d x))+120 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+384 \log (\cos (c+d x))\right )}{384 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(30*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6 + 192*Csc[c + d*x]^2 + 96*Csc[c + d*x]^
4 + 64*Csc[c + d*x]^6 + 120*Log[Cos[(c + d*x)/2]] + 384*Log[Cos[c + d*x]] - 120*Log[Sin[(c + d*x)/2]] - 384*Lo
g[Sin[c + d*x]] - 30*Sec[(c + d*x)/2]^2 - 6*Sec[(c + d*x)/2]^4 - Sec[(c + d*x)/2]^6))/(384*d)

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Maple [A]  time = 0.083, size = 112, normalized size = 0.7 \begin{align*} -{\frac{a}{32\,d \left ( 1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a}{4\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{11\,a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{32\,d}}-{\frac{a}{24\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{9\,a}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{15\,a}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{21\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{32\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04834, size = 184, normalized size = 1.13 \begin{align*} \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 ) - 96 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac{2 \,{\left (15 \, a \cos \left (d x + c\right )^{4} + 9 \, a \cos \left (d x + c\right )^{3} - 49 \, a \cos \left (d x + c\right )^{2} - 11 \, 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(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) - 96*a*log(cos(d*x + c)) + 2*(15*a*cos(d*x + c)^
4 + 9*a*cos(d*x + c)^3 - 49*a*cos(d*x + c)^2 - 11*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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Fricas [B]  time = 1.85606, size = 802, normalized size = 4.92 \begin{align*} \frac{30 \, a \cos \left (d x + c\right )^{4} + 18 \, a \cos \left (d x + c\right )^{3} - 98 \, a \cos \left (d x + c\right )^{2} - 22 \, a \cos \left (d x + c\right ) - 96 \,{\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 (-\cos \left (d x + c\right )\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(30*a*cos(d*x + c)^4 + 18*a*cos(d*x + c)^3 - 98*a*cos(d*x + c)^2 - 22*a*cos(d*x + c) - 96*(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(-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*c
os(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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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(csc(d*x+c)**7*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.67476, size = 265, normalized size = 1.63 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(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*(cos
(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/d