3.28 \(\int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=205 \[ -\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac{a^4}{64 d (a \cos (c+d x)+a)^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{9 a^3}{64 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (\cos (c+d x)+1)}{128 d} \]

[Out]

-a^6/(32*d*(a - a*Cos[c + d*x])^4) - (7*a^5)/(48*d*(a - a*Cos[c + d*x])^3) - (15*a^4)/(32*d*(a - a*Cos[c + d*x
])^2) - (51*a^3)/(32*d*(a - a*Cos[c + d*x])) + a^4/(64*d*(a + a*Cos[c + d*x])^2) + (9*a^3)/(64*d*(a + a*Cos[c
+ d*x])) + (303*a^2*Log[1 - Cos[c + d*x]])/(128*d) - (2*a^2*Log[Cos[c + d*x]])/d - (47*a^2*Log[1 + Cos[c + d*x
]])/(128*d) + (a^2*Sec[c + d*x])/d

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Rubi [A]  time = 0.238155, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2836, 12, 88} \[ -\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac{a^4}{64 d (a \cos (c+d x)+a)^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{9 a^3}{64 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (\cos (c+d x)+1)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^2,x]

[Out]

-a^6/(32*d*(a - a*Cos[c + d*x])^4) - (7*a^5)/(48*d*(a - a*Cos[c + d*x])^3) - (15*a^4)/(32*d*(a - a*Cos[c + d*x
])^2) - (51*a^3)/(32*d*(a - a*Cos[c + d*x])) + a^4/(64*d*(a + a*Cos[c + d*x])^2) + (9*a^3)/(64*d*(a + a*Cos[c
+ d*x])) + (303*a^2*Log[1 - Cos[c + d*x]])/(128*d) - (2*a^2*Log[Cos[c + d*x]])/d - (47*a^2*Log[1 + Cos[c + d*x
]])/(128*d) + (a^2*Sec[c + d*x])/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 ^9(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^9(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a^9 \operatorname{Subst}\left (\int \frac{a^2}{(-a-x)^5 x^2 (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \frac{1}{(-a-x)^5 x^2 (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \left (\frac{1}{32 a^7 (a-x)^3}+\frac{9}{64 a^8 (a-x)^2}+\frac{47}{128 a^9 (a-x)}+\frac{1}{a^8 x^2}-\frac{2}{a^9 x}+\frac{1}{8 a^5 (a+x)^5}+\frac{7}{16 a^6 (a+x)^4}+\frac{15}{16 a^7 (a+x)^3}+\frac{51}{32 a^8 (a+x)^2}+\frac{303}{128 a^9 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{a^4}{64 d (a+a \cos (c+d x))^2}+\frac{9 a^3}{64 d (a+a \cos (c+d x))}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (1+\cos (c+d x))}{128 d}+\frac{a^2 \sec (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 3.25396, size = 164, normalized size = 0.8 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (3 \csc ^8\left (\frac{1}{2} (c+d x)\right )+28 \csc ^6\left (\frac{1}{2} (c+d x)\right )+180 \csc ^4\left (\frac{1}{2} (c+d x)\right )+1224 \csc ^2\left (\frac{1}{2} (c+d x)\right )-6 \left (\sec ^4\left (\frac{1}{2} (c+d x)\right )+18 \sec ^2\left (\frac{1}{2} (c+d x)\right )+4 \left (64 \sec (c+d x)+303 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-47 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-128 \log (\cos (c+d x))\right )\right )\right )}{6144 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(1224*Csc[(c + d*x)/2]^2 + 180*Csc[(c + d*x)/2]^4 + 28*Csc[(c +
d*x)/2]^6 + 3*Csc[(c + d*x)/2]^8 - 6*(18*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 4*(-47*Log[Cos[(c + d*x)/2]
] - 128*Log[Cos[c + d*x]] + 303*Log[Sin[(c + d*x)/2]] + 64*Sec[c + d*x]))))/(6144*d)

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Maple [A]  time = 0.083, size = 157, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}\sec \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}}{64\,d \left ( 1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{2}}{64\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{47\,{a}^{2}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{{a}^{2}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{4}}}-{\frac{13\,{a}^{2}}{48\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{35\,{a}^{2}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{99\,{a}^{2}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{303\,{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{128\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^9*(a+a*sec(d*x+c))^2,x)

[Out]

a^2*sec(d*x+c)/d+1/64/d*a^2/(1+sec(d*x+c))^2-11/64/d*a^2/(1+sec(d*x+c))-47/128/d*a^2*ln(1+sec(d*x+c))-1/32/d*a
^2/(-1+sec(d*x+c))^4-13/48/d*a^2/(-1+sec(d*x+c))^3-35/32/d*a^2/(-1+sec(d*x+c))^2-99/32/d*a^2/(-1+sec(d*x+c))+3
03/128/d*a^2*ln(-1+sec(d*x+c))

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Maxima [A]  time = 1.01514, size = 266, normalized size = 1.3 \begin{align*} -\frac{141 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 909 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 768 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{2 \,{\left (525 \, a^{2} \cos \left (d x + c\right )^{6} - 858 \, a^{2} \cos \left (d x + c\right )^{5} - 734 \, a^{2} \cos \left (d x + c\right )^{4} + 1654 \, a^{2} \cos \left (d x + c\right )^{3} - 19 \, a^{2} \cos \left (d x + c\right )^{2} - 784 \, a^{2} \cos \left (d x + c\right ) + 192 \, a^{2}\right )}}{\cos \left (d x + c\right )^{7} - 2 \, \cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/384*(141*a^2*log(cos(d*x + c) + 1) - 909*a^2*log(cos(d*x + c) - 1) + 768*a^2*log(cos(d*x + c)) - 2*(525*a^2
*cos(d*x + c)^6 - 858*a^2*cos(d*x + c)^5 - 734*a^2*cos(d*x + c)^4 + 1654*a^2*cos(d*x + c)^3 - 19*a^2*cos(d*x +
c)^2 - 784*a^2*cos(d*x + c) + 192*a^2)/(cos(d*x + c)^7 - 2*cos(d*x + c)^6 - cos(d*x + c)^5 + 4*cos(d*x + c)^4
- cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c)))/d

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Fricas [B]  time = 1.92133, size = 1152, normalized size = 5.62 \begin{align*} \frac{1050 \, a^{2} \cos \left (d x + c\right )^{6} - 1716 \, a^{2} \cos \left (d x + c\right )^{5} - 1468 \, a^{2} \cos \left (d x + c\right )^{4} + 3308 \, a^{2} \cos \left (d x + c\right )^{3} - 38 \, a^{2} \cos \left (d x + c\right )^{2} - 1568 \, a^{2} \cos \left (d x + c\right ) + 384 \, a^{2} - 768 \,{\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - 141 \,{\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 909 \,{\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{384 \,{\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{5} + 4 \, d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/384*(1050*a^2*cos(d*x + c)^6 - 1716*a^2*cos(d*x + c)^5 - 1468*a^2*cos(d*x + c)^4 + 3308*a^2*cos(d*x + c)^3 -
38*a^2*cos(d*x + c)^2 - 1568*a^2*cos(d*x + c) + 384*a^2 - 768*(a^2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^
2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(-c
os(d*x + c)) - 141*(a^2*cos(d*x + c)^7 - 2*a^2*cos(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^
2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 909*(a^2*cos(d*x + c
)^7 - 2*a^2*cos(d*x + c)^6 - a^2*cos(d*x + c)^5 + 4*a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x +
c)^2 + a^2*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^7 - 2*d*cos(d*x + c)^6 - d*cos(d*x + c)
^5 + 4*d*cos(d*x + c)^4 - d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c))

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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)**9*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.48165, size = 393, normalized size = 1.92 \begin{align*} \frac{3636 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 3072 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac{120 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{6 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{{\left (3 \, a^{2} - \frac{40 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{282 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1680 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{7575 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}} + \frac{3072 \,{\left (2 \, a^{2} + \frac{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{1536 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/1536*(3636*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 3072*a^2*log(abs(-(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) - 1)) - 120*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c
) + 1)^2 - (3*a^2 - 40*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 282*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c)
+ 1)^2 - 1680*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 7575*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)
^4)*(cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4 + 3072*(2*a^2 + a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos
(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/d