3.47 \(\int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=202 \[ -\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a \cos (c+d x)+a)}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (\cos (c+d x)+1)}{64 d} \]

[Out]

-a^7/(16*d*(a - a*Cos[c + d*x])^4) - a^6/(3*d*(a - a*Cos[c + d*x])^3) - (39*a^5)/(32*d*(a - a*Cos[c + d*x])^2)
- (75*a^4)/(16*d*(a - a*Cos[c + d*x])) - a^4/(32*d*(a + a*Cos[c + d*x])) + (501*a^3*Log[1 - Cos[c + d*x]])/(6
4*d) - (8*a^3*Log[Cos[c + d*x]])/d + (11*a^3*Log[1 + Cos[c + d*x]])/(64*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec
[c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.230595, antiderivative size = 202, 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} \[ -\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a \cos (c+d x)+a)}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (\cos (c+d x)+1)}{64 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

-a^7/(16*d*(a - a*Cos[c + d*x])^4) - a^6/(3*d*(a - a*Cos[c + d*x])^3) - (39*a^5)/(32*d*(a - a*Cos[c + d*x])^2)
- (75*a^4)/(16*d*(a - a*Cos[c + d*x])) - a^4/(32*d*(a + a*Cos[c + d*x])) + (501*a^3*Log[1 - Cos[c + d*x]])/(6
4*d) - (8*a^3*Log[Cos[c + d*x]])/d + (11*a^3*Log[1 + Cos[c + d*x]])/(64*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec
[c + d*x]^2)/(2*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))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^9(c+d x) \sec ^3(c+d x) \, dx\\ &=\frac{a^9 \operatorname{Subst}\left (\int \frac{a^3}{(-a-x)^5 x^3 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^{12} \operatorname{Subst}\left (\int \frac{1}{(-a-x)^5 x^3 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^{12} \operatorname{Subst}\left (\int \left (-\frac{1}{32 a^8 (a-x)^2}-\frac{11}{64 a^9 (a-x)}-\frac{1}{a^7 x^3}+\frac{3}{a^8 x^2}-\frac{8}{a^9 x}+\frac{1}{4 a^5 (a+x)^5}+\frac{1}{a^6 (a+x)^4}+\frac{39}{16 a^7 (a+x)^3}+\frac{75}{16 a^8 (a+x)^2}+\frac{501}{64 a^9 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a+a \cos (c+d x))}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (1+\cos (c+d x))}{64 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 1.1944, size = 159, normalized size = 0.79 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (3 \csc ^8\left (\frac{1}{2} (c+d x)\right )+32 \csc ^6\left (\frac{1}{2} (c+d x)\right )+234 \csc ^4\left (\frac{1}{2} (c+d x)\right )+1800 \csc ^2\left (\frac{1}{2} (c+d x)\right )-12 \left (-\sec ^2\left (\frac{1}{2} (c+d x)\right )+32 \sec ^2(c+d x)+192 \sec (c+d x)+1002 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+22 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-512 \log (\cos (c+d x))\right )\right )}{6144 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

-(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(1800*Csc[(c + d*x)/2]^2 + 234*Csc[(c + d*x)/2]^4 + 32*Csc[(c +
d*x)/2]^6 + 3*Csc[(c + d*x)/2]^8 - 12*(22*Log[Cos[(c + d*x)/2]] - 512*Log[Cos[c + d*x]] + 1002*Log[Sin[(c + d*
x)/2]] - Sec[(c + d*x)/2]^2 + 192*Sec[c + d*x] + 32*Sec[c + d*x]^2)))/(6144*d)

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Maple [A]  time = 0.088, size = 156, normalized size = 0.8 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}}{32\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{11\,{a}^{3}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{64\,d}}-{\frac{{a}^{3}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{4}}}-{\frac{7\,{a}^{3}}{12\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{83\,{a}^{3}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{67\,{a}^{3}}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{501\,{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{64\,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))^3,x)

[Out]

1/2*a^3*sec(d*x+c)^2/d+3*a^3*sec(d*x+c)/d+1/32/d*a^3/(1+sec(d*x+c))+11/64/d*a^3*ln(1+sec(d*x+c))-1/16/d*a^3/(-
1+sec(d*x+c))^4-7/12/d*a^3/(-1+sec(d*x+c))^3-83/32/d*a^3/(-1+sec(d*x+c))^2-67/8/d*a^3/(-1+sec(d*x+c))+501/64/d
*a^3*ln(-1+sec(d*x+c))

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Maxima [A]  time = 1.01392, size = 255, normalized size = 1.26 \begin{align*} \frac{33 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 1503 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 1536 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{2 \,{\left (735 \, a^{3} \cos \left (d x + c\right )^{6} - 1821 \, a^{3} \cos \left (d x + c\right )^{5} + 563 \, a^{3} \cos \left (d x + c\right )^{4} + 1695 \, a^{3} \cos \left (d x + c\right )^{3} - 1376 \, a^{3} \cos \left (d x + c\right )^{2} + 144 \, a^{3} \cos \left (d x + c\right ) + 48 \, a^{3}\right )}}{\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{192 \, 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))^3,x, algorithm="maxima")

[Out]

1/192*(33*a^3*log(cos(d*x + c) + 1) + 1503*a^3*log(cos(d*x + c) - 1) - 1536*a^3*log(cos(d*x + c)) + 2*(735*a^3
*cos(d*x + c)^6 - 1821*a^3*cos(d*x + c)^5 + 563*a^3*cos(d*x + c)^4 + 1695*a^3*cos(d*x + c)^3 - 1376*a^3*cos(d*
x + c)^2 + 144*a^3*cos(d*x + c) + 48*a^3)/(cos(d*x + c)^7 - 3*cos(d*x + c)^6 + 2*cos(d*x + c)^5 + 2*cos(d*x +
c)^4 - 3*cos(d*x + c)^3 + cos(d*x + c)^2))/d

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Fricas [B]  time = 1.85144, size = 1064, normalized size = 5.27 \begin{align*} \frac{1470 \, a^{3} \cos \left (d x + c\right )^{6} - 3642 \, a^{3} \cos \left (d x + c\right )^{5} + 1126 \, a^{3} \cos \left (d x + c\right )^{4} + 3390 \, a^{3} \cos \left (d x + c\right )^{3} - 2752 \, a^{3} \cos \left (d x + c\right )^{2} + 288 \, a^{3} \cos \left (d x + c\right ) + 96 \, a^{3} - 1536 \,{\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \,{\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1503 \,{\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{192 \,{\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\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))^3,x, algorithm="fricas")

[Out]

1/192*(1470*a^3*cos(d*x + c)^6 - 3642*a^3*cos(d*x + c)^5 + 1126*a^3*cos(d*x + c)^4 + 3390*a^3*cos(d*x + c)^3 -
2752*a^3*cos(d*x + c)^2 + 288*a^3*cos(d*x + c) + 96*a^3 - 1536*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2
*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(-cos(d*x + c)) + 3
3*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c
)^3 + a^3*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 1503*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a^
3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1
/2))/(d*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + 2*d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^4 - 3*d*cos(d*x + c)^3 + d
*cos(d*x + c)^2)

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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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37384, size = 394, normalized size = 1.95 \begin{align*} \frac{6012 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 6144 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{12 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (3 \, a^{3} - \frac{44 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{348 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{2376 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{12525 \, a^{3}{\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{1536 \,{\left (9 \, a^{3} + \frac{14 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{6 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{768 \, 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))^3,x, algorithm="giac")

[Out]

1/768*(6012*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 6144*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(
d*x + c) + 1) - 1)) + 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (3*a^3 - 44*a^3*(cos(d*x + c) - 1)/(cos(d
*x + c) + 1) + 348*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 2376*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c)
+ 1)^3 + 12525*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)*(cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4 + 153
6*(9*a^3 + 14*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((c
os(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2)/d