3.27 \(\int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=160 \[ -\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{4 d} \]

[Out]

-a^5/(12*d*(a - a*Cos[c + d*x])^3) - (3*a^4)/(8*d*(a - a*Cos[c + d*x])^2) - (23*a^3)/(16*d*(a - a*Cos[c + d*x]
)) + a^3/(16*d*(a + a*Cos[c + d*x])) + (9*a^2*Log[1 - Cos[c + d*x]])/(4*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^
2*Log[1 + Cos[c + d*x]])/(4*d) + (a^2*Sec[c + d*x])/d

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Rubi [A]  time = 0.199207, antiderivative size = 160, 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} \[ -\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]

[Out]

-a^5/(12*d*(a - a*Cos[c + d*x])^3) - (3*a^4)/(8*d*(a - a*Cos[c + d*x])^2) - (23*a^3)/(16*d*(a - a*Cos[c + d*x]
)) + a^3/(16*d*(a + a*Cos[c + d*x])) + (9*a^2*Log[1 - Cos[c + d*x]])/(4*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^
2*Log[1 + Cos[c + d*x]])/(4*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 ^7(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^7(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a^7 \operatorname{Subst}\left (\int \frac{a^2}{(-a-x)^4 x^2 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \frac{1}{(-a-x)^4 x^2 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \left (\frac{1}{16 a^6 (a-x)^2}+\frac{1}{4 a^7 (a-x)}+\frac{1}{a^6 x^2}-\frac{2}{a^7 x}+\frac{1}{4 a^4 (a+x)^4}+\frac{3}{4 a^5 (a+x)^3}+\frac{23}{16 a^6 (a+x)^2}+\frac{9}{4 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a+a \cos (c+d x))}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (1+\cos (c+d x))}{4 d}+\frac{a^2 \sec (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 1.28363, size = 136, normalized size = 0.85 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (36 \csc ^4\left (\frac{1}{2} (c+d x)\right )+120 \csc ^2\left (\frac{1}{2} (c+d x)\right )+\csc ^6\left (\frac{1}{2} (c+d x)\right ) \left (16-3 \sec ^2\left (\frac{1}{2} (c+d x)\right ) (2 \sec (c+d x)+3)\right )+48 \left (-9 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 \log (\cos (c+d x))\right )\right )}{384 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(120*Csc[(c + d*x)/2]^2 + 36*Csc[(c + d*x)/2]^4 + 48*(Log[Cos[(c
+ d*x)/2]] + 4*Log[Cos[c + d*x]] - 9*Log[Sin[(c + d*x)/2]]) + Csc[(c + d*x)/2]^6*(16 - 3*Sec[(c + d*x)/2]^2*(
3 + 2*Sec[c + d*x]))))/(384*d)

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Maple [A]  time = 0.069, size = 121, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}\sec \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}}{16\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{{a}^{2}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{{a}^{2}}{12\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5\,{a}^{2}}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{39\,{a}^{2}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{9\,{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{4\,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))^2,x)

[Out]

a^2*sec(d*x+c)/d-1/16/d*a^2/(1+sec(d*x+c))-1/4/d*a^2*ln(1+sec(d*x+c))-1/12/d*a^2/(-1+sec(d*x+c))^3-5/8/d*a^2/(
-1+sec(d*x+c))^2-39/16/d*a^2/(-1+sec(d*x+c))+9/4/d*a^2*ln(-1+sec(d*x+c))

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Maxima [A]  time = 0.988657, size = 193, normalized size = 1.21 \begin{align*} -\frac{3 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 27 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 24 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{2 \,{\left (15 \, a^{2} \cos \left (d x + c\right )^{4} - 24 \, a^{2} \cos \left (d x + c\right )^{3} - 7 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) - 6 \, a^{2}\right )}}{\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )}}{12 \, 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))^2,x, algorithm="maxima")

[Out]

-1/12*(3*a^2*log(cos(d*x + c) + 1) - 27*a^2*log(cos(d*x + c) - 1) + 24*a^2*log(cos(d*x + c)) - 2*(15*a^2*cos(d
*x + c)^4 - 24*a^2*cos(d*x + c)^3 - 7*a^2*cos(d*x + c)^2 + 23*a^2*cos(d*x + c) - 6*a^2)/(cos(d*x + c)^5 - 2*co
s(d*x + c)^4 + 2*cos(d*x + c)^2 - cos(d*x + c)))/d

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Fricas [A]  time = 1.82971, size = 722, normalized size = 4.51 \begin{align*} \frac{30 \, a^{2} \cos \left (d x + c\right )^{4} - 48 \, a^{2} \cos \left (d x + c\right )^{3} - 14 \, a^{2} \cos \left (d x + c\right )^{2} + 46 \, a^{2} \cos \left (d x + c\right ) - 12 \, a^{2} - 24 \,{\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 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 ) - 3 \,{\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 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 ) + 27 \,{\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 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 )}{12 \,{\left (d \cos \left (d x + c\right )^{5} - 2 \, d \cos \left (d x + c\right )^{4} + 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)^7*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/12*(30*a^2*cos(d*x + c)^4 - 48*a^2*cos(d*x + c)^3 - 14*a^2*cos(d*x + c)^2 + 46*a^2*cos(d*x + c) - 12*a^2 - 2
4*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*log(-cos(d*x + c)) - 3
*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*log(1/2*cos(d*x + c) +
1/2) + 27*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 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)^5 - 2*d*cos(d*x + c)^4 + 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)**7*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.45157, size = 321, normalized size = 2.01 \begin{align*} \frac{216 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 192 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac{3 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{{\left (a^{2} - \frac{12 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{90 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{396 \, a^{2}{\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{192 \,{\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}}{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))^2,x, algorithm="giac")

[Out]

1/96*(216*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 192*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) - 1)) - 3*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + (a^2 - 12*a^2*(cos(d*x + c) - 1)/(cos(d*x + c
) + 1) + 90*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 396*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)
*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3 + 192*(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