3.26 \(\int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{8 d} \]

[Out]

-a^4/(4*d*(a - a*Cos[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Cos[c + d*x])) + (17*a^2*Log[1 - Cos[c + d*x]])/(8*d)
- (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(8*d) + (a^2*Sec[c + d*x])/d

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Rubi [A]  time = 0.171222, antiderivative size = 115, 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} \[ -\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]

[Out]

-a^4/(4*d*(a - a*Cos[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Cos[c + d*x])) + (17*a^2*Log[1 - Cos[c + d*x]])/(8*d)
- (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(8*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 ^5(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^5(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a^5 \operatorname{Subst}\left (\int \frac{a^2}{(-a-x)^3 x^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \frac{1}{(-a-x)^3 x^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \left (\frac{1}{8 a^5 (a-x)}+\frac{1}{a^4 x^2}-\frac{2}{a^5 x}+\frac{1}{2 a^3 (a+x)^3}+\frac{5}{4 a^4 (a+x)^2}+\frac{17}{8 a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (1+\cos (c+d x))}{8 d}+\frac{a^2 \sec (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 1.51333, size = 103, normalized size = 0.9 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (\csc ^4\left (\frac{1}{2} (c+d x)\right )+10 \csc ^2\left (\frac{1}{2} (c+d x)\right )+4 \left (-4 \sec (c+d x)-17 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 \log (\cos (c+d x))\right )\right )}{64 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(10*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 4*(Log[Cos[(c + d*
x)/2]] + 8*Log[Cos[c + d*x]] - 17*Log[Sin[(c + d*x)/2]] - 4*Sec[c + d*x])))/(64*d)

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Maple [A]  time = 0.072, size = 85, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}\sec \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{a}^{2}}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,{a}^{2}}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{17\,{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^5*(a+a*sec(d*x+c))^2,x)

[Out]

a^2*sec(d*x+c)/d-1/8/d*a^2*ln(1+sec(d*x+c))-1/4/d*a^2/(-1+sec(d*x+c))^2-7/4/d*a^2/(-1+sec(d*x+c))+17/8/d*a^2*l
n(-1+sec(d*x+c))

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Maxima [A]  time = 1.0235, size = 140, normalized size = 1.22 \begin{align*} -\frac{a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{2 \,{\left (9 \, a^{2} \cos \left (d x + c\right )^{2} - 14 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )}}{\cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/8*(a^2*log(cos(d*x + c) + 1) - 17*a^2*log(cos(d*x + c) - 1) + 16*a^2*log(cos(d*x + c)) - 2*(9*a^2*cos(d*x +
c)^2 - 14*a^2*cos(d*x + c) + 4*a^2)/(cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c)))/d

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Fricas [A]  time = 1.83125, size = 531, normalized size = 4.62 \begin{align*} \frac{18 \, a^{2} \cos \left (d x + c\right )^{2} - 28 \, a^{2} \cos \left (d x + c\right ) + 8 \, a^{2} - 16 \,{\left (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 ) -{\left (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 ) + 17 \,{\left (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 )}{8 \,{\left (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)^5*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/8*(18*a^2*cos(d*x + c)^2 - 28*a^2*cos(d*x + c) + 8*a^2 - 16*(a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2
*cos(d*x + c))*log(-cos(d*x + c)) - (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) + 17*(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)^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)**5*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.52379, size = 258, normalized size = 2.24 \begin{align*} \frac{34 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \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{51 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac{32 \,{\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}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/16*(34*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 32*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x +
c) + 1) - 1)) - (a^2 - 12*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 51*a^2*(cos(d*x + c) - 1)^2/(cos(d*x +
c) + 1)^2)*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2 + 32*(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