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

Optimal. Leaf size=118 \[ -\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a \cos (c+d x)+a)}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]

[Out]

-a^3/(8*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^2/(8*d*(a + a*Cos[c + d*x])) + (11*a*Lo
g[1 - Cos[c + d*x]])/(16*d) - (a*Log[Cos[c + d*x]])/d + (5*a*Log[1 + Cos[c + d*x]])/(16*d)

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Rubi [A]  time = 0.120413, antiderivative size = 118, 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} \[ -\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a \cos (c+d x)+a)}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^3/(8*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^2/(8*d*(a + a*Cos[c + d*x])) + (11*a*Lo
g[1 - Cos[c + d*x]])/(16*d) - (a*Log[Cos[c + d*x]])/d + (5*a*Log[1 + Cos[c + d*x]])/(16*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)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^5(c+d x) \sec (c+d x) \, dx\\ &=\frac{a^5 \operatorname{Subst}\left (\int \frac{a}{(-a-x)^3 x (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \frac{1}{(-a-x)^3 x (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{8 a^4 (a-x)^2}-\frac{5}{16 a^5 (a-x)}-\frac{1}{a^5 x}+\frac{1}{4 a^3 (a+x)^3}+\frac{1}{2 a^4 (a+x)^2}+\frac{11}{16 a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a+a \cos (c+d x))}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (1+\cos (c+d x))}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.336741, size = 164, normalized size = 1.39 \[ -\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{3 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{a \left (\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Lo
g[Sin[(c + d*x)/2]])/(8*d) - (a*(2*Csc[c + d*x]^2 + Csc[c + d*x]^4 + 4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]]
))/(4*d) + (3*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)

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Maple [A]  time = 0.068, size = 80, normalized size = 0.7 \begin{align*}{\frac{a}{8\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{5\,a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{a}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{11\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{16\,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)),x)

[Out]

1/8/d*a/(1+sec(d*x+c))+5/16/d*a*ln(1+sec(d*x+c))-1/8/d*a/(-1+sec(d*x+c))^2-3/4/d*a/(-1+sec(d*x+c))+11/16/d*a*l
n(-1+sec(d*x+c))

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Maxima [A]  time = 1.07625, size = 128, normalized size = 1.08 \begin{align*} \frac{5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 16 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac{2 \,{\left (3 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 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)),x, algorithm="maxima")

[Out]

1/16*(5*a*log(cos(d*x + c) + 1) + 11*a*log(cos(d*x + c) - 1) - 16*a*log(cos(d*x + c)) + 2*(3*a*cos(d*x + c)^2
+ a*cos(d*x + c) - 6*a)/(cos(d*x + c)^3 - cos(d*x + c)^2 - cos(d*x + c) + 1))/d

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Fricas [A]  time = 1.76525, size = 512, normalized size = 4.34 \begin{align*} \frac{6 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - 16 \,{\left (a \cos \left (d x + c\right )^{3} - 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 ) + 5 \,{\left (a \cos \left (d x + c\right )^{3} - 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 ) + 11 \,{\left (a \cos \left (d x + c\right )^{3} - 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 ) - 12 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{3} - 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)^5*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/16*(6*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - 16*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*lo
g(-cos(d*x + c)) + 5*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(1/2*cos(d*x + c) + 1/2) +
11*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(-1/2*cos(d*x + c) + 1/2) - 12*a)/(d*cos(d*x
+ c)^3 - 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)**5*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.46771, size = 201, normalized size = 1.7 \begin{align*} \frac{22 \, a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac{{\left (a - \frac{10 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{33 \, a{\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{2 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, 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)),x, algorithm="giac")

[Out]

1/32*(22*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 32*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - 1)) - (a - 10*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 33*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)
*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2 + 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/d