3.82 \(\int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{a^2}{32 d (a \cos (c+d x)+a)^4}-\frac{1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac{1}{32 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{a}{48 d (a \cos (c+d x)+a)^3}-\frac{1}{64 d (a-a \cos (c+d x))^2}-\frac{1}{32 d (a \cos (c+d x)+a)^2} \]

[Out]

ArcTanh[Cos[c + d*x]]/(64*a^2*d) - 1/(64*d*(a - a*Cos[c + d*x])^2) + a^2/(32*d*(a + a*Cos[c + d*x])^4) - a/(48
*d*(a + a*Cos[c + d*x])^3) - 1/(32*d*(a + a*Cos[c + d*x])^2) - 1/(64*d*(a^2 - a^2*Cos[c + d*x])) - 1/(32*d*(a^
2 + a^2*Cos[c + d*x]))

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Rubi [A]  time = 0.216907, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3872, 2836, 12, 88, 206} \[ \frac{a^2}{32 d (a \cos (c+d x)+a)^4}-\frac{1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac{1}{32 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{a}{48 d (a \cos (c+d x)+a)^3}-\frac{1}{64 d (a-a \cos (c+d x))^2}-\frac{1}{32 d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Cos[c + d*x]]/(64*a^2*d) - 1/(64*d*(a - a*Cos[c + d*x])^2) + a^2/(32*d*(a + a*Cos[c + d*x])^4) - a/(48
*d*(a + a*Cos[c + d*x])^3) - 1/(32*d*(a + a*Cos[c + d*x])^2) - 1/(64*d*(a^2 - a^2*Cos[c + d*x])) - 1/(32*d*(a^
2 + a^2*Cos[c + d*x]))

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]))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^2}{a^2 (-a-x)^3 (-a+x)^5} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{(-a-x)^3 (-a+x)^5} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{8 a (a-x)^5}-\frac{1}{16 a^2 (a-x)^4}-\frac{1}{16 a^3 (a-x)^3}-\frac{1}{32 a^4 (a-x)^2}+\frac{1}{32 a^3 (a+x)^3}+\frac{1}{64 a^4 (a+x)^2}-\frac{1}{64 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{1}{64 d (a-a \cos (c+d x))^2}+\frac{a^2}{32 d (a+a \cos (c+d x))^4}-\frac{a}{48 d (a+a \cos (c+d x))^3}-\frac{1}{32 d (a+a \cos (c+d x))^2}-\frac{1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac{1}{32 d \left (a^2+a^2 \cos (c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{64 a d}\\ &=\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{1}{64 d (a-a \cos (c+d x))^2}+\frac{a^2}{32 d (a+a \cos (c+d x))^4}-\frac{a}{48 d (a+a \cos (c+d x))^3}-\frac{1}{32 d (a+a \cos (c+d x))^2}-\frac{1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac{1}{32 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.745516, size = 152, normalized size = 1.04 \[ -\frac{\cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (6 \csc ^4\left (\frac{1}{2} (c+d x)\right )+12 \csc ^2\left (\frac{1}{2} (c+d x)\right )-3 \sec ^8\left (\frac{1}{2} (c+d x)\right )+4 \sec ^6\left (\frac{1}{2} (c+d x)\right )+12 \sec ^4\left (\frac{1}{2} (c+d x)\right )+24 \sec ^2\left (\frac{1}{2} (c+d x)\right )+24 \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{384 a^2 d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[(c + d*x)/2]^4*(12*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 + 24*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c +
d*x)/2]]) + 24*Sec[(c + d*x)/2]^2 + 12*Sec[(c + d*x)/2]^4 + 4*Sec[(c + d*x)/2]^6 - 3*Sec[(c + d*x)/2]^8)*Sec[
c + d*x]^2)/(384*a^2*d*(1 + Sec[c + d*x])^2)

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Maple [A]  time = 0.072, size = 144, normalized size = 1. \begin{align*}{\frac{1}{32\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{4}}}-{\frac{1}{48\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{1}{32\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}-{\frac{1}{32\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{128\,d{a}^{2}}}-{\frac{1}{64\,d{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{64\,d{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{128\,d{a}^{2}}} \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]

1/32/d/a^2/(cos(d*x+c)+1)^4-1/48/d/a^2/(cos(d*x+c)+1)^3-1/32/d/a^2/(cos(d*x+c)+1)^2-1/32/d/a^2/(cos(d*x+c)+1)+
1/128*ln(cos(d*x+c)+1)/a^2/d-1/64/d/a^2/(-1+cos(d*x+c))^2+1/64/d/a^2/(-1+cos(d*x+c))-1/128/d/a^2*ln(-1+cos(d*x
+c))

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Maxima [A]  time = 1.0289, size = 225, normalized size = 1.54 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 6 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right ) + 16\right )}}{a^{2} \cos \left (d x + c\right )^{6} + 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} - \frac{3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, 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]

-1/384*(2*(3*cos(d*x + c)^5 + 6*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 10*cos(d*x + c)^2 + 35*cos(d*x + c) + 16)/
(a^2*cos(d*x + c)^6 + 2*a^2*cos(d*x + c)^5 - a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^3 - a^2*cos(d*x + c)^2 +
2*a^2*cos(d*x + c) + a^2) - 3*log(cos(d*x + c) + 1)/a^2 + 3*log(cos(d*x + c) - 1)/a^2)/d

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Fricas [B]  time = 1.73821, size = 734, normalized size = 5.03 \begin{align*} -\frac{6 \, \cos \left (d x + c\right )^{5} + 12 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - 20 \, \cos \left (d x + c\right )^{2} - 3 \,{\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 70 \, \cos \left (d x + c\right ) + 32}{384 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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))^2,x, algorithm="fricas")

[Out]

-1/384*(6*cos(d*x + c)^5 + 12*cos(d*x + c)^4 - 4*cos(d*x + c)^3 - 20*cos(d*x + c)^2 - 3*(cos(d*x + c)^6 + 2*co
s(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) +
1/2) + 3*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x
+ c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 70*cos(d*x + c) + 32)/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5
- a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.3645, size = 279, normalized size = 1.91 \begin{align*} \frac{\frac{6 \,{\left (\frac{4 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{12 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac{\frac{48 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{6 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{8}}}{1536 \, 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]

1/1536*(6*(4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)*(cos(d*x
+ c) + 1)^2/(a^2*(cos(d*x + c) - 1)^2) - 12*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 + (48*a^6*(
cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*a^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 8*a^6*(cos(d*x + c) -
1)^3/(cos(d*x + c) + 1)^3 + 3*a^6*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/a^8)/d