\(\int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx\) [45]

Optimal result

Integrand size = 21, antiderivative size = 111 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]

[Out]

-1/2*a^5/d/(a-a*cos(d*x+c))^2-3*a^4/d/(a-a*cos(d*x+c))+6*a^3*ln(1-cos(d*x+c))/d-6*a^3*ln(cos(d*x+c))/d+3*a^3*s
ec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 46} \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d} \]

[In]

Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

-1/2*a^5/(d*(a - a*Cos[c + d*x])^2) - (3*a^4)/(d*(a - a*Cos[c + d*x])) + (6*a^3*Log[1 - Cos[c + d*x]])/d - (6*
a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2915

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/b)*x
)^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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^5(c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {a^3}{(-a-x)^3 x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \frac {1}{(-a-x)^3 x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \left (-\frac {1}{a^3 x^3}+\frac {3}{a^4 x^2}-\frac {6}{a^5 x}+\frac {1}{a^3 (a+x)^3}+\frac {3}{a^4 (a+x)^2}+\frac {6}{a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+48 \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-24 \sec (c+d x)-4 \sec ^2(c+d x)\right )}{64 d} \]

[In]

Integrate[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

-1/64*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(12*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 48*(Log[Cos[c
+ d*x]] - 2*Log[Sin[(c + d*x)/2]]) - 24*Sec[c + d*x] - 4*Sec[c + d*x]^2))/d

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.38

method	result	size
norman	\(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 d}-\frac {23 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 d}+\frac {75 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {12 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(153\)
parallelrisch	\(\frac {12 \left (\frac {\left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {\left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {49 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\cos \left (d x +c \right )-\frac {34 \cos \left (2 d x +2 c \right )}{49}+\frac {11 \cos \left (3 d x +3 c \right )}{49}-\frac {86}{147}\right )}{128}\right ) a^{3}}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(153\)
risch	\(\frac {4 a^{3} \left (3 \,{\mathrm e}^{7 i \left (d x +c \right )}-9 \,{\mathrm e}^{6 i \left (d x +c \right )}+13 \,{\mathrm e}^{5 i \left (d x +c \right )}-16 \,{\mathrm e}^{4 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}-9 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {12 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {6 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(154\)
derivativedivides	\(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\)	\(218\)
default	\(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\)	\(218\)

[In]

int(csc(d*x+c)^5*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

(-1/8/d*a^3-3/2*a^3/d*tan(1/2*d*x+1/2*c)^2-23/4*a^3/d*tan(1/2*d*x+1/2*c)^6+75/8*a^3/d*tan(1/2*d*x+1/2*c)^4)/ta
n(1/2*d*x+1/2*c)^4/(-1+tan(1/2*d*x+1/2*c)^2)^2+12/d*a^3*ln(tan(1/2*d*x+1/2*c))-6/d*a^3*ln(tan(1/2*d*x+1/2*c)-1
)-6/d*a^3*ln(tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.59 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3} - 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(csc(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2*(12*a^3*cos(d*x + c)^3 - 18*a^3*cos(d*x + c)^2 + 4*a^3*cos(d*x + c) + a^3 - 12*(a^3*cos(d*x + c)^4 - 2*a^3
*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(-cos(d*x + c)) + 12*(a^3*cos(d*x + c)^4 - 2*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)^4 - 2*d*cos(d*x + c)^3 + d*cos(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**5*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {12 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{2 \, d} \]

[In]

integrate(csc(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*(12*a^3*log(cos(d*x + c) - 1) - 12*a^3*log(cos(d*x + c)) + (12*a^3*cos(d*x + c)^3 - 18*a^3*cos(d*x + c)^2
+ 4*a^3*cos(d*x + c) + a^3)/(cos(d*x + c)^4 - 2*cos(d*x + c)^3 + cos(d*x + c)^2))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.68 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {48 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 48 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {a^{3} - \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {75 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {46 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}^{2}}}{8 \, d} \]

[In]

integrate(csc(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(48*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 48*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x +
c) + 1) - 1)) - (a^3 - 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 75*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c
) + 1)^2 - 46*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + (cos(d*x
+ c) - 1)^2/(cos(d*x + c) + 1)^2)^2)/d

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^3-9\,a^3\,{\cos \left (c+d\,x\right )}^2+2\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {12\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]

[In]

int((a + a/cos(c + d*x))^3/sin(c + d*x)^5,x)

[Out]

(2*a^3*cos(c + d*x) + a^3/2 - 9*a^3*cos(c + d*x)^2 + 6*a^3*cos(c + d*x)^3)/(d*(cos(c + d*x)^2 - 2*cos(c + d*x)
^3 + cos(c + d*x)^4)) - (12*a^3*atanh(2*cos(c + d*x) - 1))/d