\(\int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx\) [43]

Optimal result

Integrand size = 19, antiderivative size = 67 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2915, 12, 90} \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 a^3 \log (\cos (c+d x))}{d} \]

[In]

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

[Out]

(4*a^3*Log[1 - Cos[c + d*x]])/d - (4*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 90

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 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 (c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {a^3 (-a+x)^2}{(-a-x) x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \frac {(-a+x)^2}{(-a-x) x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \left (-\frac {a}{x^3}+\frac {3}{x^2}-\frac {4}{a x}+\frac {4}{a (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 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 = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 \left (1+6 \cos (c+d x)-4 \log (\cos (c+d x))-4 \cos (2 (c+d x)) \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sec ^2(c+d x)}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\)	\(90\)
default	\(\frac {a^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\)	\(90\)
risch	\(\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(95\)
norman	\(\frac {\frac {6 a^{3}}{d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(104\)
parallelrisch	\(\frac {-8 a^{3} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-a^{3} \left (-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-12 \cos \left (d x +c \right )-5 \cos \left (2 d x +2 c \right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-7\right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(144\)

[In]

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

[Out]

1/d*(a^3*(1/2/cos(d*x+c)^2+ln(tan(d*x+c)))+3*a^3*(1/cos(d*x+c)+ln(-cot(d*x+c)+csc(d*x+c)))+3*a^3*ln(tan(d*x+c)
)+a^3*ln(-cot(d*x+c)+csc(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]

[In]

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

[Out]

-1/2*(8*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) - 8*a^3*cos(d*x + c)^2*log(-1/2*cos(d*x + c) + 1/2) - 6*a^3*cos(
d*x + c) - a^3)/(d*cos(d*x + c)^2)

Sympy [F]

\[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \csc {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \csc {\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

a**3*(Integral(3*csc(c + d*x)*sec(c + d*x), x) + Integral(3*csc(c + d*x)*sec(c + d*x)**2, x) + Integral(csc(c
+ d*x)*sec(c + d*x)**3, x) + Integral(csc(c + d*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {8 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 8 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*(8*a^3*log(cos(d*x + c) - 1) - 8*a^3*log(cos(d*x + c)) + (6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (65) = 130\).

Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.12 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {6 \, a^{3} + \frac {8 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}\right )}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{d\,{\cos \left (c+d\,x\right )}^2}-\frac {8\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]

[In]

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

[Out]

(3*a^3*cos(c + d*x) + a^3/2)/(d*cos(c + d*x)^2) - (8*a^3*atanh(2*cos(c + d*x) - 1))/d