Integrand size = 21, antiderivative size = 86 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {5 \cos (c+d x)}{a^3 d}-\frac {3 \cos ^2(c+d x)}{2 a^3 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {2}{a^3 d (1+\cos (c+d x))}-\frac {7 \log (1+\cos (c+d x))}{a^3 d} \] Output:
5*cos(d*x+c)/a^3/d-3/2*cos(d*x+c)^2/a^3/d+1/3*cos(d*x+c)^3/a^3/d-2/a^3/d/( 1+cos(d*x+c))-7*ln(1+cos(d*x+c))/a^3/d
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (389-184 \cos (2 (c+d x))+28 \cos (3 (c+d x))-4 \cos (4 (c+d x))+1344 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (-19+1344 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{24 a^3 d (1+\cos (c+d x))^3} \] Input:
Integrate[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]
Output:
-1/24*(Cos[(c + d*x)/2]^4*(389 - 184*Cos[2*(c + d*x)] + 28*Cos[3*(c + d*x) ] - 4*Cos[4*(c + d*x)] + 1344*Log[Cos[(c + d*x)/2]] + Cos[c + d*x]*(-19 + 1344*Log[Cos[(c + d*x)/2]])))/(a^3*d*(1 + Cos[c + d*x])^3)
Time = 0.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 27, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin ^3(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^3}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^3}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int \frac {\cos ^3(c+d x) (a-a \cos (c+d x))}{(\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {a^3 \cos ^3(c+d x) (a-a \cos (c+d x))}{(\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{a^6 d}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {\int \left (-\frac {2 a^4}{(\cos (c+d x) a+a)^2}+\frac {7 a^3}{\cos (c+d x) a+a}-\cos ^2(c+d x) a^2+3 \cos (c+d x) a^2-5 a^2\right )d(a \cos (c+d x))}{a^6 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {2 a^4}{a \cos (c+d x)+a}-\frac {1}{3} a^3 \cos ^3(c+d x)+\frac {3}{2} a^3 \cos ^2(c+d x)-5 a^3 \cos (c+d x)+7 a^3 \log (a \cos (c+d x)+a)}{a^6 d}\) |
Input:
Int[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]
Output:
-((-5*a^3*Cos[c + d*x] + (3*a^3*Cos[c + d*x]^2)/2 - (a^3*Cos[c + d*x]^3)/3 + (2*a^4)/(a + a*Cos[c + d*x]) + 7*a^3*Log[a + a*Cos[c + d*x]])/(a^6*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {3 \cos \left (d x +c \right )^{2}}{2}+5 \cos \left (d x +c \right )-7 \ln \left (1+\cos \left (d x +c \right )\right )-\frac {2}{1+\cos \left (d x +c \right )}}{d \,a^{3}}\) | \(60\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {3 \cos \left (d x +c \right )^{2}}{2}+5 \cos \left (d x +c \right )-7 \ln \left (1+\cos \left (d x +c \right )\right )-\frac {2}{1+\cos \left (d x +c \right )}}{d \,a^{3}}\) | \(60\) |
parallelrisch | \(\frac {121+84 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\cos \left (3 d x +3 c \right )-9 \cos \left (2 d x +2 c \right )+63 \cos \left (d x +c \right )-12 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{12 d \,a^{3}}\) | \(66\) |
norman | \(\frac {\frac {34 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}+\frac {41}{3 a d}+\frac {24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{2}}+\frac {7 \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \,a^{3}}\) | \(109\) |
risch | \(\frac {7 i x}{a^{3}}+\frac {21 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {21 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {14 i c}{d \,a^{3}}-\frac {4 \,{\mathrm e}^{i \left (d x +c \right )}}{a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}-\frac {14 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\cos \left (3 d x +3 c \right )}{12 a^{3} d}-\frac {3 \cos \left (2 d x +2 c \right )}{4 a^{3} d}\) | \(137\) |
Input:
int(sin(d*x+c)^3/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(1/3*cos(d*x+c)^3-3/2*cos(d*x+c)^2+5*cos(d*x+c)-7*ln(1+cos(d*x+c)) -2/(1+cos(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3} + 42 \, \cos \left (d x + c\right )^{2} - 84 \, {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 69 \, \cos \left (d x + c\right ) - 15}{12 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate(sin(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/12*(4*cos(d*x + c)^4 - 14*cos(d*x + c)^3 + 42*cos(d*x + c)^2 - 84*(cos(d *x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 69*cos(d*x + c) - 15)/(a^3*d*co s(d*x + c) + a^3*d)
Timed out. \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**3/(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {12}{a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac {2 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right )}{a^{3}} + \frac {42 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{6 \, d} \] Input:
integrate(sin(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/6*(12/(a^3*cos(d*x + c) + a^3) - (2*cos(d*x + c)^3 - 9*cos(d*x + c)^2 + 30*cos(d*x + c))/a^3 + 42*log(cos(d*x + c) + 1)/a^3)/d
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {7 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} - \frac {2}{a^{3} d {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {2 \, a^{6} d^{2} \cos \left (d x + c\right )^{3} - 9 \, a^{6} d^{2} \cos \left (d x + c\right )^{2} + 30 \, a^{6} d^{2} \cos \left (d x + c\right )}{6 \, a^{9} d^{3}} \] Input:
integrate(sin(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
-7*log(abs(cos(d*x + c) + 1))/(a^3*d) - 2/(a^3*d*(cos(d*x + c) + 1)) + 1/6 *(2*a^6*d^2*cos(d*x + c)^3 - 9*a^6*d^2*cos(d*x + c)^2 + 30*a^6*d^2*cos(d*x + c))/(a^9*d^3)
Time = 11.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2}{a^3\,\cos \left (c+d\,x\right )+a^3}+\frac {7\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3}-\frac {5\,\cos \left (c+d\,x\right )}{a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^2}{2\,a^3}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^3}}{d} \] Input:
int(sin(c + d*x)^3/(a + a/cos(c + d*x))^3,x)
Output:
-(2/(a^3*cos(c + d*x) + a^3) + (7*log(cos(c + d*x) + 1))/a^3 - (5*cos(c + d*x))/a^3 + (3*cos(c + d*x)^2)/(2*a^3) - cos(c + d*x)^3/(3*a^3))/d
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {42 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+7 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+42 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+2 \sin \left (d x +c \right )^{4}-25 \sin \left (d x +c \right )^{2}-6}{6 a^{3} d \left (\cos \left (d x +c \right )+1\right )} \] Input:
int(sin(d*x+c)^3/(a+a*sec(d*x+c))^3,x)
Output:
(42*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1) + 7*cos(c + d*x)*sin(c + d*x )**2 + 6*cos(c + d*x) + 42*log(tan((c + d*x)/2)**2 + 1) + 2*sin(c + d*x)** 4 - 25*sin(c + d*x)**2 - 6)/(6*a**3*d*(cos(c + d*x) + 1))