Integrand size = 27, antiderivative size = 100 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=-\frac {6 a^3 \log (1-\sin (c+d x))}{d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {a^3 \sin ^2(c+d x)}{2 d}+\frac {a^5}{2 d (a-a \sin (c+d x))^2}-\frac {4 a^5}{d \left (a^2-a^2 \sin (c+d x)\right )} \] Output:
-6*a^3*ln(1-sin(d*x+c))/d-3*a^3*sin(d*x+c)/d-1/2*a^3*sin(d*x+c)^2/d+1/2*a^ 5/d/(a-a*sin(d*x+c))^2-4*a^5/d/(a^2-a^2*sin(d*x+c))
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.61 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=-\frac {a^3 \left (12 \log (1-\sin (c+d x))+\frac {7-8 \sin (c+d x)}{(-1+\sin (c+d x))^2}+6 \sin (c+d x)+\sin ^2(c+d x)\right )}{2 d} \] Input:
Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]
Output:
-1/2*(a^3*(12*Log[1 - Sin[c + d*x]] + (7 - 8*Sin[c + d*x])/(-1 + Sin[c + d *x])^2 + 6*Sin[c + d*x] + Sin[c + d*x]^2))/d
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 49, 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.
\(\displaystyle \int \tan ^4(c+d x) \sec (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^4 (a \sin (c+d x)+a)^3}{\cos (c+d x)^5}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {a^5 \int \frac {\sin ^4(c+d x)}{(a-a \sin (c+d x))^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {a^4 \sin ^4(c+d x)}{(a-a \sin (c+d x))^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a \int \left (\frac {a^4}{(a-a \sin (c+d x))^3}-\frac {4 a^3}{(a-a \sin (c+d x))^2}+\frac {6 a^2}{a-a \sin (c+d x)}-\sin (c+d x) a-3 a\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (\frac {a^4}{2 (a-a \sin (c+d x))^2}-\frac {4 a^3}{a-a \sin (c+d x)}-\frac {1}{2} a^2 \sin ^2(c+d x)-3 a^2 \sin (c+d x)-6 a^2 \log (a-a \sin (c+d x))\right )}{d}\) |
Input:
Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]
Output:
(a*(-6*a^2*Log[a - a*Sin[c + d*x]] - 3*a^2*Sin[c + d*x] - (a^2*Sin[c + d*x ]^2)/2 + a^4/(2*(a - a*Sin[c + d*x])^2) - (4*a^3)/(a - a*Sin[c + d*x])))/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_))^(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] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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]
Time = 7.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{2}}{2}+3 \sin \left (d x +c \right )-\frac {1}{2 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {4}{\sin \left (d x +c \right )-1}+6 \ln \left (\sin \left (d x +c \right )-1\right )\right )}{d}\) | \(63\) |
default | \(-\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{2}}{2}+3 \sin \left (d x +c \right )-\frac {1}{2 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {4}{\sin \left (d x +c \right )-1}+6 \ln \left (\sin \left (d x +c \right )-1\right )\right )}{d}\) | \(63\) |
risch | \(6 i a^{3} x +\frac {a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {12 i a^{3} c}{d}+\frac {2 i a^{3} \left (-7 i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4}}-\frac {12 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(168\) |
Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
-a^3/d*(1/2*sin(d*x+c)^2+3*sin(d*x+c)-1/2/(sin(d*x+c)-1)^2-4/(sin(d*x+c)-1 )+6*ln(sin(d*x+c)-1))
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.28 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{4} + 19 \, a^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} - 24 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="fricas" )
Output:
1/4*(2*a^3*cos(d*x + c)^4 + 19*a^3*cos(d*x + c)^2 - 8*a^3 - 24*(a^3*cos(d* x + c)^2 + 2*a^3*sin(d*x + c) - 2*a^3)*log(-sin(d*x + c) + 1) - 2*(4*a^3*c os(d*x + c)^2 - 3*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)
Timed out. \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))**3*tan(d*x+c)**4,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=-\frac {a^{3} \sin \left (d x + c\right )^{2} + 12 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, a^{3} \sin \left (d x + c\right ) - \frac {8 \, a^{3} \sin \left (d x + c\right ) - 7 \, a^{3}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{2 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="maxima" )
Output:
-1/2*(a^3*sin(d*x + c)^2 + 12*a^3*log(sin(d*x + c) - 1) + 6*a^3*sin(d*x + c) - (8*a^3*sin(d*x + c) - 7*a^3)/(sin(d*x + c)^2 - 2*sin(d*x + c) + 1))/d
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.70 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=-\frac {1}{2} \, a^{3} {\left (\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} + \frac {d \sin \left (d x + c\right )^{2} + 6 \, d \sin \left (d x + c\right )}{d^{2}} - \frac {8 \, \sin \left (d x + c\right ) - 7}{d {\left (\sin \left (d x + c\right ) - 1\right )}^{2}}\right )} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="giac")
Output:
-1/2*a^3*(12*log(abs(sin(d*x + c) - 1))/d + (d*sin(d*x + c)^2 + 6*d*sin(d* x + c))/d^2 - (8*sin(d*x + c) - 7)/(d*(sin(d*x + c) - 1)^2))
Time = 33.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.63 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {6\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-36\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+52\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-64\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+52\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-36\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+12\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {12\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d} \] Input:
int((tan(c + d*x)^4*(a + a*sin(c + d*x))^3)/cos(c + d*x),x)
Output:
(6*a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (52*a^3*tan(c/2 + (d*x)/2)^3 - 3 6*a^3*tan(c/2 + (d*x)/2)^2 - 64*a^3*tan(c/2 + (d*x)/2)^4 + 52*a^3*tan(c/2 + (d*x)/2)^5 - 36*a^3*tan(c/2 + (d*x)/2)^6 + 12*a^3*tan(c/2 + (d*x)/2)^7 + 12*a^3*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x) /2) - 12*tan(c/2 + (d*x)/2)^3 + 14*tan(c/2 + (d*x)/2)^4 - 12*tan(c/2 + (d* x)/2)^5 + 8*tan(c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x )/2)^8 + 1)) - (12*a^3*log(tan(c/2 + (d*x)/2) - 1))/d
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {a^{3} \left (12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\sin \left (d x +c \right )^{4}-4 \sin \left (d x +c \right )^{3}+12 \sin \left (d x +c \right )^{2}-6\right )}{2 d \left (\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c)^4,x)
Output:
(a**3*(12*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 - 24*log(tan((c + d *x)/2)**2 + 1)*sin(c + d*x) + 12*log(tan((c + d*x)/2)**2 + 1) - 24*log(tan ((c + d*x)/2) - 1)*sin(c + d*x)**2 + 48*log(tan((c + d*x)/2) - 1)*sin(c + d*x) - 24*log(tan((c + d*x)/2) - 1) - sin(c + d*x)**4 - 4*sin(c + d*x)**3 + 12*sin(c + d*x)**2 - 6))/(2*d*(sin(c + d*x)**2 - 2*sin(c + d*x) + 1))