Integrand size = 19, antiderivative size = 72 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=a x-\frac {b \cos (c+d x)}{d}-\frac {2 b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \] Output:
a*x-b*cos(d*x+c)/d-2*b*sec(d*x+c)/d+1/3*b*sec(d*x+c)^3/d-a*tan(d*x+c)/d+1/ 3*a*tan(d*x+c)^3/d
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\frac {a \arctan (\tan (c+d x))}{d}-\frac {b \cos (c+d x)}{d}-\frac {2 b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \] Input:
Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]
Output:
(a*ArcTan[Tan[c + d*x]])/d - (b*Cos[c + d*x])/d - (2*b*Sec[c + d*x])/d + ( b*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3201, 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.
\(\displaystyle \int \tan ^4(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x)^4 (a+b \sin (c+d x))dx\) |
\(\Big \downarrow \) 3201 |
\(\displaystyle \int \left (a \tan ^4(c+d x)+b \sin (c+d x) \tan ^4(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x-\frac {b \cos (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}-\frac {2 b \sec (c+d x)}{d}\) |
Input:
Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]
Output:
a*x - (b*Cos[c + d*x])/d - (2*b*Sec[c + d*x])/d + (b*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*( x_)])^(p_.), x_Symbol] :> Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Si n[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 1.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(98\) |
default | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+b \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(98\) |
parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {b \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(103\) |
risch | \(a x -\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {4 \left (3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a +3 \,{\mathrm e}^{i \left (d x +c \right )} b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(120\) |
Input:
int((a+b*sin(d*x+c))*tan(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/d*(a*(1/3*tan(d*x+c)^3-tan(d*x+c)+d*x+c)+b*(1/3*sin(d*x+c)^6/cos(d*x+c)^ 3-sin(d*x+c)^6/cos(d*x+c)-(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c)))
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\frac {3 \, a d x \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )^{4} - 6 \, b \cos \left (d x + c\right )^{2} - {\left (4 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) + b}{3 \, d \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+b*sin(d*x+c))*tan(d*x+c)^4,x, algorithm="fricas")
Output:
1/3*(3*a*d*x*cos(d*x + c)^3 - 3*b*cos(d*x + c)^4 - 6*b*cos(d*x + c)^2 - (4 *a*cos(d*x + c)^2 - a)*sin(d*x + c) + b)/(d*cos(d*x + c)^3)
\[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \tan ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate((a+b*sin(d*x+c))*tan(d*x+c)**4,x)
Output:
Integral((a + b*sin(c + d*x))*tan(c + d*x)**4, x)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a - b {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{3 \, d} \] Input:
integrate((a+b*sin(d*x+c))*tan(d*x+c)^4,x, algorithm="maxima")
Output:
1/3*((tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a - b*((6*cos(d*x + c )^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)))/d
Timed out. \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(d*x+c))*tan(d*x+c)^4,x, algorithm="giac")
Output:
Timed out
Time = 19.85 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.53 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=a\,x+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {32\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {16\,b}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:
int(tan(c + d*x)^4*(a + b*sin(c + d*x)),x)
Output:
a*x + ((16*b)/3 + 2*a*tan(c/2 + (d*x)/2) - (14*a*tan(c/2 + (d*x)/2)^3)/3 - (14*a*tan(c/2 + (d*x)/2)^5)/3 + 2*a*tan(c/2 + (d*x)/2)^7 - (32*b*tan(c/2 + (d*x)/2)^2)/3)/(d*(tan(c/2 + (d*x)/2)^2 - 1)^3*(tan(c/2 + (d*x)/2)^2 + 1 ))
Time = 0.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.85 \[ \int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx=\frac {-\cos \left (d x +c \right ) \tan \left (d x +c \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +\cos \left (d x +c \right ) \tan \left (d x +c \right )^{2} b +2 \cos \left (d x +c \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b -2 \cos \left (d x +c \right ) b +2 \sin \left (d x +c \right ) \tan \left (d x +c \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b -2 \sin \left (d x +c \right ) \tan \left (d x +c \right )^{3} b +2 \sin \left (d x +c \right ) \tan \left (d x +c \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b -2 \sin \left (d x +c \right ) \tan \left (d x +c \right ) b +2 \tan \left (d x +c \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tan \left (d x +c \right )^{3} a -6 \tan \left (d x +c \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +6 \tan \left (d x +c \right ) a +6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a d x +36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b -6 a d x}{6 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-1\right )} \] Input:
int((a+b*sin(d*x+c))*tan(d*x+c)^4,x)
Output:
( - cos(c + d*x)*tan(c + d*x)**2*tan((c + d*x)/2)**4*b + cos(c + d*x)*tan( c + d*x)**2*b + 2*cos(c + d*x)*tan((c + d*x)/2)**4*b - 2*cos(c + d*x)*b + 2*sin(c + d*x)*tan(c + d*x)**3*tan((c + d*x)/2)**4*b - 2*sin(c + d*x)*tan( c + d*x)**3*b + 2*sin(c + d*x)*tan(c + d*x)*tan((c + d*x)/2)**4*b - 2*sin( c + d*x)*tan(c + d*x)*b + 2*tan(c + d*x)**3*tan((c + d*x)/2)**4*a - 2*tan( c + d*x)**3*a - 6*tan(c + d*x)*tan((c + d*x)/2)**4*a + 6*tan(c + d*x)*a + 6*tan((c + d*x)/2)**4*a*d*x + 36*tan((c + d*x)/2)**4*b - 6*a*d*x)/(6*d*(ta n((c + d*x)/2)**4 - 1))