Integrand size = 25, antiderivative size = 69 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^5}{8 d \left (a^2-a^2 \sin (c+d x)\right )^2}+\frac {a^5}{8 d \left (a^4+a^4 \sin (c+d x)\right )} \] Output:
-1/8*a*arctanh(sin(d*x+c))/d+1/8*a^5/d/(a^2-a^2*sin(d*x+c))^2+1/8*a^5/d/(a ^4+a^4*sin(d*x+c))
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \sec ^4(c+d x)}{4 d}-\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Input:
Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*Tan[c + d*x],x]
Output:
-1/8*(a*ArcTanh[Sin[c + d*x]])/d + (a*Sec[c + d*x]^4)/(4*d) - (a*Sec[c + d *x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 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.
\(\displaystyle \int \tan (c+d x) \sec ^4(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) (a \sin (c+d x)+a)}{\cos (c+d x)^5}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {a^5 \int \frac {\sin (c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^4 \int \frac {a \sin (c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {a^4 \int \left (\frac {1}{4 (a-a \sin (c+d x))^3 a}-\frac {1}{8 \left (a^2-a^2 \sin ^2(c+d x)\right ) a^2}-\frac {1}{8 (\sin (c+d x) a+a)^2 a^2}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^4 \left (-\frac {\text {arctanh}(\sin (c+d x))}{8 a^3}+\frac {1}{8 a^2 (a \sin (c+d x)+a)}+\frac {1}{8 a (a-a \sin (c+d x))^2}\right )}{d}\) |
Input:
Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*Tan[c + d*x],x]
Output:
(a^4*(-1/8*ArcTanh[Sin[c + d*x]]/a^3 + 1/(8*a*(a - a*Sin[c + d*x])^2) + 1/ (8*a^2*(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_))*((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]
Time = 0.71 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(80\) |
default | \(\frac {a \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(80\) |
risch | \(\frac {i \left (2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{i \left (d x +c \right )}-10 a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+a \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{4 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}\) | \(135\) |
Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))*tan(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(1/4*sin(d*x+c)^3/cos(d*x+c)^4+1/8*sin(d*x+c)^3/cos(d*x+c)^2+1/8*si n(d*x+c)-1/8*ln(sec(d*x+c)+tan(d*x+c)))+1/4*a/cos(d*x+c)^4)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (64) = 128\).
Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.96 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=\frac {2 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a \sin \left (d x + c\right ) - 6 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))*tan(d*x+c),x, algorithm="fricas")
Output:
1/16*(2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^2*sin(d*x + c) - a*cos(d*x + c) ^2)*log(sin(d*x + c) + 1) + (a*cos(d*x + c)^2*sin(d*x + c) - a*cos(d*x + c )^2)*log(-sin(d*x + c) + 1) + 2*a*sin(d*x + c) - 6*a)/(d*cos(d*x + c)^2*si n(d*x + c) - d*cos(d*x + c)^2)
\[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=a \left (\int \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)**4*(a+a*sin(d*x+c))*tan(d*x+c),x)
Output:
a*(Integral(tan(c + d*x)*sec(c + d*x)**4, x) + Integral(sin(c + d*x)*tan(c + d*x)*sec(c + d*x)**4, x))
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))*tan(d*x+c),x, algorithm="maxima")
Output:
-1/16*(a*log(sin(d*x + c) + 1) - a*log(sin(d*x + c) - 1) - 2*(a*sin(d*x + c)^2 - a*sin(d*x + c) + 2*a)/(sin(d*x + c)^3 - sin(d*x + c)^2 - sin(d*x + c) + 1))/d
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {1}{16} \, a {\left (\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{d} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, {\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 2\right )}}{d {\left (\sin \left (d x + c\right ) + 1\right )} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}}\right )} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))*tan(d*x+c),x, algorithm="giac")
Output:
-1/16*a*(log(abs(sin(d*x + c) + 1))/d - log(abs(sin(d*x + c) - 1))/d - 2*( sin(d*x + c)^2 - sin(d*x + c) + 2)/(d*(sin(d*x + c) + 1)*(sin(d*x + c) - 1 )^2))
Time = 35.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.42 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \] Input:
int((tan(c + d*x)*(a + a*sin(c + d*x)))/cos(c + d*x)^4,x)
Output:
- (a*atanh(tan(c/2 + (d*x)/2)))/(4*d) - ((a*tan(c/2 + (d*x)/2))/4 + (3*a*t an(c/2 + (d*x)/2)^2)/2 - (3*a*tan(c/2 + (d*x)/2)^3)/2 + (3*a*tan(c/2 + (d* x)/2)^4)/2 + (a*tan(c/2 + (d*x)/2)^5)/4)/(d*(2*tan(c/2 + (d*x)/2) + tan(c/ 2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^3 + tan(c/2 + (d*x)/2)^4 + 2*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2)^6 - 1))
Time = 0.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.03 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=\frac {a \left (\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{3}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2}+1\right )}{8 d \left (\sin \left (d x +c \right )^{3}-\sin \left (d x +c \right )^{2}-\sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))*tan(d*x+c),x)
Output:
(a*(log(tan((c + d*x)/2) - 1)*sin(c + d*x)**3 - log(tan((c + d*x)/2) - 1)* sin(c + d*x)**2 - log(tan((c + d*x)/2) - 1)*sin(c + d*x) + log(tan((c + d* x)/2) - 1) - log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 + log(tan((c + d*x) /2) + 1)*sin(c + d*x)**2 + log(tan((c + d*x)/2) + 1)*sin(c + d*x) - log(ta n((c + d*x)/2) + 1) - sin(c + d*x)**3 + 2*sin(c + d*x)**2 + 1))/(8*d*(sin( c + d*x)**3 - sin(c + d*x)**2 - sin(c + d*x) + 1))