Integrand size = 25, antiderivative size = 61 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{8 d (a+a \sin (c+d x))} \]
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \sec ^4(c+d x)}{4 d}-\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
-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 = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, 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 |
\(\displaystyle \frac {a^4 \left (-\frac {\text {arctanh}(\sin (c+d x))}{8 a^3}+\frac {1}{8 a^2 (a \sin (c+d x)+a)}+\frac {1}{8 a (a-a \sin (c+d x))^2}\right )}{d}\) |
(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
3.9.56.3.1 Defintions of rubi rules used
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 = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(80\) |
default | \(\frac {a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(80\) |
risch | \(\frac {i \left (2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{i \left (d x +c \right )}-10 a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+a \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{4 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(135\) |
parallelrisch | \(-\frac {\left (\left (-1-\cos \left (2 d x +2 c \right )+\frac {\sin \left (d x +c \right )}{2}+\frac {\sin \left (3 d x +3 c \right )}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\frac {\sin \left (3 d x +3 c \right )}{2}-\frac {\sin \left (d x +c \right )}{2}+\cos \left (2 d x +2 c \right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 \cos \left (2 d x +2 c \right )+\sin \left (d x +c \right )-\sin \left (3 d x +3 c \right )-3\right ) a}{4 d \left (2-\sin \left (3 d x +3 c \right )-\sin \left (d x +c \right )+2 \cos \left (2 d x +2 c \right )\right )}\) | \(159\) |
norman | \(\frac {\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {7 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {2 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {2 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(221\) |
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. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.21 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=\frac {2 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a \sin \left (d x + c\right ) - 6 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
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)
\[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=a \left (\int \sin {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
a*(Integral(sin(c + d*x)*sec(c + d*x)**5, x) + Integral(sin(c + d*x)**2*se c(c + d*x)**5, x))
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.38 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \]
-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 = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {2 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 2 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right ) + 3 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {3 \, a \sin \left (d x + c\right )^{2} - 6 \, a \sin \left (d x + c\right ) - a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
-1/32*(2*a*log(abs(sin(d*x + c) + 1)) - 2*a*log(abs(sin(d*x + c) - 1)) - 2 *(a*sin(d*x + c) + 3*a)/(sin(d*x + c) + 1) + (3*a*sin(d*x + c)^2 - 6*a*sin (d*x + c) - a)/(sin(d*x + c) - 1)^2)/d
Time = 15.74 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.74 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
- (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))