Integrand size = 27, antiderivative size = 82 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a x}{2}+\frac {2 a \cos (c+d x)}{d}-\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d} \]
-3/2*a*x+2*a*cos(d*x+c)/d-1/3*a*cos(d*x+c)^3/d+a*sec(d*x+c)/d+3/2*a*tan(d* x+c)/d-1/2*a*sin(d*x+c)^2*tan(d*x+c)/d
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a (c+d x)}{2 d}+\frac {7 a \cos (c+d x)}{4 d}-\frac {a \cos (3 (c+d x))}{12 d}+\frac {a \sec (c+d x)}{d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \tan (c+d x)}{d} \]
(-3*a*(c + d*x))/(2*d) + (7*a*Cos[c + d*x])/(4*d) - (a*Cos[3*(c + d*x)])/( 12*d) + (a*Sec[c + d*x])/d + (a*Sin[2*(c + d*x)])/(4*d) + (a*Tan[c + d*x]) /d
Time = 0.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3317, 3042, 3070, 244, 2009, 3071, 252, 262, 216}
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.
| \(\displaystyle \int \sin ^2(c+d x) \tan ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 (a \sin (c+d x)+a)}{\cos (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \sin ^3(c+d x) \tan ^2(c+d x)dx+a \int \sin ^2(c+d x) \tan ^2(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin (c+d x)^2 \tan (c+d x)^2dx+a \int \sin (c+d x)^3 \tan (c+d x)^2dx\) |
| \(\Big \downarrow \) 3070 |
| \(\displaystyle a \int \sin (c+d x)^2 \tan (c+d x)^2dx-\frac {a \int \left (1-\cos ^2(c+d x)\right )^2 \sec ^2(c+d x)d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle a \int \sin (c+d x)^2 \tan (c+d x)^2dx-\frac {a \int \left (\cos ^2(c+d x)+\sec ^2(c+d x)-2\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \sin (c+d x)^2 \tan (c+d x)^2dx-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-2 \cos (c+d x)-\sec (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {a \int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right )^2}d\tan (c+d x)}{d}-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-2 \cos (c+d x)-\sec (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {a \left (\frac {3}{2} \int \frac {\tan ^2(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)-\frac {\tan ^3(c+d x)}{2 \left (\tan ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-2 \cos (c+d x)-\sec (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {a \left (\frac {3}{2} \left (\tan (c+d x)-\int \frac {1}{\tan ^2(c+d x)+1}d\tan (c+d x)\right )-\frac {\tan ^3(c+d x)}{2 \left (\tan ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-2 \cos (c+d x)-\sec (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {a \left (\frac {3}{2} (\tan (c+d x)-\arctan (\tan (c+d x)))-\frac {\tan ^3(c+d x)}{2 \left (\tan ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-2 \cos (c+d x)-\sec (c+d x)\right )}{d}\) |
-((a*(-2*Cos[c + d*x] + Cos[c + d*x]^3/3 - Sec[c + d*x]))/d) + (a*((3*(-Ar cTan[Tan[c + d*x]] + Tan[c + d*x]))/2 - Tan[c + d*x]^3/(2*(1 + Tan[c + d*x ]^2))))/d
3.8.51.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-\frac {3 a x}{2}+\frac {7 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {7 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {2 a}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}-\frac {a \cos \left (3 d x +3 c \right )}{12 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(86\) |
| derivativedivides | \(\frac {a \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(104\) |
| default | \(\frac {a \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(104\) |
| parallelrisch | \(-\frac {a \left (36 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-36 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+2 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-18 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+2 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+18 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+53 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+37 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(143\) |
| norman | \(\frac {\frac {3 a x}{2}-\frac {16 a}{3 d}-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {5 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {32 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(171\) |
-3/2*a*x+7/8*a/d*exp(I*(d*x+c))+7/8*a/d*exp(-I*(d*x+c))+2*a/d/(exp(I*(d*x+ c))-I)-1/12*a/d*cos(3*d*x+3*c)+1/4*a/d*sin(2*d*x+2*c)
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.59 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {2 \, a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{3} + 9 \, a d x - 12 \, a \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, a d x - 5 \, a\right )} \cos \left (d x + c\right ) - {\left (2 \, a \cos \left (d x + c\right )^{3} + 9 \, a d x + 3 \, a \cos \left (d x + c\right )^{2} - 9 \, a \cos \left (d x + c\right ) + 6 \, a\right )} \sin \left (d x + c\right ) - 6 \, a}{6 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
-1/6*(2*a*cos(d*x + c)^4 - a*cos(d*x + c)^3 + 9*a*d*x - 12*a*cos(d*x + c)^ 2 + 3*(3*a*d*x - 5*a)*cos(d*x + c) - (2*a*cos(d*x + c)^3 + 9*a*d*x + 3*a*c os(d*x + c)^2 - 9*a*cos(d*x + c) + 6*a)*sin(d*x + c) - 6*a)/(d*cos(d*x + c ) - d*sin(d*x + c) + d)
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=a \left (\int \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
a*(Integral(sin(c + d*x)**4*sec(c + d*x)**2, x) + Integral(sin(c + d*x)**5 *sec(c + d*x)**2, x))
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a + 3 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a}{6 \, d} \]
-1/6*(2*(cos(d*x + c)^3 - 3/cos(d*x + c) - 6*cos(d*x + c))*a + 3*(3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2*tan(d*x + c))*a)/d
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.28 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {9 \, {\left (d x + c\right )} a + \frac {12 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
-1/6*(9*(d*x + c)*a + 12*a/(tan(1/2*d*x + 1/2*c) - 1) + 2*(3*a*tan(1/2*d*x + 1/2*c)^5 - 6*a*tan(1/2*d*x + 1/2*c)^4 - 24*a*tan(1/2*d*x + 1/2*c)^2 - 3 *a*tan(1/2*d*x + 1/2*c) - 10*a)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 16.24 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.13 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {\left (\frac {a\,\left (9\,c+9\,d\,x-18\right )}{6}-\frac {3\,a\,\left (c+d\,x\right )}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {9\,a\,\left (c+d\,x\right )}{2}-\frac {a\,\left (27\,c+27\,d\,x-18\right )}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {a\,\left (27\,c+27\,d\,x-48\right )}{6}-\frac {9\,a\,\left (c+d\,x\right )}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {9\,a\,\left (c+d\,x\right )}{2}-\frac {a\,\left (27\,c+27\,d\,x-48\right )}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {a\,\left (27\,c+27\,d\,x-78\right )}{6}-\frac {9\,a\,\left (c+d\,x\right )}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {3\,a\,\left (c+d\,x\right )}{2}-\frac {a\,\left (9\,c+9\,d\,x-14\right )}{6}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,\left (9\,c+9\,d\,x-32\right )}{6}-\frac {3\,a\,\left (c+d\,x\right )}{2}}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {3\,a\,x}{2} \]
((a*(9*c + 9*d*x - 32))/6 - tan(c/2 + (d*x)/2)*((a*(9*c + 9*d*x - 14))/6 - (3*a*(c + d*x))/2) - (3*a*(c + d*x))/2 + tan(c/2 + (d*x)/2)^6*((a*(9*c + 9*d*x - 18))/6 - (3*a*(c + d*x))/2) - tan(c/2 + (d*x)/2)^5*((a*(27*c + 27* d*x - 18))/6 - (9*a*(c + d*x))/2) - tan(c/2 + (d*x)/2)^3*((a*(27*c + 27*d* x - 48))/6 - (9*a*(c + d*x))/2) + tan(c/2 + (d*x)/2)^4*((a*(27*c + 27*d*x - 48))/6 - (9*a*(c + d*x))/2) + tan(c/2 + (d*x)/2)^2*((a*(27*c + 27*d*x - 78))/6 - (9*a*(c + d*x))/2))/(d*(tan(c/2 + (d*x)/2) - 1)*(tan(c/2 + (d*x)/ 2)^2 + 1)^3) - (3*a*x)/2