Integrand size = 29, antiderivative size = 240 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (35 a^2+57 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\left (35 a^2-57 a b+24 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {a^8 \log (a+b \sin (c+d x))}{b^3 \left (a^2-b^2\right )^3 d}+\frac {a \sin (c+d x)}{b^2 d}-\frac {\sin ^2(c+d x)}{2 b d}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) \left (4 b \left (4 a^2-3 b^2\right )-a \left (13 a^2-9 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \]
-1/16*(35*a^2+57*a*b+24*b^2)*ln(1-sin(d*x+c))/(a+b)^3/d+1/16*(35*a^2-57*a* b+24*b^2)*ln(1+sin(d*x+c))/(a-b)^3/d-a^8*ln(a+b*sin(d*x+c))/b^3/(a^2-b^2)^ 3/d+a*sin(d*x+c)/b^2/d-1/2*sin(d*x+c)^2/b/d-1/4*sec(d*x+c)^4*(b-a*sin(d*x+ c))/(a^2-b^2)/d+1/8*sec(d*x+c)^2*(4*b*(4*a^2-3*b^2)-a*(13*a^2-9*b^2)*sin(d *x+c))/(a^2-b^2)^2/d
Time = 2.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-\frac {\left (35 a^2+57 a b+24 b^2\right ) \log (1-\sin (c+d x))}{(a+b)^3}+\frac {\left (35 a^2-57 a b+24 b^2\right ) \log (1+\sin (c+d x))}{(a-b)^3}-\frac {16 a^8 \log (a+b \sin (c+d x))}{(a-b)^3 b^3 (a+b)^3}+\frac {1}{(a+b) (-1+\sin (c+d x))^2}+\frac {13 a+11 b}{(a+b)^2 (-1+\sin (c+d x))}+\frac {16 a \sin (c+d x)}{b^2}-\frac {8 \sin ^2(c+d x)}{b}-\frac {1}{(a-b) (1+\sin (c+d x))^2}+\frac {13 a-11 b}{(a-b)^2 (1+\sin (c+d x))}}{16 d} \]
(-(((35*a^2 + 57*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(a + b)^3) + ((35*a^ 2 - 57*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 - (16*a^8*Log[a + b* Sin[c + d*x]])/((a - b)^3*b^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^ 2) + (13*a + 11*b)/((a + b)^2*(-1 + Sin[c + d*x])) + (16*a*Sin[c + d*x])/b ^2 - (8*Sin[c + d*x]^2)/b - 1/((a - b)*(1 + Sin[c + d*x])^2) + (13*a - 11* b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)
Time = 0.91 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3042, 3316, 27, 601, 2178, 25, 2160, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^8}{\cos (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {b^5 \int \frac {\sin ^8(c+d x)}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^8 \sin ^8(c+d x)}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{b^3 d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {-\frac {\int \frac {-\frac {3 a \sin (c+d x) b^9}{a^2-b^2}+4 \sin ^6(c+d x) b^8+4 \sin ^4(c+d x) b^8+4 \sin ^2(c+d x) b^8+\frac {a^2 b^8}{a^2-b^2}}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{4 b^2}-\frac {b^6 \left (\frac {b^2}{a^2-b^2}-\frac {a b \sin (c+d x)}{a^2-b^2}\right )}{4 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}}{b^3 d}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-\frac {a \left (13 a^2-9 b^2\right ) \sin (c+d x) b^9}{\left (a^2-b^2\right )^2}+8 \sin ^4(c+d x) b^8+16 \sin ^2(c+d x) b^8+\frac {a^2 \left (11 a^2-7 b^2\right ) b^8}{\left (a^2-b^2\right )^2}}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{2 b^2}-\frac {b^6 \left (4 b^2 \left (4 a^2-3 b^2\right )-a b \left (13 a^2-9 b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 \left (b^2-b^2 \sin ^2(c+d x)\right )}}{4 b^2}-\frac {b^6 \left (\frac {b^2}{a^2-b^2}-\frac {a b \sin (c+d x)}{a^2-b^2}\right )}{4 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}}{b^3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-\frac {a \left (13 a^2-9 b^2\right ) \sin (c+d x) b^9}{\left (a^2-b^2\right )^2}+8 \sin ^4(c+d x) b^8+16 \sin ^2(c+d x) b^8+\frac {a^2 \left (11 a^2-7 b^2\right ) b^8}{\left (a^2-b^2\right )^2}}{(a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{2 b^2}-\frac {b^6 \left (4 b^2 \left (4 a^2-3 b^2\right )-a b \left (13 a^2-9 b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 \left (b^2-b^2 \sin ^2(c+d x)\right )}}{4 b^2}-\frac {b^6 \left (\frac {b^2}{a^2-b^2}-\frac {a b \sin (c+d x)}{a^2-b^2}\right )}{4 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}}{b^3 d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {-\frac {-\frac {\int \left (-\frac {8 b^4 a^8}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+8 b^4 a-8 b^5 \sin (c+d x)+\frac {b^7 \left (35 a^2+57 b a+24 b^2\right )}{2 (a+b)^3 (b-b \sin (c+d x))}+\frac {b^7 \left (35 a^2-57 b a+24 b^2\right )}{2 (a-b)^3 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{2 b^2}-\frac {b^6 \left (4 b^2 \left (4 a^2-3 b^2\right )-a b \left (13 a^2-9 b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 \left (b^2-b^2 \sin ^2(c+d x)\right )}}{4 b^2}-\frac {b^6 \left (\frac {b^2}{a^2-b^2}-\frac {a b \sin (c+d x)}{a^2-b^2}\right )}{4 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}}{b^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^6 \left (\frac {b^2}{a^2-b^2}-\frac {a b \sin (c+d x)}{a^2-b^2}\right )}{4 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}-\frac {-\frac {b^6 \left (4 b^2 \left (4 a^2-3 b^2\right )-a b \left (13 a^2-9 b^2\right ) \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 \left (b^2-b^2 \sin ^2(c+d x)\right )}-\frac {-\frac {b^7 \left (35 a^2+57 a b+24 b^2\right ) \log (b-b \sin (c+d x))}{2 (a+b)^3}+\frac {b^7 \left (35 a^2-57 a b+24 b^2\right ) \log (b \sin (c+d x)+b)}{2 (a-b)^3}-\frac {8 a^8 b^4 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3}+8 a b^5 \sin (c+d x)-4 b^6 \sin ^2(c+d x)}{2 b^2}}{4 b^2}}{b^3 d}\) |
(-1/4*(b^6*(b^2/(a^2 - b^2) - (a*b*Sin[c + d*x])/(a^2 - b^2)))/(b^2 - b^2* Sin[c + d*x]^2)^2 - (-1/2*(b^6*(4*b^2*(4*a^2 - 3*b^2) - a*b*(13*a^2 - 9*b^ 2)*Sin[c + d*x]))/((a^2 - b^2)^2*(b^2 - b^2*Sin[c + d*x]^2)) - (-1/2*(b^7* (35*a^2 + 57*a*b + 24*b^2)*Log[b - b*Sin[c + d*x]])/(a + b)^3 - (8*a^8*b^4 *Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^3 + (b^7*(35*a^2 - 57*a*b + 24*b^2)* Log[b + b*Sin[c + d*x]])/(2*(a - b)^3) + 8*a*b^5*Sin[c + d*x] - 4*b^6*Sin[ c + d*x]^2)/(2*b^2))/(4*b^2))/(b^3*d)
3.14.59.3.1 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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 1.62 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) b}{2}+a \sin \left (d x +c \right )}{b^{2}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-13 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (35 a^{2}-57 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-13 a -11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-35 a^{2}-57 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {a^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} \left (a +b \right )^{3} \left (a -b \right )^{3}}}{d}\) | \(217\) |
| default | \(\frac {\frac {-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) b}{2}+a \sin \left (d x +c \right )}{b^{2}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-13 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (35 a^{2}-57 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-13 a -11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-35 a^{2}-57 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {a^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} \left (a +b \right )^{3} \left (a -b \right )^{3}}}{d}\) | \(217\) |
| parallelrisch | \(\frac {-32 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a^{8} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+32 \left (a^{2}+3 b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{3} \left (a -b \right )^{3} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 b \left (-\frac {35 b^{2} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{3} \left (a^{2}+\frac {57}{35} a b +\frac {24}{35} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{3}+\left (\frac {35 b^{2} \left (a^{2}-\frac {57}{35} a b +\frac {24}{35} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{3}+\left (\left (-\frac {1}{12} a^{4} b +\frac {5}{6} a^{2} b^{3}-\frac {3}{4} b^{5}\right ) \cos \left (2 d x +2 c \right )+\frac {b \left (a^{4}-9 a^{2} b^{2}+6 b^{4}\right ) \cos \left (4 d x +4 c \right )}{6}+\frac {b \left (a -b \right )^{2} \left (a +b \right )^{2} \cos \left (6 d x +6 c \right )}{12}+a \left (a^{4}-\frac {25}{6} a^{2} b^{2}+\frac {5}{2} b^{4}\right ) \sin \left (3 d x +3 c \right )+\frac {a \left (a -b \right )^{2} \left (a +b \right )^{2} \sin \left (5 d x +5 c \right )}{3}+\left (\frac {2}{3} a^{5}-\frac {13}{6} a^{3} b^{2}+\frac {5}{6} a \,b^{4}\right ) \sin \left (d x +c \right )-\frac {b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right )}{6}\right ) \left (a -b \right )\right ) \left (a +b \right )\right )}{8 \left (a -b \right )^{3} \left (a +b \right )^{3} b^{3} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(471\) |
| norman | \(\frac {\frac {\left (-4 a^{4}-16 a^{2} b^{2}+16 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\left (-4 a^{4}-16 a^{2} b^{2}+16 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\left (8 a^{4}+7 a^{2} b^{2}-11 b^{4}\right ) a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (2 a^{4}-10 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (2 a^{4}-10 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (-6 a^{4}+10 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (-6 a^{4}+10 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a \left (8 a^{4}-27 a^{2} b^{2}+15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a \left (8 a^{4}-27 a^{2} b^{2}+15 b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (8 a^{4}-59 a^{2} b^{2}+47 b^{4}\right ) a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (8 a^{4}-59 a^{2} b^{2}+47 b^{4}\right ) a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (8 a^{4}-9 a^{2} b^{2}+5 b^{4}\right ) a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\left (8 a^{4}-9 a^{2} b^{2}+5 b^{4}\right ) a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) 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 )^{4}}+\frac {\left (a^{2}+3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3} d}+\frac {\left (35 a^{2}-57 a b +24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {\left (35 a^{2}+57 a b +24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {a^{8} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{3} d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(955\) |
| risch | \(\text {Expression too large to display}\) | \(1099\) |
1/d*(1/b^2*(-1/2*sin(d*x+c)^2*b+a*sin(d*x+c))-1/2/(8*a-8*b)/(1+sin(d*x+c)) ^2-1/16*(-13*a+11*b)/(a-b)^2/(1+sin(d*x+c))+1/16*(35*a^2-57*a*b+24*b^2)/(a -b)^3*ln(1+sin(d*x+c))+1/2/(8*a+8*b)/(sin(d*x+c)-1)^2-1/16*(-13*a-11*b)/(a +b)^2/(sin(d*x+c)-1)+1/16/(a+b)^3*(-35*a^2-57*a*b-24*b^2)*ln(sin(d*x+c)-1) -1/b^3*a^8/(a+b)^3/(a-b)^3*ln(a+b*sin(d*x+c)))
Time = 0.67 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.79 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {16 \, a^{8} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, a^{4} b^{4} - 8 \, a^{2} b^{6} + 4 \, b^{8} - 8 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (d x + c\right )^{6} - {\left (35 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 42 \, a^{3} b^{5} - 64 \, a^{2} b^{6} + 15 \, a b^{7} + 24 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (35 \, a^{5} b^{3} - 48 \, a^{4} b^{4} - 42 \, a^{3} b^{5} + 64 \, a^{2} b^{6} + 15 \, a b^{7} - 24 \, b^{8}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (4 \, a^{4} b^{4} - 7 \, a^{2} b^{6} + 3 \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (2 \, a^{5} b^{3} - 4 \, a^{3} b^{5} + 2 \, a b^{7} + 8 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )^{4} - {\left (13 \, a^{5} b^{3} - 22 \, a^{3} b^{5} + 9 \, a b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{4}} \]
-1/16*(16*a^8*cos(d*x + c)^4*log(b*sin(d*x + c) + a) + 4*a^4*b^4 - 8*a^2*b ^6 + 4*b^8 - 8*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(d*x + c)^6 - (3 5*a^5*b^3 + 48*a^4*b^4 - 42*a^3*b^5 - 64*a^2*b^6 + 15*a*b^7 + 24*b^8)*cos( d*x + c)^4*log(sin(d*x + c) + 1) + (35*a^5*b^3 - 48*a^4*b^4 - 42*a^3*b^5 + 64*a^2*b^6 + 15*a*b^7 - 24*b^8)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 4 *(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(d*x + c)^4 - 8*(4*a^4*b^4 - 7 *a^2*b^6 + 3*b^8)*cos(d*x + c)^2 - 2*(2*a^5*b^3 - 4*a^3*b^5 + 2*a*b^7 + 8* (a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*cos(d*x + c)^4 - (13*a^5*b^3 - 22* a^3*b^5 + 9*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))/((a^6*b^3 - 3*a^4*b^5 + 3 *a^2*b^7 - b^9)*d*cos(d*x + c)^4)
Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, a^{8} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}} - \frac {{\left (35 \, a^{2} - 57 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (35 \, a^{2} + 57 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left ({\left (13 \, a^{3} - 9 \, a b^{2}\right )} \sin \left (d x + c\right )^{3} + 14 \, a^{2} b - 10 \, b^{3} - 4 \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} \sin \left (d x + c\right )^{2} - {\left (11 \, a^{3} - 7 \, a b^{2}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}} + \frac {8 \, {\left (b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{b^{2}}}{16 \, d} \]
-1/16*(16*a^8*log(b*sin(d*x + c) + a)/(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b ^9) - (35*a^2 - 57*a*b + 24*b^2)*log(sin(d*x + c) + 1)/(a^3 - 3*a^2*b + 3* a*b^2 - b^3) + (35*a^2 + 57*a*b + 24*b^2)*log(sin(d*x + c) - 1)/(a^3 + 3*a ^2*b + 3*a*b^2 + b^3) - 2*((13*a^3 - 9*a*b^2)*sin(d*x + c)^3 + 14*a^2*b - 10*b^3 - 4*(4*a^2*b - 3*b^3)*sin(d*x + c)^2 - (11*a^3 - 7*a*b^2)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*( a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c)^2) + 8*(b*sin(d*x + c)^2 - 2*a*sin(d*x + c))/b^2)/d
Time = 0.44 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.68 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, a^{8} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}} - \frac {{\left (35 \, a^{2} - 57 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (35 \, a^{2} + 57 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {8 \, {\left (b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{b^{2}} + \frac {2 \, {\left (36 \, a^{4} b \sin \left (d x + c\right )^{4} - 48 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} + 18 \, b^{5} \sin \left (d x + c\right )^{4} - 13 \, a^{5} \sin \left (d x + c\right )^{3} + 22 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} - 9 \, a b^{4} \sin \left (d x + c\right )^{3} - 56 \, a^{4} b \sin \left (d x + c\right )^{2} + 68 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} - 24 \, b^{5} \sin \left (d x + c\right )^{2} + 11 \, a^{5} \sin \left (d x + c\right ) - 18 \, a^{3} b^{2} \sin \left (d x + c\right ) + 7 \, a b^{4} \sin \left (d x + c\right ) + 22 \, a^{4} b - 24 \, a^{2} b^{3} + 8 \, b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
-1/16*(16*a^8*log(abs(b*sin(d*x + c) + a))/(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^ 7 - b^9) - (35*a^2 - 57*a*b + 24*b^2)*log(abs(sin(d*x + c) + 1))/(a^3 - 3* a^2*b + 3*a*b^2 - b^3) + (35*a^2 + 57*a*b + 24*b^2)*log(abs(sin(d*x + c) - 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 8*(b*sin(d*x + c)^2 - 2*a*sin(d*x + c))/b^2 + 2*(36*a^4*b*sin(d*x + c)^4 - 48*a^2*b^3*sin(d*x + c)^4 + 18*b^5 *sin(d*x + c)^4 - 13*a^5*sin(d*x + c)^3 + 22*a^3*b^2*sin(d*x + c)^3 - 9*a* b^4*sin(d*x + c)^3 - 56*a^4*b*sin(d*x + c)^2 + 68*a^2*b^3*sin(d*x + c)^2 - 24*b^5*sin(d*x + c)^2 + 11*a^5*sin(d*x + c) - 18*a^3*b^2*sin(d*x + c) + 7 *a*b^4*sin(d*x + c) + 22*a^4*b - 24*a^2*b^3 + 8*b^5)/((a^6 - 3*a^4*b^2 + 3 *a^2*b^4 - b^6)*(sin(d*x + c)^2 - 1)^2))/d
Time = 14.22 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.36 \[ \int \frac {\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {b^2}{4\,{\left (a-b\right )}^3}+\frac {13\,b}{8\,{\left (a-b\right )}^2}+\frac {35}{8\,\left (a-b\right )}\right )}{d}-\frac {\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-3\,a^4+a^2\,b^2+b^4\right )}{b\,{\left (a^2-b^2\right )}^2}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^4-5\,a^2\,b^2+3\,b^4\right )}{b\,{\left (a^2-b^2\right )}^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (8\,a^5+7\,a^3\,b^2-11\,a\,b^4\right )}{2\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (8\,a^5+7\,a^3\,b^2-11\,a\,b^4\right )}{2\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (8\,a^5-27\,a^3\,b^2+15\,a\,b^4\right )}{4\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (24\,a^5-45\,a^3\,b^2+25\,a\,b^4\right )}{4\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (24\,a^5-45\,a^3\,b^2+25\,a\,b^4\right )}{4\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^2-3\,b^2\right )}{b\,\left (a^2-b^2\right )}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (2\,a^2-3\,b^2\right )}{b\,\left (a^2-b^2\right )}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (a^4-5\,a^2\,b^2+3\,b^4\right )}{b\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^4-27\,a^2\,b^2+15\,b^4\right )}{4\,b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {35}{8\,\left (a+b\right )}-\frac {13\,b}{8\,{\left (a+b\right )}^2}+\frac {b^2}{4\,{\left (a+b\right )}^3}\right )}{d}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (a^2+3\,b^2\right )}{b^3\,d}+\frac {a^8\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{d\,\left (-a^6\,b^3+3\,a^4\,b^5-3\,a^2\,b^7+b^9\right )} \]
(log(tan(c/2 + (d*x)/2) + 1)*(b^2/(4*(a - b)^3) + (13*b)/(8*(a - b)^2) + 3 5/(8*(a - b))))/d - ((4*tan(c/2 + (d*x)/2)^6*(b^4 - 3*a^4 + a^2*b^2))/(b*( a^2 - b^2)^2) - (2*tan(c/2 + (d*x)/2)^2*(a^4 + 3*b^4 - 5*a^2*b^2))/(b*(a^2 - b^2)^2) + (tan(c/2 + (d*x)/2)^5*(8*a^5 - 11*a*b^4 + 7*a^3*b^2))/(2*b^2* (a^4 + b^4 - 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^7*(8*a^5 - 11*a*b^4 + 7*a^3 *b^2))/(2*b^2*(a^4 + b^4 - 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^11*(15*a*b^4 + 8*a^5 - 27*a^3*b^2))/(4*b^2*(a^4 + b^4 - 2*a^2*b^2)) - (tan(c/2 + (d*x)/ 2)^3*(25*a*b^4 + 24*a^5 - 45*a^3*b^2))/(4*b^2*(a^4 + b^4 - 2*a^2*b^2)) - ( tan(c/2 + (d*x)/2)^9*(25*a*b^4 + 24*a^5 - 45*a^3*b^2))/(4*b^2*(a^4 + b^4 - 2*a^2*b^2)) + (4*tan(c/2 + (d*x)/2)^4*(2*a^2 - 3*b^2))/(b*(a^2 - b^2)) + (4*tan(c/2 + (d*x)/2)^8*(2*a^2 - 3*b^2))/(b*(a^2 - b^2)) - (2*tan(c/2 + (d *x)/2)^10*(a^4 + 3*b^4 - 5*a^2*b^2))/(b*(a^4 + b^4 - 2*a^2*b^2)) + (a*tan( c/2 + (d*x)/2)*(8*a^4 + 15*b^4 - 27*a^2*b^2))/(4*b^2*(a^4 + b^4 - 2*a^2*b^ 2)))/(d*(2*tan(c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x) /2)^6 + tan(c/2 + (d*x)/2)^8 + 2*tan(c/2 + (d*x)/2)^10 - tan(c/2 + (d*x)/2 )^12 - 1)) - (log(tan(c/2 + (d*x)/2) - 1)*(35/(8*(a + b)) - (13*b)/(8*(a + b)^2) + b^2/(4*(a + b)^3)))/d + (log(tan(c/2 + (d*x)/2)^2 + 1)*(a^2 + 3*b ^2))/(b^3*d) + (a^8*log(a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^ 2))/(d*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))