Integrand size = 21, antiderivative size = 261 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) x}{8 a^6}-\frac {2 \sqrt {a-b} b \sqrt {a+b} \left (2 a^2-5 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d}+\frac {b \left (11 a^2-15 b^2\right ) \sin (c+d x)}{3 a^5 d}-\frac {\left (13 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^4 d}+\frac {\left (3 a^2-5 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^3 b d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}-\frac {\left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{a^2 b d (b+a \cos (c+d x))} \]
1/8*(3*a^4-36*a^2*b^2+40*b^4)*x/a^6+1/3*b*(11*a^2-15*b^2)*sin(d*x+c)/a^5/d -1/8*(13*a^2-20*b^2)*cos(d*x+c)*sin(d*x+c)/a^4/d+1/3*(3*a^2-5*b^2)*cos(d*x +c)^2*sin(d*x+c)/a^3/b/d+1/4*cos(d*x+c)^3*sin(d*x+c)/a^2/d-(a^2-b^2)*cos(d *x+c)^3*sin(d*x+c)/a^2/b/d/(b+a*cos(d*x+c))-2*b*(2*a^2-5*b^2)*arctanh((a-b )^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))*(a-b)^(1/2)*(a+b)^(1/2)/a^6/d
Time = 3.06 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.08 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {384 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {72 a^4 b c-864 a^2 b^3 c+960 b^5 c+72 a^4 b d x-864 a^2 b^3 d x+960 b^5 d x+24 a \left (3 a^4-36 a^2 b^2+40 b^4\right ) (c+d x) \cos (c+d x)-24 a \left (a^4-31 a^2 b^2+40 b^4\right ) \sin (c+d x)+176 a^4 b \sin (2 (c+d x))-240 a^2 b^3 \sin (2 (c+d x))-21 a^5 \sin (3 (c+d x))+40 a^3 b^2 \sin (3 (c+d x))-10 a^4 b \sin (4 (c+d x))+3 a^5 \sin (5 (c+d x))}{b+a \cos (c+d x)}}{192 a^6 d} \]
((384*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sq rt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (72*a^4*b*c - 864*a^2*b^3*c + 960*b^5*c + 72*a^4*b*d*x - 864*a^2*b^3*d*x + 960*b^5*d*x + 24*a*(3*a^4 - 36*a^2*b^2 + 40*b^4)*(c + d*x)*Cos[c + d*x] - 24*a*(a^4 - 31*a^2*b^2 + 40*b^4)*Sin[c + d*x] + 176*a^4*b*Sin[2*(c + d*x)] - 240*a^2*b^3*Sin[2*(c + d*x)] - 21*a^ 5*Sin[3*(c + d*x)] + 40*a^3*b^2*Sin[3*(c + d*x)] - 10*a^4*b*Sin[4*(c + d*x )] + 3*a^5*Sin[5*(c + d*x)])/(b + a*Cos[c + d*x]))/(192*a^6*d)
Time = 1.73 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.15, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4360, 3042, 3371, 3042, 3528, 25, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^4}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^2(c+d x)}{(-a \cos (c+d x)-b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \cos \left (c+d x+\frac {\pi }{2}\right )^4}{\left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}dx\) |
| \(\Big \downarrow \) 3371 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (8 a^2+b \cos (c+d x) a-15 b^2-4 \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)\right )}{b+a \cos (c+d x)}dx}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (8 a^2+b \sin \left (c+d x+\frac {\pi }{2}\right ) a-15 b^2-4 \left (3 a^2-5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\int -\frac {\cos (c+d x) \left (5 a \cos (c+d x) b^2-3 \left (13 a^2-20 b^2\right ) \cos ^2(c+d x) b+8 \left (3 a^2-5 b^2\right ) b\right )}{b+a \cos (c+d x)}dx}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (5 a \cos (c+d x) b^2-3 \left (13 a^2-20 b^2\right ) \cos ^2(c+d x) b+8 \left (3 a^2-5 b^2\right ) b\right )}{b+a \cos (c+d x)}dx}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (5 a \sin \left (c+d x+\frac {\pi }{2}\right ) b^2-3 \left (13 a^2-20 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b+8 \left (3 a^2-5 b^2\right ) b\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {-8 \left (11 a^2-15 b^2\right ) \cos ^2(c+d x) b^2+3 \left (13 a^2-20 b^2\right ) b^2-a \left (9 a^2-20 b^2\right ) \cos (c+d x) b}{b+a \cos (c+d x)}dx}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-8 \left (11 a^2-15 b^2\right ) \cos ^2(c+d x) b^2+3 \left (13 a^2-20 b^2\right ) b^2-a \left (9 a^2-20 b^2\right ) \cos (c+d x) b}{b+a \cos (c+d x)}dx}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-8 \left (11 a^2-15 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+3 \left (13 a^2-20 b^2\right ) b^2-a \left (9 a^2-20 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {3 \left (a b^2 \left (13 a^2-20 b^2\right )-b \left (3 a^4-36 b^2 a^2+40 b^4\right ) \cos (c+d x)\right )}{b+a \cos (c+d x)}dx}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {3 \int \frac {a b^2 \left (13 a^2-20 b^2\right )-b \left (3 a^4-36 b^2 a^2+40 b^4\right ) \cos (c+d x)}{b+a \cos (c+d x)}dx}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {3 \int \frac {a b^2 \left (13 a^2-20 b^2\right )-b \left (3 a^4-36 b^2 a^2+40 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {3 \left (\frac {8 b^2 \left (2 a^4-7 a^2 b^2+5 b^4\right ) \int \frac {1}{b+a \cos (c+d x)}dx}{a}-\frac {b x \left (3 a^4-36 a^2 b^2+40 b^4\right )}{a}\right )}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {3 \left (\frac {8 b^2 \left (2 a^4-7 a^2 b^2+5 b^4\right ) \int \frac {1}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {b x \left (3 a^4-36 a^2 b^2+40 b^4\right )}{a}\right )}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {3 \left (\frac {16 b^2 \left (2 a^4-7 a^2 b^2+5 b^4\right ) \int \frac {1}{-\left ((a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}-\frac {b x \left (3 a^4-36 a^2 b^2+40 b^4\right )}{a}\right )}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}}{4 a^2 b}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac {-\frac {4 \left (3 a^2-5 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {-\frac {3 b \left (13 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {3 \left (\frac {16 b^2 \left (2 a^4-7 a^2 b^2+5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}-\frac {b x \left (3 a^4-36 a^2 b^2+40 b^4\right )}{a}\right )}{a}-\frac {8 b^2 \left (11 a^2-15 b^2\right ) \sin (c+d x)}{a d}}{2 a}}{3 a}}{4 a^2 b}\) |
(Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) - ((a^2 - b^2)*Cos[c + d*x]^3*Sin[ c + d*x])/(a^2*b*d*(b + a*Cos[c + d*x])) - ((-4*(3*a^2 - 5*b^2)*Cos[c + d* x]^2*Sin[c + d*x])/(3*a*d) - ((-3*b*(13*a^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((3*(-((b*(3*a^4 - 36*a^2*b^2 + 40*b^4)*x)/a) + (16*b^2*( 2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a - (8*b^2*(11*a^2 - 15*b^2)*Sin[c + d*x])/(a*d))/(2*a))/(3*a))/(4*a^2*b)
3.3.17.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1))), x] + (-Simp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*((d*Sin[e + f* x])^(n + 1)/(b^2*d*f*(m + n + 4))), x] - Simp[1/(a*b^2*(m + 1)*(m + n + 4)) Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2 )*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; Fre eQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && Lt Q[m, -1] && !LtQ[n, -1] && NeQ[m + n + 4, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.64 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (a -b \right ) \left (a +b \right ) b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-5 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+2 a^{3} b -\frac {3}{2} a^{2} b^{2}-4 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {26}{3} a^{3} b -\frac {3}{2} a^{2} b^{2}-12 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+\frac {3}{2} a^{2} b^{2}+\frac {26}{3} a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a^{3} b -4 a \,b^{3}-\frac {3}{8} a^{4}+\frac {3}{2} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4}-36 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{6}}}{d}\) | \(325\) |
| default | \(\frac {\frac {2 \left (a -b \right ) \left (a +b \right ) b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-5 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+2 a^{3} b -\frac {3}{2} a^{2} b^{2}-4 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {26}{3} a^{3} b -\frac {3}{2} a^{2} b^{2}-12 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+\frac {3}{2} a^{2} b^{2}+\frac {26}{3} a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a^{3} b -4 a \,b^{3}-\frac {3}{8} a^{4}+\frac {3}{2} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4}-36 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{6}}}{d}\) | \(325\) |
| risch | \(\frac {3 x}{8 a^{2}}-\frac {9 x \,b^{2}}{2 a^{4}}+\frac {5 x \,b^{4}}{a^{6}}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{2} d}-\frac {5 i b \,{\mathrm e}^{i \left (d x +c \right )}}{4 a^{3} d}-\frac {2 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{a^{5} d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{2} d}+\frac {2 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{a^{5} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{8 a^{4} d}+\frac {5 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{4 a^{3} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{8 a^{4} d}+\frac {2 i \left (a^{2}-b^{2}\right ) b^{2} \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{6} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {2 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{4}}-\frac {5 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{6}}-\frac {2 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{4}}+\frac {5 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{6}}+\frac {\sin \left (4 d x +4 c \right )}{32 a^{2} d}-\frac {b \sin \left (3 d x +3 c \right )}{6 d \,a^{3}}\) | \(494\) |
1/d*(2*(a-b)*(a+b)*b/a^6*(-a*b*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a- tan(1/2*d*x+1/2*c)^2*b-a-b)-(2*a^2-5*b^2)/((a-b)*(a+b))^(1/2)*arctanh((a-b )*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))+2/a^6*(((3/8*a^4+2*a^3*b-3/2*a^ 2*b^2-4*a*b^3)*tan(1/2*d*x+1/2*c)^7+(26/3*a^3*b-3/2*a^2*b^2-12*a*b^3+11/8* a^4)*tan(1/2*d*x+1/2*c)^5+(-11/8*a^4+3/2*a^2*b^2+26/3*a^3*b-12*a*b^3)*tan( 1/2*d*x+1/2*c)^3+(2*a^3*b-4*a*b^3-3/8*a^4+3/2*a^2*b^2)*tan(1/2*d*x+1/2*c)) /(1+tan(1/2*d*x+1/2*c)^2)^4+1/8*(3*a^4-36*a^2*b^2+40*b^4)*arctan(tan(1/2*d *x+1/2*c))))
Time = 0.34 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.23 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\left [\frac {3 \, {\left (3 \, a^{5} - 36 \, a^{3} b^{2} + 40 \, a b^{4}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (3 \, a^{4} b - 36 \, a^{2} b^{3} + 40 \, b^{5}\right )} d x - 12 \, {\left (2 \, a^{2} b^{2} - 5 \, b^{4} + {\left (2 \, a^{3} b - 5 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (6 \, a^{5} \cos \left (d x + c\right )^{4} - 10 \, a^{4} b \cos \left (d x + c\right )^{3} + 88 \, a^{3} b^{2} - 120 \, a b^{4} - 5 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (49 \, a^{4} b - 60 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} d \cos \left (d x + c\right ) + a^{6} b d\right )}}, \frac {3 \, {\left (3 \, a^{5} - 36 \, a^{3} b^{2} + 40 \, a b^{4}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (3 \, a^{4} b - 36 \, a^{2} b^{3} + 40 \, b^{5}\right )} d x - 24 \, {\left (2 \, a^{2} b^{2} - 5 \, b^{4} + {\left (2 \, a^{3} b - 5 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (6 \, a^{5} \cos \left (d x + c\right )^{4} - 10 \, a^{4} b \cos \left (d x + c\right )^{3} + 88 \, a^{3} b^{2} - 120 \, a b^{4} - 5 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (49 \, a^{4} b - 60 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} d \cos \left (d x + c\right ) + a^{6} b d\right )}}\right ] \]
[1/24*(3*(3*a^5 - 36*a^3*b^2 + 40*a*b^4)*d*x*cos(d*x + c) + 3*(3*a^4*b - 3 6*a^2*b^3 + 40*b^5)*d*x - 12*(2*a^2*b^2 - 5*b^4 + (2*a^3*b - 5*a*b^3)*cos( d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2 )/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (6*a^5*cos(d*x + c)^4 - 10*a^4*b*cos(d*x + c)^3 + 88*a^3*b^2 - 120*a*b^4 - 5*(3*a^5 - 4*a^3*b^2 )*cos(d*x + c)^2 + (49*a^4*b - 60*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/(a^ 7*d*cos(d*x + c) + a^6*b*d), 1/24*(3*(3*a^5 - 36*a^3*b^2 + 40*a*b^4)*d*x*c os(d*x + c) + 3*(3*a^4*b - 36*a^2*b^3 + 40*b^5)*d*x - 24*(2*a^2*b^2 - 5*b^ 4 + (2*a^3*b - 5*a*b^3)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (6*a^5*cos(d*x + c)^4 - 10*a^4*b*cos(d*x + c)^3 + 88*a^3*b^2 - 120*a*b^4 - 5*(3*a^5 - 4*a^3 *b^2)*cos(d*x + c)^2 + (49*a^4*b - 60*a^2*b^3)*cos(d*x + c))*sin(d*x + c)) /(a^7*d*cos(d*x + c) + a^6*b*d)]
\[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\sin ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.34 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.85 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (3 \, a^{4} - 36 \, a^{2} b^{2} + 40 \, b^{4}\right )} {\left (d x + c\right )}}{a^{6}} - \frac {48 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} + 5 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{6}} - \frac {48 \, {\left (a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} a^{5}} + \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 208 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 208 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{5}}}{24 \, d} \]
1/24*(3*(3*a^4 - 36*a^2*b^2 + 40*b^4)*(d*x + c)/a^6 - 48*(2*a^4*b - 7*a^2* b^3 + 5*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-( a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(sqrt( -a^2 + b^2)*a^6) - 48*(a^2*b^2*tan(1/2*d*x + 1/2*c) - b^4*tan(1/2*d*x + 1/ 2*c))/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)*a^5) + 2*(9*a^3*tan(1/2*d*x + 1/2*c)^7 + 48*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 36*a *b^2*tan(1/2*d*x + 1/2*c)^7 - 96*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*a^3*tan(1 /2*d*x + 1/2*c)^5 + 208*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 36*a*b^2*tan(1/2*d* x + 1/2*c)^5 - 288*b^3*tan(1/2*d*x + 1/2*c)^5 - 33*a^3*tan(1/2*d*x + 1/2*c )^3 + 208*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 36*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*b^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^3*tan(1/2*d*x + 1/2*c) + 48*a^2*b*ta n(1/2*d*x + 1/2*c) + 36*a*b^2*tan(1/2*d*x + 1/2*c) - 96*b^3*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^5))/d
Time = 15.42 (sec) , antiderivative size = 2804, normalized size of antiderivative = 10.74 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^5*(33*a^4 - 360*b^4 + 244*a^2*b^2))/(6*a^5) - (tan(c/ 2 + (d*x)/2)^3*(60*a*b^3 - 59*a^3*b + 12*a^4 + 240*b^4 - 176*a^2*b^2))/(6* a^5) - (tan(c/2 + (d*x)/2)^7*(59*a^3*b - 60*a*b^3 + 12*a^4 + 240*b^4 - 176 *a^2*b^2))/(6*a^5) + (tan(c/2 + (d*x)/2)^9*(a - b)*(20*a*b^2 - 16*a^2*b - 3*a^3 + 40*b^3))/(4*a^5) + (tan(c/2 + (d*x)/2)*(a + b)*(20*a*b^2 + 16*a^2* b - 3*a^3 - 40*b^3))/(4*a^5))/(d*(a + b - tan(c/2 + (d*x)/2)^10*(a - b) + tan(c/2 + (d*x)/2)^2*(3*a + 5*b) + tan(c/2 + (d*x)/2)^4*(2*a + 10*b) - tan (c/2 + (d*x)/2)^8*(3*a - 5*b) - tan(c/2 + (d*x)/2)^6*(2*a - 10*b))) + (ata n(((((((76*a^17*b - 12*a^18 - 160*a^12*b^6 + 240*a^13*b^5 + 144*a^14*b^4 - 316*a^15*b^3 + 28*a^16*b^2)/a^15 - (tan(c/2 + (d*x)/2)*(a^4*3i + b^4*40i - a^2*b^2*36i)*(128*a^14*b + 128*a^12*b^3 - 256*a^13*b^2))/(16*a^16))*(a^4 *3i + b^4*40i - a^2*b^2*36i))/(8*a^6) + (tan(c/2 + (d*x)/2)*(6400*a*b^10 - 27*a^10*b + 9*a^11 - 3200*b^11 + 2560*a^2*b^9 - 11520*a^3*b^8 + 2688*a^4* b^7 + 6144*a^5*b^6 - 2600*a^6*b^5 - 904*a^7*b^4 + 383*a^8*b^3 + 67*a^9*b^2 ))/(2*a^10))*(a^4*3i + b^4*40i - a^2*b^2*36i)*1i)/(8*a^6) - (((((76*a^17*b - 12*a^18 - 160*a^12*b^6 + 240*a^13*b^5 + 144*a^14*b^4 - 316*a^15*b^3 + 2 8*a^16*b^2)/a^15 + (tan(c/2 + (d*x)/2)*(a^4*3i + b^4*40i - a^2*b^2*36i)*(1 28*a^14*b + 128*a^12*b^3 - 256*a^13*b^2))/(16*a^16))*(a^4*3i + b^4*40i - a ^2*b^2*36i))/(8*a^6) - (tan(c/2 + (d*x)/2)*(6400*a*b^10 - 27*a^10*b + 9*a^ 11 - 3200*b^11 + 2560*a^2*b^9 - 11520*a^3*b^8 + 2688*a^4*b^7 + 6144*a^5...