Integrand size = 21, antiderivative size = 178 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {3}{8} \left (a^2-4 b^2\right ) x+\frac {2 a b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \left (28 a^2+b^2\right ) \sin (c+d x)}{6 a d}-\frac {\left (39 a^2+2 b^2\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {\left (12 a^2+b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{12 a b d}+\frac {(b+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {(b+a \cos (c+d x))^3 \tan (c+d x)}{b d} \]
3/8*(a^2-4*b^2)*x+2*a*b*arctanh(sin(d*x+c))/d-1/6*b*(28*a^2+b^2)*sin(d*x+c )/a/d-1/24*(39*a^2+2*b^2)*cos(d*x+c)*sin(d*x+c)/d-1/12*(12*a^2+b^2)*(b+a*c os(d*x+c))^2*sin(d*x+c)/a/b/d+1/4*(b+a*cos(d*x+c))^3*sin(d*x+c)/a/d+(b+a*c os(d*x+c))^3*tan(d*x+c)/b/d
Time = 1.46 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.88 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {12 \left (3 \left (a^2-4 b^2\right ) (c+d x)-16 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+16 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-208 a b \sin (c+d x)+\left (-6 \left (3 a^2-4 b^2\right ) \cos (2 (c+d x))+16 a b \cos (3 (c+d x))+3 \left (-7 a^2+40 b^2+a^2 \cos (4 (c+d x))\right )\right ) \tan (c+d x)}{96 d} \]
(12*(3*(a^2 - 4*b^2)*(c + d*x) - 16*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d* x)/2]] + 16*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 208*a*b*Sin[c + d*x] + (-6*(3*a^2 - 4*b^2)*Cos[2*(c + d*x)] + 16*a*b*Cos[3*(c + d*x)] + 3*(-7*a^2 + 40*b^2 + a^2*Cos[4*(c + d*x)]))*Tan[c + d*x])/(96*d)
Time = 1.43 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 4360, 3042, 3373, 25, 3042, 3528, 3042, 3512, 3042, 3502, 27, 3042, 3214, 3042, 4257}
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.
| \(\displaystyle \int \sin ^4(c+d x) (a+b \sec (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^4 \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^2(c+d x) \tan ^2(c+d x) (-a \cos (c+d x)-b)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^4 \left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3373 |
| \(\displaystyle -\frac {\int -(b+a \cos (c+d x))^2 \left (8 a^2-5 b \cos (c+d x) a-\left (12 a^2+b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (b+a \cos (c+d x))^2 \left (8 a^2-5 b \cos (c+d x) a-\left (12 a^2+b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (8 a^2-5 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-12 a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{3} \int (b+a \cos (c+d x)) \left (24 b a^2-17 b^2 \cos (c+d x) a-b \left (39 a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (24 b a^2-17 b^2 \sin \left (c+d x+\frac {\pi }{2}\right ) a-b \left (39 a^2+2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \left (48 a^2 b^2-4 \left (28 a^2+b^2\right ) \cos ^2(c+d x) b^2+9 a \left (a^2-4 b^2\right ) \cos (c+d x) b\right ) \sec (c+d x)dx-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {48 a^2 b^2-4 \left (28 a^2+b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+9 a \left (a^2-4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (16 a^2 b^2+3 a \left (a^2-4 b^2\right ) \cos (c+d x) b\right ) \sec (c+d x)dx-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (16 a^2 b^2+3 a \left (a^2-4 b^2\right ) \cos (c+d x) b\right ) \sec (c+d x)dx-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {16 a^2 b^2+3 a \left (a^2-4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (16 a^2 b^2 \int \sec (c+d x)dx+3 a b x \left (a^2-4 b^2\right )\right )-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (16 a^2 b^2 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+3 a b x \left (a^2-4 b^2\right )\right )-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {16 a^2 b^2 \text {arctanh}(\sin (c+d x))}{d}+3 a b x \left (a^2-4 b^2\right )\right )-\frac {4 b^2 \left (28 a^2+b^2\right ) \sin (c+d x)}{d}\right )-\frac {a b \left (39 a^2+2 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\left (12 a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}}{4 a b}+\frac {\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac {\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}\) |
((b + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d) + (-1/3*((12*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/d + (-1/2*(a*b*(39*a^2 + 2*b^2)*Cos[c + d *x]*Sin[c + d*x])/d + (3*(3*a*b*(a^2 - 4*b^2)*x + (16*a^2*b^2*ArcTanh[Sin[ c + d*x]])/d) - (4*b^2*(28*a^2 + b^2)*Sin[c + d*x])/d)/2)/3)/(4*a*b) + ((b + a*Cos[c + d*x])^3*Tan[c + d*x])/(b*d)
3.2.80.3.1 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_.)*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[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 2)/(b*d ^2*f*(m + n + 4))), x] + Simp[1/(a*b*d*(n + 1)*(m + n + 4)) Int[(a + b*Si n[e + f*x])^m*(d*Sin[e + f*x])^(n + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 3)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2* (m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && ! 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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 = 1.80 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+b^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(133\) |
| default | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+b^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(133\) |
| parts | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {b^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}+\frac {2 a b \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(138\) |
| parallelrisch | \(\frac {72 a^{2} x d \cos \left (d x +c \right )-288 b^{2} d x \cos \left (d x +c \right )+384 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )-384 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-24 a^{2} \sin \left (d x +c \right )+216 b^{2} \sin \left (d x +c \right )-21 a^{2} \sin \left (3 d x +3 c \right )+24 b^{2} \sin \left (3 d x +3 c \right )+3 a^{2} \sin \left (5 d x +5 c \right )-224 a b \sin \left (2 d x +2 c \right )+16 a b \sin \left (4 d x +4 c \right )}{192 d \cos \left (d x +c \right )}\) | \(175\) |
| risch | \(\frac {3 a^{2} x}{8}-\frac {3 b^{2} x}{2}+\frac {i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{8 d}+\frac {5 i a b \,{\mathrm e}^{i \left (d x +c \right )}}{4 d}-\frac {5 i a b \,{\mathrm e}^{-i \left (d x +c \right )}}{4 d}-\frac {i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{8 d}+\frac {2 i b^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a b \sin \left (3 d x +3 c \right )}{6 d}\) | \(215\) |
| norman | \(\frac {\left (-\frac {3 a^{2}}{8}+\frac {3 b^{2}}{2}\right ) x +\left (-\frac {9 a^{2}}{8}+\frac {9 b^{2}}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {3 a^{2}}{4}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {3 a^{2}}{4}-3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {3 a^{2}}{8}-\frac {3 b^{2}}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {9 a^{2}}{8}-\frac {9 b^{2}}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (11 a^{2}+20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 d}+\frac {2 \left (3 a^{2}-20 a b -12 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {\left (3 a^{2}-16 a b -12 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {\left (3 a^{2}+16 a b -12 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {2 \left (3 a^{2}+20 a b -12 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(354\) |
1/d*(a^2*(-1/4*(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)+3/8*d*x+3/8*c)+2*a *b*(-1/3*sin(d*x+c)^3-sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+b^2*(sin(d*x+c )^5/cos(d*x+c)+(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)-3/2*d*x-3/2*c))
Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {9 \, {\left (a^{2} - 4 \, b^{2}\right )} d x \cos \left (d x + c\right ) + 24 \, a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 24 \, a b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a b \cos \left (d x + c\right )^{3} - 64 \, a b \cos \left (d x + c\right ) - 3 \, {\left (5 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 24 \, b^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
1/24*(9*(a^2 - 4*b^2)*d*x*cos(d*x + c) + 24*a*b*cos(d*x + c)*log(sin(d*x + c) + 1) - 24*a*b*cos(d*x + c)*log(-sin(d*x + c) + 1) + (6*a^2*cos(d*x + c )^4 + 16*a*b*cos(d*x + c)^3 - 64*a*b*cos(d*x + c) - 3*(5*a^2 - 4*b^2)*cos( d*x + c)^2 + 24*b^2)*sin(d*x + c))/(d*cos(d*x + c))
\[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \sin ^{4}{\left (c + d x \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 32 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a b - 48 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} b^{2}}{96 \, d} \]
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) - 8*sin(2*d*x + 2*c))*a^2 - 32*( 2*sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1) + 6*s in(d*x + c))*a*b - 48*(3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2 *tan(d*x + c))*b^2)/d
Time = 0.37 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.60 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {48 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 48 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 9 \, {\left (a^{2} - 4 \, b^{2}\right )} {\left (d x + c\right )} - \frac {48 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 208 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 33 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 208 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(48*a*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 48*a*b*log(abs(tan(1/2*d *x + 1/2*c) - 1)) + 9*(a^2 - 4*b^2)*(d*x + c) - 48*b^2*tan(1/2*d*x + 1/2*c )/(tan(1/2*d*x + 1/2*c)^2 - 1) + 2*(9*a^2*tan(1/2*d*x + 1/2*c)^7 - 48*a*b* tan(1/2*d*x + 1/2*c)^7 - 12*b^2*tan(1/2*d*x + 1/2*c)^7 + 33*a^2*tan(1/2*d* x + 1/2*c)^5 - 208*a*b*tan(1/2*d*x + 1/2*c)^5 - 12*b^2*tan(1/2*d*x + 1/2*c )^5 - 33*a^2*tan(1/2*d*x + 1/2*c)^3 - 208*a*b*tan(1/2*d*x + 1/2*c)^3 + 12* b^2*tan(1/2*d*x + 1/2*c)^3 - 9*a^2*tan(1/2*d*x + 1/2*c) - 48*a*b*tan(1/2*d *x + 1/2*c) + 12*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4) /d
Time = 14.82 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.16 \[ \int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {3\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}-\frac {3\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {8\,a\,b\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {5\,a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]
(3*a^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(4*d) - (3*b^2*atan(si n(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (a^2*cos(c + d*x)^3*sin(c + d*x) )/(4*d) + (b^2*sin(c + d*x))/(d*cos(c + d*x)) - (8*a*b*sin(c + d*x))/(3*d) + (4*a*b*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (5*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (b^2*cos(c + d*x)*sin(c + d*x))/(2*d) + (2*a*b *cos(c + d*x)^2*sin(c + d*x))/(3*d)