Integrand size = 33, antiderivative size = 163 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=3 a b^2 C x+\frac {b \left (2 A b^2+3 a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d}+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {A b (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
3*a*b^2*C*x+1/2*b*(2*A*b^2+3*a^2*(A+2*C))*arctanh(sin(d*x+c))/d-1/6*b^3*(5 *A-6*C)*sin(d*x+c)/d+1/3*a*(3*A*b^2+a^2*(2*A+3*C))*tan(d*x+c)/d+1/2*A*b*(a +b*cos(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*A*(a+b*cos(d*x+c))^3*sec(d*x+ c)^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(163)=326\).
Time = 6.76 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.31 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {36 a b^2 C (c+d x)-6 b \left (2 A b^2+3 a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 b \left (2 A b^2+3 a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 A (a+9 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 a \left (9 A b^2+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {a^2 A (a+9 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a \left (9 A b^2+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+12 b^3 C \sin (c+d x)}{12 d} \]
(36*a*b^2*C*(c + d*x) - 6*b*(2*A*b^2 + 3*a^2*(A + 2*C))*Log[Cos[(c + d*x)/ 2] - Sin[(c + d*x)/2]] + 6*b*(2*A*b^2 + 3*a^2*(A + 2*C))*Log[Cos[(c + d*x) /2] + Sin[(c + d*x)/2]] + (a^2*A*(a + 9*b))/(Cos[(c + d*x)/2] - Sin[(c + d *x)/2])^2 + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2 ])^3 + (4*a*(9*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2 ] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin [(c + d*x)/2])^3 - (a^2*A*(a + 9*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) ^2 + (4*a*(9*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*b^3*C*Sin[c + d*x])/(12*d)
Time = 1.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 3527, 3042, 3526, 3042, 3510, 25, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{3} \int (a+b \cos (c+d x))^2 \left (-b (A-3 C) \cos ^2(c+d x)+a (2 A+3 C) \cos (c+d x)+3 A b\right ) \sec ^3(c+d x)dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int (a+b \cos (c+d x)) \left (-b^2 (5 A-6 C) \cos ^2(c+d x)+a b (5 A+12 C) \cos (c+d x)+2 \left (\frac {1}{2} (4 A+6 C) a^2+3 A b^2\right )\right ) \sec ^2(c+d x)dx+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b^2 (5 A-6 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a b (5 A+12 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (\frac {1}{2} (4 A+6 C) a^2+3 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\int -\left (\left (-\left ((5 A-6 C) \cos ^2(c+d x) b^3\right )+18 a C \cos (c+d x) b^2+3 \left (3 (A+2 C) a^2+2 A b^2\right ) b\right ) \sec (c+d x)\right )dx\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \left (-\left ((5 A-6 C) \cos ^2(c+d x) b^3\right )+18 a C \cos (c+d x) b^2+3 \left (3 (A+2 C) a^2+2 A b^2\right ) b\right ) \sec (c+d x)dx+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \frac {-\left ((5 A-6 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3\right )+18 a C \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+3 \left (3 (A+2 C) a^2+2 A b^2\right ) b}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (6 a C \cos (c+d x) b^2+\left (3 (A+2 C) a^2+2 A b^2\right ) b\right ) \sec (c+d x)dx+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (6 a C \cos (c+d x) b^2+\left (3 (A+2 C) a^2+2 A b^2\right ) b\right ) \sec (c+d x)dx+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {6 a C \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (3 (A+2 C) a^2+2 A b^2\right ) b}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (b \left (3 a^2 (A+2 C)+2 A b^2\right ) \int \sec (c+d x)dx+6 a b^2 C x\right )+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (b \left (3 a^2 (A+2 C)+2 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+6 a b^2 C x\right )+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {b \left (3 a^2 (A+2 C)+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+6 a b^2 C x\right )+\frac {2 a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{d}\right )+\frac {3 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
(A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(6*a*b^2*C*x + ( b*(2*A*b^2 + 3*a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/d) - (b^3*(5*A - 6*C) *Sin[c + d*x])/d + (2*a*(3*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/d)/2)/3
3.6.45.3.1 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[((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[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, 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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 8.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94
| method | result | size |
| parts | \(-\frac {A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 C \,a^{2} b \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 A a \,b^{2}+C \,a^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 C a \,b^{2} \left (d x +c \right )}{d}\) | \(153\) |
| derivativedivides | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A a \,b^{2} \tan \left (d x +c \right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(155\) |
| default | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A a \,b^{2} \tan \left (d x +c \right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(155\) |
| parallelrisch | \(\frac {-27 b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (\frac {2 A \,b^{2}}{3}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+27 b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (\frac {2 A \,b^{2}}{3}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+18 C a \,b^{2} d x \cos \left (3 d x +3 c \right )+\left (18 A a \,b^{2}+4 a^{3} \left (A +\frac {3 C}{2}\right )\right ) \sin \left (3 d x +3 c \right )+\left (18 A \,a^{2} b +6 C \,b^{3}\right ) \sin \left (2 d x +2 c \right )+3 C \sin \left (4 d x +4 c \right ) b^{3}+12 \left (\frac {9 C \,b^{2} d x \cos \left (d x +c \right )}{2}+\left (\frac {3 A \,b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right )\right ) \sin \left (d x +c \right )\right ) a}{6 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(251\) |
| risch | \(3 a \,b^{2} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} C \,b^{3}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{3}}{2 d}-\frac {i a \left (9 A a b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 A a b \,{\mathrm e}^{i \left (d x +c \right )}-4 A \,a^{2}-18 A \,b^{2}-6 a^{2} C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {3 A \,a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} C}{d}+\frac {3 A \,a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} C}{d}\) | \(325\) |
| norman | \(\frac {-\frac {\left (2 A \,a^{3}-3 A \,a^{2} b +6 A a \,b^{2}+2 C \,a^{3}-2 C \,b^{3}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 A \,a^{3}+3 A \,a^{2} b +6 A a \,b^{2}+2 C \,a^{3}+2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (26 A \,a^{3}-45 A \,a^{2} b +54 A a \,b^{2}+18 C \,a^{3}-6 C \,b^{3}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (26 A \,a^{3}+45 A \,a^{2} b +54 A a \,b^{2}+18 C \,a^{3}+6 C \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (46 A \,a^{3}-81 A \,a^{2} b +18 A a \,b^{2}+6 C \,a^{3}+18 C \,b^{3}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (46 A \,a^{3}+81 A \,a^{2} b +18 A a \,b^{2}+6 C \,a^{3}-18 C \,b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (50 A \,a^{3}-45 A \,a^{2} b -90 A a \,b^{2}-30 C \,a^{3}+18 C \,b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (50 A \,a^{3}+45 A \,a^{2} b -90 A a \,b^{2}-30 C \,a^{3}-18 C \,b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-3 a \,b^{2} C x -6 a \,b^{2} C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \,b^{2} C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 a \,b^{2} C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 a \,b^{2} C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 a \,b^{2} C x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \,b^{2} C x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \,b^{2} C x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {b \left (3 A \,a^{2}+2 A \,b^{2}+6 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (3 A \,a^{2}+2 A \,b^{2}+6 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(639\) |
-A*a^3/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(A*b^3+3*C*a^2*b)/d*ln(sec(d*x +c)+tan(d*x+c))+(3*A*a*b^2+C*a^3)/d*tan(d*x+c)+1/d*sin(d*x+c)*C*b^3+3*A*a^ 2*b/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+3*C*a*b^2/ d*(d*x+c)
Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {36 \, C a b^{2} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, {\left (A + 2 \, C\right )} a^{2} b + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, {\left (A + 2 \, C\right )} a^{2} b + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 9 \, A a^{2} b \cos \left (d x + c\right ) + 2 \, A a^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
1/12*(36*C*a*b^2*d*x*cos(d*x + c)^3 + 3*(3*(A + 2*C)*a^2*b + 2*A*b^3)*cos( d*x + c)^3*log(sin(d*x + c) + 1) - 3*(3*(A + 2*C)*a^2*b + 2*A*b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(6*C*b^3*cos(d*x + c)^3 + 9*A*a^2*b*cos (d*x + c) + 2*A*a^3 + 2*((2*A + 3*C)*a^3 + 9*A*a*b^2)*cos(d*x + c)^2)*sin( d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.11 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 36 \, {\left (d x + c\right )} C a b^{2} - 9 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, C a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C b^{3} \sin \left (d x + c\right ) + 12 \, C a^{3} \tan \left (d x + c\right ) + 36 \, A a b^{2} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 36*(d*x + c)*C*a*b^2 - 9 *A*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + lo g(sin(d*x + c) - 1)) + 18*C*a^2*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c ) - 1)) + 6*A*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 12*C*b ^3*sin(d*x + c) + 12*C*a^3*tan(d*x + c) + 36*A*a*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (153) = 306\).
Time = 0.32 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.98 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {18 \, {\left (d x + c\right )} C a b^{2} + \frac {12 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (3 \, A a^{2} b + 6 \, C a^{2} b + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{2} b + 6 \, C a^{2} b + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(18*(d*x + c)*C*a*b^2 + 12*C*b^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1 /2*c)^2 + 1) + 3*(3*A*a^2*b + 6*C*a^2*b + 2*A*b^3)*log(abs(tan(1/2*d*x + 1 /2*c) + 1)) - 3*(3*A*a^2*b + 6*C*a^2*b + 2*A*b^3)*log(abs(tan(1/2*d*x + 1/ 2*c) - 1)) - 2*(6*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^3*tan(1/2*d*x + 1/2 *c)^5 - 9*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^2*tan(1/2*d*x + 1/2*c) ^5 - 4*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 36 *A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^3*tan(1/2*d*x + 1/2*c) + 6*C*a^3*t an(1/2*d*x + 1/2*c) + 9*A*a^2*b*tan(1/2*d*x + 1/2*c) + 18*A*a*b^2*tan(1/2* d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d
Time = 3.40 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.85 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\frac {A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {3\,A\,a\,b^2\,\sin \left (c+d\,x\right )}{4}-\frac {A\,b^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{2}+\frac {3\,A\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,A\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{2}-\frac {A\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{4}+\frac {3\,C\,a\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {C\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{2}-\frac {A\,a^2\,b\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}}{4}+\frac {9\,C\,a\,b^2\,\cos \left (c+d\,x\right )\,\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 )}{2}-\frac {C\,a^2\,b\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}}{2}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
((A*a^3*sin(3*c + 3*d*x))/6 + (C*a^3*sin(3*c + 3*d*x))/4 + (C*b^3*sin(2*c + 2*d*x))/4 + (C*b^3*sin(4*c + 4*d*x))/8 + (A*a^3*sin(c + d*x))/2 + (C*a^3 *sin(c + d*x))/4 + (3*A*a*b^2*sin(c + d*x))/4 - (A*b^3*cos(c + d*x)*atan(( sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*3i)/2 + (3*A*a^2*b*sin(2*c + 2* d*x))/4 + (3*A*a*b^2*sin(3*c + 3*d*x))/4 - (A*b^3*atan((sin(c/2 + (d*x)/2) *1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*1i)/2 - (A*a^2*b*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/4 + (3*C*a*b^2*atan (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 - (C*a^2*b*ata n((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/2 - (A* a^2*b*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i)/4 + (9*C*a*b^2*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - (C*a^2*b*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i )/2)/(d*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4))