Integrand size = 21, antiderivative size = 79 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=b^3 x+\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^2 b \tan (c+d x)}{2 d}+\frac {a^2 (a+b \cos (c+d x)) \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
b^3*x+1/2*a*(a^2+6*b^2)*arctanh(sin(d*x+c))/d+5/2*a^2*b*tan(d*x+c)/d+1/2*a ^2*(a+b*cos(d*x+c))*sec(d*x+c)*tan(d*x+c)/d
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.85 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {2 b^3 d x+6 a b^2 \coth ^{-1}(\sin (c+d x))+a^3 \text {arctanh}(\sin (c+d x))+6 a^2 b \tan (c+d x)+a^3 \sec (c+d x) \tan (c+d x)}{2 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3,x]
Output:
(2*b^3*d*x + 6*a*b^2*ArcCoth[Sin[c + d*x]] + a^3*ArcTanh[Sin[c + d*x]] + 6 *a^2*b*Tan[c + d*x] + a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)
Time = 0.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3271, 3042, 3500, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{2} \int \left (2 \cos ^2(c+d x) b^3+5 a^2 b+a \left (a^2+6 b^2\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {2 \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+5 a^2 b+a \left (a^2+6 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{2} \left (\int \left (2 \cos (c+d x) b^3+a \left (a^2+6 b^2\right )\right ) \sec (c+d x)dx+\frac {5 a^2 b \tan (c+d x)}{d}\right )+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {2 \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+a \left (a^2+6 b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^2 b \tan (c+d x)}{d}\right )+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{2} \left (a \left (a^2+6 b^2\right ) \int \sec (c+d x)dx+\frac {5 a^2 b \tan (c+d x)}{d}+2 b^3 x\right )+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (a \left (a^2+6 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^2 b \tan (c+d x)}{d}+2 b^3 x\right )+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{2} \left (\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^2 b \tan (c+d x)}{d}+2 b^3 x\right )+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3,x]
Output:
(a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (2*b^3*x + (a *(a^2 + 6*b^2)*ArcTanh[Sin[c + d*x]])/d + (5*a^2*b*Tan[c + d*x])/d)/2
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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((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 - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 5.72 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \tan \left (d x +c \right )+3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+b^{3} \left (d x +c \right )}{d}\) | \(82\) |
| default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \tan \left (d x +c \right )+3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+b^{3} \left (d x +c \right )}{d}\) | \(82\) |
| parts | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{3} \left (d x +c \right )}{d}+\frac {3 a^{2} b \tan \left (d x +c \right )}{d}+\frac {3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(90\) |
| parallelrisch | \(\frac {-a \left (a^{2}+6 b^{2}\right ) \left (\cos \left (2 d x +2 c \right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+a \left (a^{2}+6 b^{2}\right ) \left (\cos \left (2 d x +2 c \right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 b^{3} d x \cos \left (2 d x +2 c \right )+2 b^{3} d x +2 \sin \left (d x +c \right ) a^{3}+6 \sin \left (2 d x +2 c \right ) a^{2} b}{2 d \left (\cos \left (2 d x +2 c \right )+1\right )}\) | \(138\) |
| risch | \(b^{3} x -\frac {i a^{2} \left ({\mathrm e}^{3 i \left (d x +c \right )} a -6 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}-6 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}\) | \(154\) |
| norman | \(\frac {b^{3} x +b^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+b^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {a^{2} \left (a -6 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {a^{2} \left (a +6 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-2 b^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2 b^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {4 a^{2} \left (a -3 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {4 a^{2} \left (a +3 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}-\frac {a \left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a \left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(286\) |
Input:
int((a+cos(d*x+c)*b)^3*sec(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+3*a^2*b *tan(d*x+c)+3*b^2*a*ln(sec(d*x+c)+tan(d*x+c))+b^3*(d*x+c))
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.42 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {4 \, b^{3} d x \cos \left (d x + c\right )^{2} + {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^3,x, algorithm="fricas")
Output:
1/4*(4*b^3*d*x*cos(d*x + c)^2 + (a^3 + 6*a*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (a^3 + 6*a*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(6* a^2*b*cos(d*x + c) + a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{3} \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate((a+b*cos(d*x+c))**3*sec(d*x+c)**3,x)
Output:
Integral((a + b*cos(c + d*x))**3*sec(c + d*x)**3, x)
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} b^{3} - a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} b \tan \left (d x + c\right )}{4 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^3,x, algorithm="maxima")
Output:
1/4*(4*(d*x + c)*b^3 - a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin( d*x + c) + 1) + log(sin(d*x + c) - 1)) + 6*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 12*a^2*b*tan(d*x + c))/d
Time = 0.52 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.81 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {2 \, {\left (d x + c\right )} b^{3} + {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^3,x, algorithm="giac")
Output:
1/2*(2*(d*x + c)*b^3 + (a^3 + 6*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (a^3 + 6*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(a^3*tan(1/2*d*x + 1/2*c)^3 - 6*a^2*b*tan(1/2*d*x + 1/2*c)^3 + a^3*tan(1/2*d*x + 1/2*c) + 6 *a^2*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2)/d
Time = 34.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.72 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {6\,a\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a^2\,b\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \] Input:
int((a + b*cos(c + d*x))^3/cos(c + d*x)^3,x)
Output:
(a^3*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*b^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (a^3*sin(c + d*x))/(2*d*cos(c + d*x)^ 2) + (6*a*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (3*a^2*b*s in(c + d*x))/(d*cos(c + d*x))
Time = 0.17 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.06 \[ \int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx=\frac {-\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a^{3}-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a \,b^{2}+\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{3}+6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{2}+\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a^{3}+6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a \,b^{2}-\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{3}-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{2}+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{3} d x -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3}-2 \cos \left (d x +c \right ) b^{3} d x +6 \sin \left (d x +c \right )^{3} a^{2} b -6 \sin \left (d x +c \right ) a^{2} b}{2 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((a+b*cos(d*x+c))^3*sec(d*x+c)^3,x)
Output:
( - cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3 - 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**2 + cos(c + d*x)*log (tan((c + d*x)/2) - 1)*a**3 + 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b **2 + cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**3 + 6*cos( c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*b**2 - cos(c + d*x)*l og(tan((c + d*x)/2) + 1)*a**3 - 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a *b**2 + 2*cos(c + d*x)*sin(c + d*x)**2*b**3*d*x - cos(c + d*x)*sin(c + d*x )*a**3 - 2*cos(c + d*x)*b**3*d*x + 6*sin(c + d*x)**3*a**2*b - 6*sin(c + d* x)*a**2*b)/(2*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))