Integrand size = 19, antiderivative size = 64 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=\frac {-2 a^3 \cos (c+d x)+b \left (-6 a^2 \log (\cos (c+d x))+6 a b \sec (c+d x)+b^2 \sec ^2(c+d x)\right )}{2 d} \]
(-2*a^3*Cos[c + d*x] + b*(-6*a^2*Log[Cos[c + d*x]] + 6*a*b*Sec[c + d*x] + b^2*Sec[c + d*x]^2))/(2*d)
Time = 0.35 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 4360, 25, 25, 3042, 25, 3312, 27, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin (c+d x) (a+b \sec (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right ) \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \tan (c+d x) \sec ^2(c+d x) \left (-(-a \cos (c+d x)-b)^3\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -(b+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^3}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle -\frac {\int (b+a \cos (c+d x))^3 \sec ^3(c+d x)d(a \cos (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \int \frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {a^2 \int \left (\frac {b^3 \sec ^3(c+d x)}{a^3}+\frac {3 b^2 \sec ^2(c+d x)}{a^2}+\frac {3 b \sec (c+d x)}{a}+1\right )d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 \left (-\frac {b^3 \sec ^2(c+d x)}{2 a^2}-\frac {3 b^2 \sec (c+d x)}{a}+3 b \log (a \cos (c+d x))+a \cos (c+d x)\right )}{d}\) |
-((a^2*(a*Cos[c + d*x] + 3*b*Log[a*Cos[c + d*x]] - (3*b^2*Sec[c + d*x])/a - (b^3*Sec[c + d*x]^2)/(2*a^2)))/d)
3.2.87.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 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 = 0.82 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right ) a \,b^{2}+3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )-\frac {a^{3}}{\sec \left (d x +c \right )}}{d}\) | \(57\) |
default | \(\frac {\frac {b^{3} \sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right ) a \,b^{2}+3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )-\frac {a^{3}}{\sec \left (d x +c \right )}}{d}\) | \(57\) |
parts | \(-\frac {a^{3} \cos \left (d x +c \right )}{d}+\frac {b^{3} \sec \left (d x +c \right )^{2}}{2 d}+\frac {3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \sec \left (d x +c \right )}{d}\) | \(63\) |
risch | \(3 i a^{2} b x -\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {6 i b \,a^{2} c}{d}+\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )} a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}\) | \(133\) |
norman | \(\frac {\frac {\left (4 a^{3}+2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {2 a^{3}-6 a \,b^{2}}{d}-\frac {\left (2 a^{3}+6 a \,b^{2}-2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {3 a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {3 a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {3 a^{2} b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(175\) |
parallelrisch | \(\frac {6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-2 a^{3}+6 a \,b^{2}-b^{3}\right ) \cos \left (2 d x +2 c \right )-a^{3} \cos \left (3 d x +3 c \right )+\left (-3 a^{3}+12 a \,b^{2}\right ) \cos \left (d x +c \right )-2 a^{3}+6 a \,b^{2}+b^{3}}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(181\) |
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) - b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
-1/2*(2*a^3*cos(d*x + c)^3 + 6*a^2*b*cos(d*x + c)^2*log(-cos(d*x + c)) - 6 *a*b^2*cos(d*x + c) - b^3)/(d*cos(d*x + c)^2)
\[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sin {\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{2} b \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a b^{2}}{\cos \left (d x + c\right )} - \frac {b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
-1/2*(2*a^3*cos(d*x + c) + 6*a^2*b*log(cos(d*x + c)) - 6*a*b^2/cos(d*x + c ) - b^3/cos(d*x + c)^2)/d
Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^{3} \cos \left (d x + c\right )}{d} - \frac {3 \, a^{2} b \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
-a^3*cos(d*x + c)/d - 3*a^2*b*log(abs(cos(d*x + c))/abs(d))/d + 1/2*(6*a*b ^2*cos(d*x + c) + b^3)/(d*cos(d*x + c)^2)
Time = 13.58 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3\,\cos \left (c+d\,x\right )-\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}+3\,a^2\,b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]