Integrand size = 19, antiderivative size = 102 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {(a+b)^3 \log (1-\cos (c+d x))}{2 d}-\frac {b \left (3 a^2+b^2\right ) \log (\cos (c+d x))}{d}-\frac {(a-b)^3 \log (1+\cos (c+d x))}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
1/2*(a+b)^3*ln(1-cos(d*x+c))/d-b*(3*a^2+b^2)*ln(cos(d*x+c))/d-1/2*(a-b)^3* ln(1+cos(d*x+c))/d+3*a*b^2*sec(d*x+c)/d+1/2*b^3*sec(d*x+c)^2/d
Time = 0.96 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {-2 (a-b)^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 b \left (3 a^2+b^2\right ) \log (\cos (c+d x))+2 (a+b)^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a b^2 \sec (c+d x)+b^3 \sec ^2(c+d x)}{2 d} \]
(-2*(a - b)^3*Log[Cos[(c + d*x)/2]] - 2*b*(3*a^2 + b^2)*Log[Cos[c + d*x]] + 2*(a + b)^3*Log[Sin[(c + d*x)/2]] + 6*a*b^2*Sec[c + d*x] + b^3*Sec[c + d *x]^2)/(2*d)
Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {3042, 4360, 25, 25, 3042, 25, 3316, 25, 27, 526, 2333, 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.
| \(\displaystyle \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc (c+d x) \sec ^3(c+d x) \left (-(-a \cos (c+d x)-b)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(b+a \cos (c+d x))^3 \csc (c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right ) \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {a \int -\frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^2-a^2 \cos ^2(c+d x)}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^2-a^2 \cos ^2(c+d x)}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \int \frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 526 |
| \(\displaystyle -\frac {a^4 \left (\frac {\int \frac {\left (\cos ^2(c+d x) a^4+3 b^2 a^2+b \left (3 a^2+b^2\right ) \cos (c+d x) a\right ) \sec ^2(c+d x)}{a^2 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle -\frac {a^4 \left (\frac {\int \left (\frac {(a-b)^3}{2 a^2 (\cos (c+d x) a+a)}+\frac {3 b^2 \sec ^2(c+d x)}{a^2}+\frac {\left (b^3+3 a^2 b\right ) \sec (c+d x)}{a^3}+\frac {(a+b)^3}{2 a^2 (a-a \cos (c+d x))}\right )d(a \cos (c+d x))}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 \left (\frac {\frac {b \left (3 a^2+b^2\right ) \log (a \cos (c+d x))}{a^2}+\frac {(a-b)^3 \log (a \cos (c+d x)+a)}{2 a^2}-\frac {(a+b)^3 \log (a-a \cos (c+d x))}{2 a^2}-\frac {3 b^2 \sec (c+d x)}{a}}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
-((a^4*(-1/2*(b^3*Sec[c + d*x]^2)/a^4 + ((b*(3*a^2 + b^2)*Log[a*Cos[c + d* x]])/a^2 - ((a + b)^3*Log[a - a*Cos[c + d*x]])/(2*a^2) + ((a - b)^3*Log[a + a*Cos[c + d*x]])/(2*a^2) - (3*b^2*Sec[c + d*x])/a)/a^2))/d)
3.2.88.3.1 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[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_Symbol] : > With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemainder [(c + d*x)^n, x, x]}, Simp[R*(x^(m + 1)/(a*(m + 1))), x] + Simp[1/a Int[x ^(m + 1)*(ExpandToSum[a*Qx - b*R*x, x]/(a + b*x^2)), x], x]] /; FreeQ[{a, b , c, d}, x] && IGtQ[n, 1] && ILtQ[m, -1]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 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 = 0.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
| default | \(\frac {a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
| norman | \(\frac {\frac {6 a \,b^{2}}{d}-\frac {2 \left (3 a \,b^{2}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(143\) |
| parallelrisch | \(\frac {-3 \left (a^{2}+\frac {b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3 \left (a^{2}+\frac {b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (a +b \right )^{3} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\frac {\left (a -\frac {b}{6}\right ) \cos \left (2 d x +2 c \right )}{2}+a \cos \left (d x +c \right )+\frac {a}{2}+\frac {b}{12}\right ) b^{2}}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(152\) |
| risch | \(\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )} a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a^{2} b}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a^{2} b}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{3}}{d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(261\) |
1/d*(a^3*ln(-cot(d*x+c)+csc(d*x+c))+3*a^2*b*ln(tan(d*x+c))+3*a*b^2*(1/cos( d*x+c)+ln(-cot(d*x+c)+csc(d*x+c)))+b^3*(1/2/cos(d*x+c)^2+ln(tan(d*x+c))))
Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {6 \, a b^{2} \cos \left (d x + c\right ) - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
1/2*(6*a*b^2*cos(d*x + c) - 2*(3*a^2*b + b^3)*cos(d*x + c)^2*log(-cos(d*x + c)) - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(d*x + c)^2*log(1/2*cos(d*x + c ) + 1/2) + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2*log(-1/2*cos(d*x + c) + 1/2) + b^3)/(d*cos(d*x + c)^2)
\[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \csc {\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.10 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
-1/2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(cos(d*x + c) + 1) - (a^3 + 3*a^2 *b + 3*a*b^2 + b^3)*log(cos(d*x + c) - 1) + 2*(3*a^2*b + b^3)*log(cos(d*x + c)) - (6*a*b^2*cos(d*x + c) + b^3)/cos(d*x + c)^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (96) = 192\).
Time = 0.34 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.45 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {9 \, a^{2} b + 12 \, a b^{2} + 3 \, b^{3} + \frac {18 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]
1/2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(-cos(d*x + c) + 1)/abs(cos(d* x + c) + 1)) - 2*(3*a^2*b + b^3)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (9*a^2*b + 12*a*b^2 + 3*b^3 + 18*a^2*b*(cos(d*x + c) - 1)/(c os(d*x + c) + 1) + 12*a*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*b^3* (cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a^2*b*(cos(d*x + c) - 1)^2/(cos( d*x + c) + 1)^2 + 3*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((cos(d *x + c) - 1)/(cos(d*x + c) + 1) + 1)^2)/d
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^3}{2}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^3}{2}+\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}-\ln \left (\cos \left (c+d\,x\right )\right )\,\left (3\,a^2\,b+b^3\right )}{d} \]