Integrand size = 19, antiderivative size = 73 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^2}{2 d (a-a \cos (c+d x))}+\frac {3 a \log (1-\cos (c+d x))}{4 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {a \log (1+\cos (c+d x))}{4 d} \]
-1/2*a^2/d/(a-a*cos(d*x+c))+3/4*a*ln(1-cos(d*x+c))/d-a*ln(cos(d*x+c))/d+1/ 4*a*ln(1+cos(d*x+c))/d
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \]
-1/8*(a*Csc[(c + d*x)/2]^2)/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Cos[(c + d*x)/2]])/(2*d) - (a*Log[Cos[c + d*x]])/d + (a*Log[Sin[(c + d*x)/2]])/(2* d) + (a*Log[Sin[c + d*x]])/d + (a*Sec[(c + d*x)/2]^2)/(8*d)
Time = 0.37 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 25, 27, 93, 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.
\(\displaystyle \int \csc ^3(c+d x) (a \sec (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\left (\csc ^3(c+d x) \sec (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\left ((\cos (c+d x) a+a) \csc ^3(c+d x) \sec (c+d x)\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \csc ^3(c+d x) \sec (c+d x) (a \cos (c+d x)+a)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \cos \left (c+d x-\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {a^3 \int -\frac {\sec (c+d x)}{(a-a \cos (c+d x))^2 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^3 \int \frac {\sec (c+d x)}{(a-a \cos (c+d x))^2 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^4 \int \frac {\sec (c+d x)}{a (a-a \cos (c+d x))^2 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 93 |
\(\displaystyle -\frac {a^4 \int \left (\frac {\sec (c+d x)}{a^4}+\frac {3}{4 a^3 (a-a \cos (c+d x))}-\frac {1}{4 a^3 (\cos (c+d x) a+a)}+\frac {1}{2 a^2 (a-a \cos (c+d x))^2}\right )d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^4 \left (\frac {\log (a \cos (c+d x))}{a^3}-\frac {3 \log (a-a \cos (c+d x))}{4 a^3}-\frac {\log (a \cos (c+d x)+a)}{4 a^3}+\frac {1}{2 a^2 (a-a \cos (c+d x))}\right )}{d}\) |
-((a^4*(1/(2*a^2*(a - a*Cos[c + d*x])) + Log[a*Cos[c + d*x]]/a^3 - (3*Log[ a - a*Cos[c + d*x]])/(4*a^3) - Log[a + a*Cos[c + d*x]]/(4*a^3)))/d)
3.1.7.3.1 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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[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 = 1.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{4 d}\) | \(59\) |
derivativedivides | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(61\) |
default | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(61\) |
norman | \(-\frac {a}{4 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(71\) |
risch | \(\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(83\) |
-1/4*a*(cot(1/2*d*x+1/2*c)^2-6*ln(tan(1/2*d*x+1/2*c))+4*ln(tan(1/2*d*x+1/2 *c)-1)+4*ln(tan(1/2*d*x+1/2*c)+1))/d
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {4 \, {\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a \cos \left (d x + c\right ) - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, a}{4 \, {\left (d \cos \left (d x + c\right ) - d\right )}} \]
-1/4*(4*(a*cos(d*x + c) - a)*log(-cos(d*x + c)) - (a*cos(d*x + c) - a)*log (1/2*cos(d*x + c) + 1/2) - 3*(a*cos(d*x + c) - a)*log(-1/2*cos(d*x + c) + 1/2) - 2*a)/(d*cos(d*x + c) - d)
\[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=a \left (\int \csc ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 4 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, a}{\cos \left (d x + c\right ) - 1}}{4 \, d} \]
1/4*(a*log(cos(d*x + c) + 1) + 3*a*log(cos(d*x + c) - 1) - 4*a*log(cos(d*x + c)) + 2*a/(cos(d*x + c) - 1))/d
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=\frac {3 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 4 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (a - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{4 \, d} \]
1/4*(3*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 4*a*log(abs(- (cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (a - 3*a*(cos(d*x + c) - 1)/ (cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(cos(d*x + c) - 1))/d
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.73 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx=\frac {\frac {a}{2\,\left (\cos \left (c+d\,x\right )-1\right )}-a\,\ln \left (\cos \left (c+d\,x\right )\right )+\frac {3\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{4}+\frac {a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{4}}{d} \]