Integrand size = 27, antiderivative size = 72 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))} \] Output:
-a^3*arctanh(cos(d*x+c))/d+2/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))^2+5/3*a^3*c os(d*x+c)/d/(1-sin(d*x+c))
Time = 7.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.00 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) (-7+5 \sin (c+d x))}{\left (-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(-3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/ 2]] + 2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/2]*(-7 + 5*Sin[c + d*x]))/(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3351, 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.
\(\displaystyle \int \csc (c+d x) \sec ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\sin (c+d x) \cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3351 |
\(\displaystyle a^4 \int \left (\frac {\csc (c+d x)}{a}+\frac {1}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^4 \left (-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {5 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}\right )\) |
Input:
Int[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
a^4*(-(ArcTanh[Cos[c + d*x]]/(a*d)) + (2*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])^2) + (5*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Time = 0.71 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99
method | result | size |
parallelrisch | \(\frac {\left (\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {14}{3}\right ) a^{3}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(71\) |
risch | \(\frac {2 a^{3} \left (-12 i {\mathrm e}^{i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}-5\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}\) | \(88\) |
derivativedivides | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+\frac {a^{3}}{\cos \left (d x +c \right )^{3}}-3 a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(101\) |
default | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+\frac {a^{3}}{\cos \left (d x +c \right )^{3}}-3 a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(101\) |
norman | \(\frac {-\frac {14 a^{3}}{3 d}-\frac {10 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {68 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {50 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {50 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {10 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}-\frac {12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {26 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(266\) |
Input:
int(csc(d*x+c)*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
((tan(1/2*d*x+1/2*c)-1)^3*ln(tan(1/2*d*x+1/2*c))-6*tan(1/2*d*x+1/2*c)^2+8* tan(1/2*d*x+1/2*c)-14/3)*a^3/d/(tan(1/2*d*x+1/2*c)-1)^3
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (64) = 128\).
Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.21 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {10 \, a^{3} \cos \left (d x + c\right )^{2} + 14 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3} + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (5 \, a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate(csc(d*x+c)*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/6*(10*a^3*cos(d*x + c)^2 + 14*a^3*cos(d*x + c) + 4*a^3 + 3*(a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - 2*a^3 + (a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - 2*a^3 + (a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(5*a^3*cos(d*x + c) - 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
Timed out. \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)*sec(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.43 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 \, a^{3} \tan \left (d x + c\right )^{3} + 6 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {6 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(csc(d*x+c)*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/6*(2*a^3*tan(d*x + c)^3 + 6*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^3 + a^3* (2*(3*cos(d*x + c)^2 + 1)/cos(d*x + c)^3 - 3*log(cos(d*x + c) + 1) + 3*log (cos(d*x + c) - 1)) + 6*a^3/cos(d*x + c)^3)/d
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \] Input:
integrate(csc(d*x+c)*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/3*(3*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(9*a^3*tan(1/2*d*x + 1/2*c)^ 2 - 12*a^3*tan(1/2*d*x + 1/2*c) + 7*a^3)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 31.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.36 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-8\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {14\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \] Input:
int((a + a*sin(c + d*x))^3/(cos(c + d*x)^4*sin(c + d*x)),x)
Output:
(a^3*log(tan(c/2 + (d*x)/2)))/d - (6*a^3*tan(c/2 + (d*x)/2)^2 + (14*a^3)/3 - 8*a^3*tan(c/2 + (d*x)/2))/(d*(3*tan(c/2 + (d*x)/2) - 3*tan(c/2 + (d*x)/ 2)^2 + tan(c/2 + (d*x)/2)^3 - 1))
Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.11 \[ \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8\right )}{3 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:
int(csc(d*x+c)*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(3*log(tan((c + d*x)/2))*tan((c + d*x)/2)**3 - 9*log(tan((c + d*x)/2 ))*tan((c + d*x)/2)**2 + 9*log(tan((c + d*x)/2))*tan((c + d*x)/2) - 3*log( tan((c + d*x)/2)) - 6*tan((c + d*x)/2)**3 + 6*tan((c + d*x)/2) - 8))/(3*d* (tan((c + d*x)/2)**3 - 3*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2) - 1))