Integrand size = 29, antiderivative size = 87 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a^2 b \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \sec (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d} \]
-3*a^2*b*arctanh(cos(d*x+c))/d-a^3*cot(d*x+c)/d+3*a^2*b*sec(d*x+c)/d+b^3*s ec(d*x+c)/d+a^3*tan(d*x+c)/d+3*a*b^2*tan(d*x+c)/d
Time = 1.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (\left (2 a^3+3 a b^2\right ) \cos (2 (c+d x))-2 b \left (3 a^2+b^2\right ) \sin (c+d x)-3 a b \left (b+a \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (2 (c+d x))\right )\right )}{4 d} \]
-1/4*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sec[c + d*x]*((2*a^3 + 3*a*b^2)*Co s[2*(c + d*x)] - 2*b*(3*a^2 + b^2)*Sin[c + d*x] - 3*a*b*(b + a*(-Log[Cos[( c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)])))/d
Time = 0.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3391, 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.
\(\displaystyle \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \sin (c+d x))^3}{\sin (c+d x)^2 \cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3391 |
\(\displaystyle \int \left (a^3 \csc ^2(c+d x) \sec ^2(c+d x)+3 a^2 b \csc (c+d x) \sec ^2(c+d x)+3 a b^2 \sec ^2(c+d x)+b^3 \tan (c+d x) \sec (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {3 a^2 b \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^2 b \sec (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d}\) |
(-3*a^2*b*ArcTanh[Cos[c + d*x]])/d - (a^3*Cot[c + d*x])/d + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d*x])/d
3.15.61.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (G tQ[m, 0] || IntegerQ[n])
Time = 0.90 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{2} b \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a \,b^{2} \tan \left (d x +c \right )+\frac {b^{3}}{\cos \left (d x +c \right )}}{d}\) | \(91\) |
default | \(\frac {a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{2} b \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a \,b^{2} \tan \left (d x +c \right )+\frac {b^{3}}{\cos \left (d x +c \right )}}{d}\) | \(91\) |
parallelrisch | \(\frac {6 b \,a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (-a^{3}-2 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{3} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-12 a^{2} b -4 b^{3}}{2 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(113\) |
risch | \(-\frac {2 i \left (3 i a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-3 i a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-i b^{3} {\mathrm e}^{i \left (d x +c \right )}-3 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{3}+3 a \,b^{2}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(165\) |
norman | \(\frac {\frac {a^{3}}{2 d}+\frac {a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 \left (3 a^{2} b +b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (6 a^{2} b +2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \left (9 a^{2} b +3 b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \left (a^{2}+4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {3 a \left (a^{2}+4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (7 a^{2}+18 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (7 a^{2}+18 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 b \left (3 a^{2}+b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(305\) |
1/d*(a^3*(1/sin(d*x+c)/cos(d*x+c)-2*cot(d*x+c))+3*a^2*b*(1/cos(d*x+c)+ln(c sc(d*x+c)-cot(d*x+c)))+3*a*b^2*tan(d*x+c)+b^3/cos(d*x+c))
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 \, a^{2} b \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, a^{2} b \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, a^{3} - 6 \, a b^{2} + 2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
-1/2*(3*a^2*b*cos(d*x + c)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*a^ 2*b*cos(d*x + c)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*a^3 - 6*a*b ^2 + 2*(2*a^3 + 3*a*b^2)*cos(d*x + c)^2 - 2*(3*a^2*b + b^3)*sin(d*x + c))/ (d*cos(d*x + c)*sin(d*x + c))
Timed out. \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3 \, a^{2} b {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, a^{3} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + 6 \, a b^{2} \tan \left (d x + c\right ) + \frac {2 \, b^{3}}{\cos \left (d x + c\right )}}{2 \, d} \]
1/2*(3*a^2*b*(2/cos(d*x + c) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 2*a^3*(1/tan(d*x + c) - tan(d*x + c)) + 6*a*b^2*tan(d*x + c) + 2*b^3 /cos(d*x + c))/d
Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.70 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {6 \, a^{2} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
1/2*(6*a^2*b*log(abs(tan(1/2*d*x + 1/2*c))) + a^3*tan(1/2*d*x + 1/2*c) - ( 2*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 5*a^3*tan(1/2*d*x + 1/2*c)^2 + 12*a*b^2*t an(1/2*d*x + 1/2*c)^2 + 10*a^2*b*tan(1/2*d*x + 1/2*c) + 4*b^3*tan(1/2*d*x + 1/2*c) - a^3)/(tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c)))/d
Time = 11.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38 \[ \int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^2\,b+4\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (5\,a^3+12\,a\,b^2\right )-a^3}{d\,\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {3\,a^2\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]