Integrand size = 19, antiderivative size = 86 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {b^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
-a^3*arctanh(cos(d*x+c))/d+3*a^2*b*arctanh(sin(d*x+c))/d-1/2*b^3*arctanh(s in(d*x+c))/d+3*a*b^2*sec(d*x+c)/d+1/2*b^3*sec(d*x+c)*tan(d*x+c)/d
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 a^2 b \coth ^{-1}(\sin (c+d x))}{d}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {b^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec (c+d x) \tan (c+d x)}{2 d} \] Input:
Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^3,x]
Output:
(3*a^2*b*ArcCoth[Sin[c + d*x]])/d - (a^3*ArcTanh[Cos[c + d*x]])/d - (b^3*A rcTanh[Sin[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x] *Tan[c + d*x])/(2*d)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 4000, 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.
| \(\displaystyle \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 4000 |
| \(\displaystyle \int \left (a^3 \csc (c+d x)+3 a^2 b \sec (c+d x)+3 a b^2 \tan (c+d x) \sec (c+d x)+b^3 \tan ^2(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}-\frac {b^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^3 \tan (c+d x) \sec (c+d x)}{2 d}\) |
Input:
Int[Csc[c + d*x]*(a + b*Tan[c + d*x])^3,x]
Output:
-((a^3*ArcTanh[Cos[c + d*x]])/d) + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (b^ 3*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d *x]*Tan[c + d*x])/(2*d)
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Expand[Sin[e + f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]
Time = 0.88 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {b^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a \,b^{2}}{\cos \left (d x +c \right )}+3 a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}\) | \(107\) |
| default | \(\frac {b^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a \,b^{2}}{\cos \left (d x +c \right )}+3 a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}\) | \(107\) |
| risch | \(-\frac {i b^{2} {\mathrm e}^{i \left (d x +c \right )} \left (6 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i a -b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}\) | \(196\) |
Input:
int(csc(d*x+c)*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(b^3*(1/2*sin(d*x+c)^3/cos(d*x+c)^2+1/2*sin(d*x+c)-1/2*ln(sec(d*x+c)+t an(d*x+c)))+3*a*b^2/cos(d*x+c)+3*a^2*b*ln(sec(d*x+c)+tan(d*x+c))+a^3*ln(cs c(d*x+c)-cot(d*x+c)))
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a b^{2} \cos \left (d x + c\right ) - {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, b^{3} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate(csc(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/4*(2*a^3*cos(d*x + c)^2*log(1/2*cos(d*x + c) + 1/2) - 2*a^3*cos(d*x + c )^2*log(-1/2*cos(d*x + c) + 1/2) - 12*a*b^2*cos(d*x + c) - (6*a^2*b - b^3) *cos(d*x + c)^2*log(sin(d*x + c) + 1) + (6*a^2*b - b^3)*cos(d*x + c)^2*log (-sin(d*x + c) + 1) - 2*b^3*sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \csc {\left (c + d x \right )}\, dx \] Input:
integrate(csc(d*x+c)*(a+b*tan(d*x+c))**3,x)
Output:
Integral((a + b*tan(c + d*x))**3*csc(c + d*x), x)
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.29 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, a^{3} \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right ) - \frac {12 \, a b^{2}}{\cos \left (d x + c\right )}}{4 \, d} \] Input:
integrate(csc(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
-1/4*(b^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) + log(sin(d*x + c) + 1) - l og(sin(d*x + c) - 1)) - 6*a^2*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 4*a^3*log(cot(d*x + c) + csc(d*x + c)) - 12*a*b^2/cos(d*x + c))/d
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.67 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + {\left (6 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (6 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \] Input:
integrate(csc(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
1/2*(2*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + (6*a^2*b - b^3)*log(abs(tan(1/ 2*d*x + 1/2*c) + 1)) - (6*a^2*b - b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(b^3*tan(1/2*d*x + 1/2*c)^3 - 6*a*b^2*tan(1/2*d*x + 1/2*c)^2 + b^3*tan (1/2*d*x + 1/2*c) + 6*a*b^2)/(tan(1/2*d*x + 1/2*c)^2 - 1)^2)/d
Time = 1.33 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.23 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {2\,\left (\frac {a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-\frac {b^3\,\mathrm {atan}\left (\frac {2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3-6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}{2{}\mathrm {i}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3-6{}\mathrm {i}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+1{}\mathrm {i}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,1{}\mathrm {i}}{2}+a^2\,b\,\mathrm {atan}\left (\frac {2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3-6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}{2{}\mathrm {i}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3-6{}\mathrm {i}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b+1{}\mathrm {i}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,3{}\mathrm {i}\right )}{d}+\frac {\frac {\sin \left (c+d\,x\right )\,b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \] Input:
int((a + b*tan(c + d*x))^3/sin(c + d*x),x)
Output:
(2*((a^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - (b^3*atan((b^3*co s(c/2 + (d*x)/2) + 2*a^3*sin(c/2 + (d*x)/2) - 6*a^2*b*cos(c/2 + (d*x)/2))/ (a^3*cos(c/2 + (d*x)/2)*2i + b^3*sin(c/2 + (d*x)/2)*1i - a^2*b*sin(c/2 + ( d*x)/2)*6i))*1i)/2 + a^2*b*atan((b^3*cos(c/2 + (d*x)/2) + 2*a^3*sin(c/2 + (d*x)/2) - 6*a^2*b*cos(c/2 + (d*x)/2))/(a^3*cos(c/2 + (d*x)/2)*2i + b^3*si n(c/2 + (d*x)/2)*1i - a^2*b*sin(c/2 + (d*x)/2)*6i))*3i))/d + ((b^3*sin(c + d*x))/2 + 3*a*b^2*cos(c + d*x))/(d*(cos(2*c + 2*d*x)/2 + 1/2))
Time = 0.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.13 \[ \int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {-6 \cos \left (d x +c \right ) a \,b^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a^{2} b +\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} b^{3}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2} b -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a^{2} b -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} b^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2} b +\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{3}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-6 \sin \left (d x +c \right )^{2} a \,b^{2}-\sin \left (d x +c \right ) b^{3}+6 a \,b^{2}}{2 d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(csc(d*x+c)*(a+b*tan(d*x+c))^3,x)
Output:
( - 6*cos(c + d*x)*a*b**2 - 6*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a* *2*b + log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**3 + 6*log(tan((c + d*x )/2) - 1)*a**2*b - log(tan((c + d*x)/2) - 1)*b**3 + 6*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**2*b - log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*b **3 - 6*log(tan((c + d*x)/2) + 1)*a**2*b + log(tan((c + d*x)/2) + 1)*b**3 + 2*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**3 - 2*log(tan((c + d*x)/2))*a **3 - 6*sin(c + d*x)**2*a*b**2 - sin(c + d*x)*b**3 + 6*a*b**2)/(2*d*(sin(c + d*x)**2 - 1))