Integrand size = 28, antiderivative size = 39 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\left (a^2-b^2\right ) x-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d} \] Output:
(a^2-b^2)*x-2*a*b*ln(cos(d*x+c))/d+b^2*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.77 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {-i \left ((a+i b)^2 \log (i-\tan (c+d x))-(a-i b)^2 \log (i+\tan (c+d x))\right )+2 b^2 \tan (c+d x)}{2 d} \] Input:
Integrate[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]
Output:
((-I)*((a + I*b)^2*Log[I - Tan[c + d*x]] - (a - I*b)^2*Log[I + Tan[c + d*x ]]) + 2*b^2*Tan[c + d*x])/(2*d)
Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 3565, 3042, 3958, 3042, 3956}
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.
\(\displaystyle \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^2}{\cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3565 |
\(\displaystyle \int (a+b \tan (c+d x))^2dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \tan (c+d x))^2dx\) |
\(\Big \downarrow \) 3958 |
\(\displaystyle 2 a b \int \tan (c+d x)dx+x \left (a^2-b^2\right )+\frac {b^2 \tan (c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a b \int \tan (c+d x)dx+x \left (a^2-b^2\right )+\frac {b^2 \tan (c+d x)}{d}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle x \left (a^2-b^2\right )-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}\) |
Input:
Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]
Output:
(a^2 - b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d
Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0 ]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {a^{2} \left (d x +c \right )-2 a b \ln \left (\cos \left (d x +c \right )\right )+b^{2} \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(44\) |
default | \(\frac {a^{2} \left (d x +c \right )-2 a b \ln \left (\cos \left (d x +c \right )\right )+b^{2} \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(44\) |
parts | \(\frac {a^{2} \left (d x +c \right )}{d}+\frac {b^{2} \left (\tan \left (d x +c \right )-d x -c \right )}{d}+\frac {2 a b \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(49\) |
risch | \(2 i a b x +a^{2} x -b^{2} x +\frac {4 i a b c}{d}+\frac {2 i b^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(69\) |
parallelrisch | \(\frac {2 \cos \left (d x +c \right ) a b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2 \cos \left (d x +c \right ) a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2 \cos \left (d x +c \right ) a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+d x \left (a -b \right ) \left (a +b \right ) \cos \left (d x +c \right )+b^{2} \sin \left (d x +c \right )}{d \cos \left (d x +c \right )}\) | \(107\) |
norman | \(\frac {\left (-a^{2}+b^{2}\right ) x +\left (-a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (a^{2}-b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a^{2}-b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(225\) |
Input:
int(sec(d*x+c)^2*(a*cos(d*x+c)+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(d*x+c)-2*a*b*ln(cos(d*x+c))+b^2*(tan(d*x+c)-d*x-c))
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {{\left (a^{2} - b^{2}\right )} d x \cos \left (d x + c\right ) - 2 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + b^{2} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \] Input:
integrate(sec(d*x+c)^2*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
((a^2 - b^2)*d*x*cos(d*x + c) - 2*a*b*cos(d*x + c)*log(-cos(d*x + c)) + b^ 2*sin(d*x + c))/(d*cos(d*x + c))
\[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2} \sec ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**2*(a*cos(d*x+c)+b*sin(d*x+c))**2,x)
Output:
Integral((a*cos(c + d*x) + b*sin(c + d*x))**2*sec(c + d*x)**2, x)
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {{\left (d x + c\right )} a^{2} - {\left (d x + c - \tan \left (d x + c\right )\right )} b^{2} - a b \log \left (-\sin \left (d x + c\right )^{2} + 1\right )}{d} \] Input:
integrate(sec(d*x+c)^2*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
((d*x + c)*a^2 - (d*x + c - tan(d*x + c))*b^2 - a*b*log(-sin(d*x + c)^2 + 1))/d
Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + b^{2} \tan \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{d} \] Input:
integrate(sec(d*x+c)^2*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
(a*b*log(tan(d*x + c)^2 + 1) + b^2*tan(d*x + c) + (a^2 - b^2)*(d*x + c))/d
Time = 16.91 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.03 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,a\,b\,\ln \left (\frac {\cos \left (c+d\,x\right )}{\cos \left (c+d\,x\right )+1}\right )}{d}+\frac {2\,a\,b\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^2/cos(c + d*x)^2,x)
Output:
(b^2*tan(c + d*x))/d + (2*a^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))) /d - (2*b^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (2*a*b*log(co s(c + d*x)/(cos(c + d*x) + 1)))/d + (2*a*b*log(1/cos(c/2 + (d*x)/2)^2))/d
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.97 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a b -2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a b -2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a b +\cos \left (d x +c \right ) a^{2} d x -\cos \left (d x +c \right ) b^{2} d x +\sin \left (d x +c \right ) b^{2}}{\cos \left (d x +c \right ) d} \] Input:
int(sec(d*x+c)^2*(a*cos(d*x+c)+b*sin(d*x+c))^2,x)
Output:
(2*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a*b - 2*cos(c + d*x)*log(tan( (c + d*x)/2) - 1)*a*b - 2*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a*b + cos (c + d*x)*a**2*d*x - cos(c + d*x)*b**2*d*x + sin(c + d*x)*b**2)/(cos(c + d *x)*d)