Integrand size = 28, antiderivative size = 85 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {a^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}+\frac {\left (a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {a b \tan ^4(c+d x)}{2 d}+\frac {b^2 \tan ^5(c+d x)}{5 d} \] Output:
a^2*tan(d*x+c)/d+a*b*tan(d*x+c)^2/d+1/3*(a^2+b^2)*tan(d*x+c)^3/d+1/2*a*b*t an(d*x+c)^4/d+1/5*b^2*tan(d*x+c)^5/d
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {(a+b \tan (c+d x))^3 \left (a^2+10 b^2-3 a b \tan (c+d x)+6 b^2 \tan ^2(c+d x)\right )}{30 b^3 d} \] Input:
Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]
Output:
((a + b*Tan[c + d*x])^3*(a^2 + 10*b^2 - 3*a*b*Tan[c + d*x] + 6*b^2*Tan[c + d*x]^2))/(30*b^3*d)
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3567, 522, 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.
\(\displaystyle \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^2}{\cos (c+d x)^6}dx\) |
\(\Big \downarrow \) 3567 |
\(\displaystyle -\frac {\int (b+a \cot (c+d x))^2 \left (\cot ^2(c+d x)+1\right ) \tan ^6(c+d x)d\cot (c+d x)}{d}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\frac {\int \left (b^2 \tan ^6(c+d x)+2 a b \tan ^5(c+d x)+\left (a^2+b^2\right ) \tan ^4(c+d x)+2 a b \tan ^3(c+d x)+a^2 \tan ^2(c+d x)\right )d\cot (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} \left (a^2+b^2\right ) \tan ^3(c+d x)-a^2 \tan (c+d x)-\frac {1}{2} a b \tan ^4(c+d x)-a b \tan ^2(c+d x)-\frac {1}{5} b^2 \tan ^5(c+d x)}{d}\) |
Input:
Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]
Output:
-((-(a^2*Tan[c + d*x]) - a*b*Tan[c + d*x]^2 - ((a^2 + b^2)*Tan[c + d*x]^3) /3 - (a*b*Tan[c + d*x]^4)/2 - (b^2*Tan[c + d*x]^5)/5)/d)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[x^m*((b + a*x)^n/(1 + x^2)^((m + n + 2)/2)), x], x, Cot[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[ n, 0] && GtQ[m, 1])
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+\frac {a b}{2 \cos \left (d x +c \right )^{4}}+b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )}{d}\) | \(82\) |
default | \(\frac {-a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+\frac {a b}{2 \cos \left (d x +c \right )^{4}}+b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )}{d}\) | \(82\) |
parts | \(-\frac {a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )}{d}+\frac {a b \sec \left (d x +c \right )^{4}}{2 d}\) | \(87\) |
risch | \(\frac {4 i \left (-30 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+15 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-15 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-30 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+35 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+5 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+25 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{2}-b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}\) | \(143\) |
parallelrisch | \(-\frac {2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a b +\frac {4 \left (-2 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a b +\frac {2 \left (5 a^{2}+\frac {4 b^{2}}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a b +\frac {4 \left (-2 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a b +a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(186\) |
norman | \(\frac {\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {4 \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {2 \left (5 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {2 \left (5 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{5 d}-\frac {8 \left (5 a^{2}+14 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{15 d}-\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}\) | \(272\) |
Input:
int(sec(d*x+c)^6*(a*cos(d*x+c)+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-a^2*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+1/2*a*b/cos(d*x+c)^4+b^2*(1/5 *sin(d*x+c)^3/cos(d*x+c)^5+2/15*sin(d*x+c)^3/cos(d*x+c)^3))
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {15 \, a b \cos \left (d x + c\right ) + 2 \, {\left (2 \, {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{30 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate(sec(d*x+c)^6*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/30*(15*a*b*cos(d*x + c) + 2*(2*(5*a^2 - b^2)*cos(d*x + c)^4 + (5*a^2 - b ^2)*cos(d*x + c)^2 + 3*b^2)*sin(d*x + c))/(d*cos(d*x + c)^5)
Timed out. \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**6*(a*cos(d*x+c)+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {10 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + 2 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} b^{2} + \frac {15 \, a b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{30 \, d} \] Input:
integrate(sec(d*x+c)^6*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
1/30*(10*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^2 + 2*(3*tan(d*x + c)^5 + 5*t an(d*x + c)^3)*b^2 + 15*a*b/(sin(d*x + c)^2 - 1)^2)/d
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {6 \, b^{2} \tan \left (d x + c\right )^{5} + 15 \, a b \tan \left (d x + c\right )^{4} + 10 \, a^{2} \tan \left (d x + c\right )^{3} + 10 \, b^{2} \tan \left (d x + c\right )^{3} + 30 \, a b \tan \left (d x + c\right )^{2} + 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \] Input:
integrate(sec(d*x+c)^6*(a*cos(d*x+c)+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/30*(6*b^2*tan(d*x + c)^5 + 15*a*b*tan(d*x + c)^4 + 10*a^2*tan(d*x + c)^3 + 10*b^2*tan(d*x + c)^3 + 30*a*b*tan(d*x + c)^2 + 30*a^2*tan(d*x + c))/d
Time = 16.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.15 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {\frac {b^2\,\sin \left (c+d\,x\right )}{5}+{\cos \left (c+d\,x\right )}^2\,\left (\frac {a^2\,\sin \left (c+d\,x\right )}{3}-\frac {b^2\,\sin \left (c+d\,x\right )}{15}\right )+{\cos \left (c+d\,x\right )}^4\,\left (\frac {2\,a^2\,\sin \left (c+d\,x\right )}{3}-\frac {2\,b^2\,\sin \left (c+d\,x\right )}{15}\right )+\frac {a\,b\,\cos \left (c+d\,x\right )}{2}}{d\,{\cos \left (c+d\,x\right )}^5} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^2/cos(c + d*x)^6,x)
Output:
((b^2*sin(c + d*x))/5 + cos(c + d*x)^2*((a^2*sin(c + d*x))/3 - (b^2*sin(c + d*x))/15) + cos(c + d*x)^4*((2*a^2*sin(c + d*x))/3 - (2*b^2*sin(c + d*x) )/15) + (a*b*cos(c + d*x))/2)/(d*cos(c + d*x)^5)
Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx=\frac {\sin \left (d x +c \right ) \left (-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +30 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b +20 \sin \left (d x +c \right )^{4} a^{2}-4 \sin \left (d x +c \right )^{4} b^{2}-50 \sin \left (d x +c \right )^{2} a^{2}+10 \sin \left (d x +c \right )^{2} b^{2}+30 a^{2}\right )}{30 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int(sec(d*x+c)^6*(a*cos(d*x+c)+b*sin(d*x+c))^2,x)
Output:
(sin(c + d*x)*( - 15*cos(c + d*x)*sin(c + d*x)**3*a*b + 30*cos(c + d*x)*si n(c + d*x)*a*b + 20*sin(c + d*x)**4*a**2 - 4*sin(c + d*x)**4*b**2 - 50*sin (c + d*x)**2*a**2 + 10*sin(c + d*x)**2*b**2 + 30*a**2))/(30*cos(c + d*x)*d *(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))