Integrand size = 17, antiderivative size = 133 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\frac {\cot (c+d x)}{2 a^3 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot ^5(c+d x)}{6 a^3 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\log (\sin (c+d x)) \tan (c+d x)}{a^3 d \sqrt {-a \tan ^2(c+d x)}} \] Output:
1/2*cot(d*x+c)/a^3/d/(-a*tan(d*x+c)^2)^(1/2)-1/4*cot(d*x+c)^3/a^3/d/(-a*ta n(d*x+c)^2)^(1/2)+1/6*cot(d*x+c)^5/a^3/d/(-a*tan(d*x+c)^2)^(1/2)+ln(sin(d* x+c))*tan(d*x+c)/a^3/d/(-a*tan(d*x+c)^2)^(1/2)
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\frac {\left (18 \csc ^2(c+d x)-9 \csc ^4(c+d x)+2 \csc ^6(c+d x)+12 \log (\sin (c+d x))\right ) \tan (c+d x)}{12 a^3 d \sqrt {-a \tan ^2(c+d x)}} \] Input:
Integrate[(a - a*Sec[c + d*x]^2)^(-7/2),x]
Output:
((18*Csc[c + d*x]^2 - 9*Csc[c + d*x]^4 + 2*Csc[c + d*x]^6 + 12*Log[Sin[c + d*x]])*Tan[c + d*x])/(12*a^3*d*Sqrt[-(a*Tan[c + d*x]^2)])
Time = 0.53 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.118, Rules used = {3042, 4609, 3042, 4141, 3042, 25, 3954, 25, 3042, 25, 3954, 25, 3042, 25, 3954, 25, 3042, 25, 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.
| \(\displaystyle \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \sec (c+d x)^2\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4609 |
| \(\displaystyle \int \frac {1}{\left (-a \tan ^2(c+d x)\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-a \tan (c+d x)^2\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle -\frac {\tan (c+d x) \int \cot ^7(c+d x)dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\tan (c+d x) \int -\tan \left (c+d x+\frac {\pi }{2}\right )^7dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^7dx}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\tan (c+d x) \left (\frac {\cot ^6(c+d x)}{6 d}-\int -\cot ^5(c+d x)dx\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int \cot ^5(c+d x)dx+\frac {\cot ^6(c+d x)}{6 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int -\tan \left (c+d x+\frac {\pi }{2}\right )^5dx+\frac {\cot ^6(c+d x)}{6 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (\frac {\cot ^6(c+d x)}{6 d}-\int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^5dx\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int -\cot ^3(c+d x)dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (-\int \cot ^3(c+d x)dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \left (-\int -\tan \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\tan (c+d x) \left (-\int -\cot (c+d x)dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {\cot ^2(c+d x)}{2 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int \cot (c+d x)dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {\cot ^2(c+d x)}{2 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \left (\int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {\cot ^2(c+d x)}{2 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \left (-\int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {\cot ^2(c+d x)}{2 d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\tan (c+d x) \left (\frac {\cot ^6(c+d x)}{6 d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {\cot ^2(c+d x)}{2 d}+\frac {\log (-\sin (c+d x))}{d}\right )}{a^3 \sqrt {-a \tan ^2(c+d x)}}\) |
Input:
Int[(a - a*Sec[c + d*x]^2)^(-7/2),x]
Output:
((Cot[c + d*x]^2/(2*d) - Cot[c + d*x]^4/(4*d) + Cot[c + d*x]^6/(6*d) + Log [-Sin[c + d*x]]/d)*Tan[c + d*x])/(a^3*Sqrt[-(a*Tan[c + d*x]^2)])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Time = 1.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(\frac {\left (-\ln \left (\frac {2}{\cos \left (d x +c \right )+1}\right ) \sin \left (d x +c \right )^{6}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sin \left (d x +c \right )^{6}+\frac {25 \cos \left (d x +c \right )^{6}}{48}-\frac {\cos \left (d x +c \right )^{4}}{16}-\frac {11 \cos \left (d x +c \right )^{2}}{16}+\frac {19}{48}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )^{5}}{d \sqrt {-a \tan \left (d x +c \right )^{2}}\, a^{3}}\) | \(115\) |
| risch | \(\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) x}{a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}}-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (d x +c \right )}{a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}+\frac {2 i \left (9 \,{\mathrm e}^{10 i \left (d x +c \right )}-18 \,{\mathrm e}^{8 i \left (d x +c \right )}+34 \,{\mathrm e}^{6 i \left (d x +c \right )}-18 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}\) | \(324\) |
Input:
int(1/(a-sec(d*x+c)^2*a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/d*(-ln(2/(cos(d*x+c)+1))*sin(d*x+c)^6+ln(csc(d*x+c)-cot(d*x+c))*sin(d*x+ c)^6+25/48*cos(d*x+c)^6-1/16*cos(d*x+c)^4-11/16*cos(d*x+c)^2+19/48)/(-a*ta n(d*x+c)^2)^(1/2)/a^3*sec(d*x+c)*csc(d*x+c)^5
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\frac {{\left (18 \, \cos \left (d x + c\right )^{5} - 27 \, \cos \left (d x + c\right )^{3} - 12 \, {\left (\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 11 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{12 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(7/2),x, algorithm="fricas")
Output:
1/12*(18*cos(d*x + c)^5 - 27*cos(d*x + c)^3 - 12*(cos(d*x + c)^7 - 3*cos(d *x + c)^5 + 3*cos(d*x + c)^3 - cos(d*x + c))*log(1/2*sin(d*x + c)) + 11*co s(d*x + c))*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))
\[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(1/(a-a*sec(d*x+c)**2)**(7/2),x)
Output:
Integral((-a*sec(c + d*x)**2 + a)**(-7/2), x)
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\frac {6 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a} a^{3}} - \frac {12 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a} a^{3}} + \frac {6 \, \sqrt {-a} \tan \left (d x + c\right )^{4} - 3 \, \sqrt {-a} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {-a}}{a^{4} \tan \left (d x + c\right )^{6}}}{12 \, d} \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(7/2),x, algorithm="maxima")
Output:
-1/12*(6*log(tan(d*x + c)^2 + 1)/(sqrt(-a)*a^3) - 12*log(tan(d*x + c))/(sq rt(-a)*a^3) + (6*sqrt(-a)*tan(d*x + c)^4 - 3*sqrt(-a)*tan(d*x + c)^2 + 2*s qrt(-a))/(a^4*tan(d*x + c)^6))/d
Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\frac {\frac {384 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{\sqrt {-a} a^{3}} - \frac {192 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )}{\sqrt {-a} a^{3}} + \frac {352 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 87 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}{\sqrt {-a} a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 87 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\sqrt {-a} a^{10}}}{384 \, d} \] Input:
integrate(1/(a-a*sec(d*x+c)^2)^(7/2),x, algorithm="giac")
Output:
1/384*(384*log(tan(1/2*d*x + 1/2*c)^2 + 1)/(sqrt(-a)*a^3) - 192*log(tan(1/ 2*d*x + 1/2*c)^2)/(sqrt(-a)*a^3) + (352*tan(1/2*d*x + 1/2*c)^6 - 87*tan(1/ 2*d*x + 1/2*c)^4 + 12*tan(1/2*d*x + 1/2*c)^2 - 1)/(sqrt(-a)*a^3*tan(1/2*d* x + 1/2*c)^6) - (a^7*tan(1/2*d*x + 1/2*c)^6 - 12*a^7*tan(1/2*d*x + 1/2*c)^ 4 + 87*a^7*tan(1/2*d*x + 1/2*c)^2)/(sqrt(-a)*a^10))/d
Timed out. \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{7/2}} \,d x \] Input:
int(1/(a - a/cos(c + d*x)^2)^(7/2),x)
Output:
int(1/(a - a/cos(c + d*x)^2)^(7/2), x)
\[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{7/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sec \left (d x +c \right )^{2}+1}}{\sec \left (d x +c \right )^{8}-4 \sec \left (d x +c \right )^{6}+6 \sec \left (d x +c \right )^{4}-4 \sec \left (d x +c \right )^{2}+1}d x \right )}{a^{4}} \] Input:
int(1/(a-a*sec(d*x+c)^2)^(7/2),x)
Output:
(sqrt(a)*int(sqrt( - sec(c + d*x)**2 + 1)/(sec(c + d*x)**8 - 4*sec(c + d*x )**6 + 6*sec(c + d*x)**4 - 4*sec(c + d*x)**2 + 1),x))/a**4