Integrand size = 23, antiderivative size = 135 \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {5 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {5 a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{64 d}+\frac {5 a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{7/2}}{6 d} \]
5/64*a^2*sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2)/d+5/48*a*sec(d*x+c)^4*(a+a*si n(d*x+c))^(5/2)/d+1/6*sec(d*x+c)^6*(a+a*sin(d*x+c))^(7/2)/d+5/128*a^(7/2)* arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d
Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.89 \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {15 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^6+2 a^3 \sqrt {a (1+\sin (c+d x))} \left (67-50 \sin (c+d x)+15 \sin ^2(c+d x)\right )}{384 d (-1+\sin (c+d x))^3} \]
-1/384*(15*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqr t[a])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6 + 2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(67 - 50*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d*x]) ^3)
Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3154, 3042, 3154, 3042, 3154, 3042, 3146, 73, 219}
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.
| \(\displaystyle \int \sec ^7(c+d x) (a \sin (c+d x)+a)^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{7/2}}{\cos (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {5}{12} a \int \sec ^5(c+d x) (\sin (c+d x) a+a)^{5/2}dx+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{12} a \int \frac {(\sin (c+d x) a+a)^{5/2}}{\cos (c+d x)^5}dx+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \int \sec ^3(c+d x) (\sin (c+d x) a+a)^{3/2}dx+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \int \frac {(\sin (c+d x) a+a)^{3/2}}{\cos (c+d x)^3}dx+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \left (\frac {1}{4} a \int \sec (c+d x) \sqrt {\sin (c+d x) a+a}dx+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\right )+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \left (\frac {1}{4} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\cos (c+d x)}dx+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\right )+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \left (\frac {a^2 \int \frac {1}{(a-a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}d(a \sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\right )+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \left (\frac {a^2 \int \frac {1}{2 a-a^2 \sin ^2(c+d x)}d\sqrt {\sin (c+d x) a+a}}{2 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\right )+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{8} a \left (\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\right )+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}\right )+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}\) |
(Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2))/(6*d) + (5*a*((Sec[c + d*x]^4* (a + a*Sin[c + d*x])^(5/2))/(4*d) + (3*a*((a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[2]])/(2*Sqrt[2]*d) + (Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/ 2))/(2*d)))/8))/12
3.2.52.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p + 1))), x] + Simp[a*((m + p + 1)/(g^2*(p + 1))) Int[(g* Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ [m + 1/2, 2*p]
Time = 0.66 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07
\[\frac {2 a^{7} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{12 a \left (a \sin \left (d x +c \right )-a \right )^{3}}-\frac {5 \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{8 a \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {3 \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \left (a \sin \left (d x +c \right )-a \right )}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{8 a}\right )}{12 a}\right )}{d}\]
2*a^7*(-1/12*(a+a*sin(d*x+c))^(1/2)/a/(a*sin(d*x+c)-a)^3-5/12/a*(-1/8*(a+a *sin(d*x+c))^(1/2)/a/(a*sin(d*x+c)-a)^2-3/8/a*(-1/4*(a+a*sin(d*x+c))^(1/2) /a/(a*sin(d*x+c)-a)+1/8/a^(3/2)*2^(1/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2) *2^(1/2)/a^(1/2)))))/d
Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.43 \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {15 \, {\left (3 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - 4 \, \sqrt {2} a^{3} - {\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - 4 \, \sqrt {2} a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{2} + 50 \, a^{3} \sin \left (d x + c\right ) - 82 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{768 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \]
1/768*(15*(3*sqrt(2)*a^3*cos(d*x + c)^2 - 4*sqrt(2)*a^3 - (sqrt(2)*a^3*cos (d*x + c)^2 - 4*sqrt(2)*a^3)*sin(d*x + c))*sqrt(a)*log(-(a*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a) + 3*a)/(sin(d*x + c) - 1)) + 4* (15*a^3*cos(d*x + c)^2 + 50*a^3*sin(d*x + c) - 82*a^3)*sqrt(a*sin(d*x + c) + a))/(3*d*cos(d*x + c)^2 - (d*cos(d*x + c)^2 - 4*d)*sin(d*x + c) - 4*d)
Timed out. \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.24 \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {15 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{5} - 80 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{6} + 132 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{7}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{2} - 8 \, a^{3}}}{768 \, a d} \]
-1/768*(15*sqrt(2)*a^(9/2)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a ))/(sqrt(2)*sqrt(a) + sqrt(a*sin(d*x + c) + a))) + 4*(15*(a*sin(d*x + c) + a)^(5/2)*a^5 - 80*(a*sin(d*x + c) + a)^(3/2)*a^6 + 132*sqrt(a*sin(d*x + c ) + a)*a^7)/((a*sin(d*x + c) + a)^3 - 6*(a*sin(d*x + c) + a)^2*a + 12*(a*s in(d*x + c) + a)*a^2 - 8*a^3))/(a*d)
Time = 0.32 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.95 \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {2 \, {\left (15 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - 15 \, \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 15 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{768 \, d} \]
-1/768*sqrt(2)*a^(7/2)*(2*(15*cos(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 40*cos(-1 /4*pi + 1/2*d*x + 1/2*c)^3 + 33*cos(-1/4*pi + 1/2*d*x + 1/2*c))/(cos(-1/4* pi + 1/2*d*x + 1/2*c)^2 - 1)^3 - 15*log(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1 ) + 15*log(-cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d
Timed out. \[ \int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^7} \,d x \]