Integrand size = 23, antiderivative size = 89 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {64 a^3 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d} \] Output:
64/3*a^3*sec(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d-16/3*a^2*sec(d*x+c)*(a+a*sin( d*x+c))^(3/2)/d-2/3*a*sec(d*x+c)*(a+a*sin(d*x+c))^(5/2)/d
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.54 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {a^3 \sec (c+d x) (45+\cos (2 (c+d x))-20 \sin (c+d x)) \sqrt {a (1+\sin (c+d x))}}{3 d} \] Input:
Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(a^3*Sec[c + d*x]*(45 + Cos[2*(c + d*x)] - 20*Sin[c + d*x])*Sqrt[a*(1 + Si n[c + d*x])])/(3*d)
Time = 0.52 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3153, 3042, 3153, 3042, 3152}
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.
\(\displaystyle \int \sec ^2(c+d x) (a \sin (c+d x)+a)^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{7/2}}{\cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3153 |
\(\displaystyle \frac {8}{3} a \int \sec ^2(c+d x) (\sin (c+d x) a+a)^{5/2}dx-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{3} a \int \frac {(\sin (c+d x) a+a)^{5/2}}{\cos (c+d x)^2}dx-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}\) |
\(\Big \downarrow \) 3153 |
\(\displaystyle \frac {8}{3} a \left (4 a \int \sec ^2(c+d x) (\sin (c+d x) a+a)^{3/2}dx-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\right )-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{3} a \left (4 a \int \frac {(\sin (c+d x) a+a)^{3/2}}{\cos (c+d x)^2}dx-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\right )-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}\) |
\(\Big \downarrow \) 3152 |
\(\displaystyle \frac {8}{3} a \left (\frac {8 a^2 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d}-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\right )-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}\) |
Input:
Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(-2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(3*d) + (8*a*((8*a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x] )^(3/2))/d))/3
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 - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
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 - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Time = 4.94 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {2 a^{4} \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )^{2}+10 \sin \left (d x +c \right )-23\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(55\) |
Input:
int(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/3*a^4*(1+sin(d*x+c))*(sin(d*x+c)^2+10*sin(d*x+c)-23)/cos(d*x+c)/(a+a*si n(d*x+c))^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 10 \, a^{3} \sin \left (d x + c\right ) + 22 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, d \cos \left (d x + c\right )} \] Input:
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
2/3*(a^3*cos(d*x + c)^2 - 10*a^3*sin(d*x + c) + 22*a^3)*sqrt(a*sin(d*x + c ) + a)/(d*cos(d*x + c))
Timed out. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**2*(a+a*sin(d*x+c))**(7/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (77) = 154\).
Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.66 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2 \, {\left (23 \, a^{\frac {7}{2}} - \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {88 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {130 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {88 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {23 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \] Input:
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
-2/3*(23*a^(7/2) - 20*a^(7/2)*sin(d*x + c)/(cos(d*x + c) + 1) + 88*a^(7/2) *sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 60*a^(7/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 130*a^(7/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 60*a^(7/2) *sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 88*a^(7/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 20*a^(7/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 23*a^(7/2)* sin(d*x + c)^8/(cos(d*x + c) + 1)^8)/(d*(sin(d*x + c)/(cos(d*x + c) + 1) - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(7/2))
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.18 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {4 \, \sqrt {2} {\left (a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )} \sqrt {a}}{3 \, d} \] Input:
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Output:
4/3*sqrt(2)*(a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 6*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2* d*x + 1/2*c) - 3*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/sin(-1/4*pi + 1/2 *d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:
int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^2,x)
Output:
int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^2, x)
\[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\sqrt {a}\, a^{3} \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}d x +3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2} \sin \left (d x +c \right )d x \right )+\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2}d x \right ) \] Input:
int(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x)
Output:
sqrt(a)*a**3*(int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**2*sin(c + d*x)**3,x ) + 3*int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**2*sin(c + d*x)**2,x) + 3*in t(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**2*sin(c + d*x),x) + int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**2,x))