Integrand size = 23, antiderivative size = 61 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {8 a^2 \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d} \] Output:
-8/3*a^2*sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2)/d+2*a*sec(d*x+c)^3*(a+a*sin(d *x+c))^(5/2)/d
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 a^2 \sec ^3(c+d x) (a (1+\sin (c+d x)))^{3/2} (-1+3 \sin (c+d x))}{3 d} \] Input:
Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(2*a^2*Sec[c + d*x]^3*(a*(1 + Sin[c + d*x]))^(3/2)*(-1 + 3*Sin[c + d*x]))/ (3*d)
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {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.
\(\displaystyle \int \sec ^4(c+d x) (a \sin (c+d x)+a)^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{7/2}}{\cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3153 |
\(\displaystyle \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-4 a \int \sec ^4(c+d x) (\sin (c+d x) a+a)^{5/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-4 a \int \frac {(\sin (c+d x) a+a)^{5/2}}{\cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3152 |
\(\displaystyle \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac {8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\) |
Input:
Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(-8*a^2*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d) + (2*a*Sec[c + d* x]^3*(a + a*Sin[c + d*x])^(5/2))/d
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 = 121.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {2 a^{4} \left (1+\sin \left (d x +c \right )\right ) \left (3 \sin \left (d x +c \right )-1\right )}{3 \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(57\) |
Input:
int(sec(d*x+c)^4*(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)-1)*(3*sin(d*x+c)-1)/cos(d*x+c)/(a+a*si n(d*x+c))^(1/2)/d
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2 \, {\left (3 \, a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-2/3*(3*a^3*sin(d*x + c) - a^3)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c)*s in(d*x + c) - d*cos(d*x + c))
Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**4*(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. 320 vs. \(2 (55) = 110\).
Time = 0.17 (sec) , antiderivative size = 320, normalized size of antiderivative = 5.25 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (a^{\frac {7}{2}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {10 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {a^{\frac {7}{2}} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )}}{3 \, d {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
2/3*(a^(7/2) - 6*a^(7/2)*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a^(7/2)*sin(d *x + c)^2/(cos(d*x + c) + 1)^2 - 24*a^(7/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*a^(7/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^(7/2)*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 + 10*a^(7/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 24*a^(7/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 5*a^(7/2)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 6*a^(7/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^ 9 + a^(7/2)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)/(d*(3*sin(d*x + c)/(cos (d*x + c) + 1) - 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(c os(d*x + c) + 1)^3 - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(7/2))
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {2} {\left (3 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{3 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Output:
1/3*sqrt(2)*(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)^2 - a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))*sqrt(a)/(d*sin(-1 /4*pi + 1/2*d*x + 1/2*c)^3)
Time = 37.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.93 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {a^3\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\left (3-3\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}\right )\,4{}\mathrm {i}}{3\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )\,{\left (1+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )}^3} \] Input:
int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^4,x)
Output:
-(a^3*exp(c*1i + d*x*1i)*(a + a*((exp(- c*1i - d*x*1i)*1i)/2 - (exp(c*1i + d*x*1i)*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*2i - 3*exp(c*2i + d*x*2i) + 3)* 4i)/(3*d*(exp(c*1i + d*x*1i) + 1i)*(exp(c*1i + d*x*1i)*1i + 1)^3)
\[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\sqrt {a}\, a^{3} \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{4} \sin \left (d x +c \right )^{3}d x +3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{4} \sin \left (d x +c \right )d x \right )+\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{4}d x \right ) \] Input:
int(sec(d*x+c)^4*(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)**4*sin(c + d*x)**3,x ) + 3*int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**4*sin(c + d*x)**2,x) + 3*in t(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**4*sin(c + d*x),x) + int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**4,x))