Integrand size = 12, antiderivative size = 103 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {10 c^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \] Output:
10/21*c^4*InverseJacobiAM(1/2*a-1/4*Pi+1/2*b*x,2^(1/2))*sin(b*x+a)^(1/2)/b /(c*sin(b*x+a))^(1/2)-10/21*c^3*cos(b*x+a)*(c*sin(b*x+a))^(1/2)/b-2/7*c*co s(b*x+a)*(c*sin(b*x+a))^(5/2)/b
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {c^3 \left (-20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+(-23 \cos (a+b x)+3 \cos (3 (a+b x))) \sqrt {\sin (a+b x)}\right ) \sqrt {c \sin (a+b x)}}{42 b \sqrt {\sin (a+b x)}} \] Input:
Integrate[(c*Sin[a + b*x])^(7/2),x]
Output:
(c^3*(-20*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + (-23*Cos[a + b*x] + 3*Cos[ 3*(a + b*x)])*Sqrt[Sin[a + b*x]])*Sqrt[c*Sin[a + b*x]])/(42*b*Sqrt[Sin[a + b*x]])
Time = 0.72 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3115, 3042, 3115, 3042, 3121, 3042, 3120}
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.
\(\displaystyle \int (c \sin (a+b x))^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c \sin (a+b x))^{7/2}dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{7} c^2 \int (c \sin (a+b x))^{3/2}dx-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} c^2 \int (c \sin (a+b x))^{3/2}dx-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{7} c^2 \left (\frac {1}{3} c^2 \int \frac {1}{\sqrt {c \sin (a+b x)}}dx-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\right )-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} c^2 \left (\frac {1}{3} c^2 \int \frac {1}{\sqrt {c \sin (a+b x)}}dx-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\right )-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {5}{7} c^2 \left (\frac {c^2 \sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx}{3 \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\right )-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} c^2 \left (\frac {c^2 \sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx}{3 \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\right )-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {5}{7} c^2 \left (\frac {2 c^2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\right )-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\) |
Input:
Int[(c*Sin[a + b*x])^(7/2),x]
Output:
(-2*c*Cos[a + b*x]*(c*Sin[a + b*x])^(5/2))/(7*b) + (5*c^2*((2*c^2*Elliptic F[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*b*Sqrt[c*Sin[a + b*x]]) - (2*c*Cos[a + b*x]*Sqrt[c*Sin[a + b*x]])/(3*b)))/7
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Time = 3.66 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {c^{4} \left (-6 \sin \left (b x +a \right )^{5}+5 \sqrt {1-\sin \left (b x +a \right )}\, \sqrt {2+2 \sin \left (b x +a \right )}\, \sqrt {\sin \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-4 \sin \left (b x +a \right )^{3}+10 \sin \left (b x +a \right )\right )}{21 \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\) | \(108\) |
Input:
int((c*sin(b*x+a))^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/21*c^4*(-6*sin(b*x+a)^5+5*(1-sin(b*x+a))^(1/2)*(2+2*sin(b*x+a))^(1/2)*s in(b*x+a)^(1/2)*EllipticF((1-sin(b*x+a))^(1/2),1/2*2^(1/2))-4*sin(b*x+a)^3 +10*sin(b*x+a))/cos(b*x+a)/(c*sin(b*x+a))^(1/2)/b
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.95 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {2 \, {\left (5 \, \sqrt {-\frac {1}{2} i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 \, \sqrt {\frac {1}{2} i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (3 \, c^{3} \cos \left (b x + a\right )^{3} - 8 \, c^{3} \cos \left (b x + a\right )\right )} \sqrt {c \sin \left (b x + a\right )}\right )}}{21 \, b} \] Input:
integrate((c*sin(b*x+a))^(7/2),x, algorithm="fricas")
Output:
2/21*(5*sqrt(-1/2*I*c)*c^3*weierstrassPInverse(4, 0, cos(b*x + a) + I*sin( b*x + a)) + 5*sqrt(1/2*I*c)*c^3*weierstrassPInverse(4, 0, cos(b*x + a) - I *sin(b*x + a)) + (3*c^3*cos(b*x + a)^3 - 8*c^3*cos(b*x + a))*sqrt(c*sin(b* x + a)))/b
Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((c*sin(b*x+a))**(7/2),x)
Output:
Timed out
\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*sin(b*x+a))^(7/2),x, algorithm="maxima")
Output:
integrate((c*sin(b*x + a))^(7/2), x)
\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*sin(b*x+a))^(7/2),x, algorithm="giac")
Output:
integrate((c*sin(b*x + a))^(7/2), x)
Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^{7/2} \,d x \] Input:
int((c*sin(a + b*x))^(7/2),x)
Output:
int((c*sin(a + b*x))^(7/2), x)
\[ \int (c \sin (a+b x))^{7/2} \, dx=\sqrt {c}\, \left (\int \sqrt {\sin \left (b x +a \right )}\, \sin \left (b x +a \right )^{3}d x \right ) c^{3} \] Input:
int((c*sin(b*x+a))^(7/2),x)
Output:
sqrt(c)*int(sqrt(sin(a + b*x))*sin(a + b*x)**3,x)*c**3