\(\int (c \sin (a+b x))^{7/2} \, dx\) [25]

Optimal result

Integrand size = 12, antiderivative size = 103 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {10 c^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]

[Out]

-2/7*c*cos(b*x+a)*(c*sin(b*x+a))^(5/2)/b-10/21*c^4*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*
x)*EllipticF(cos(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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {10 c^4 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{21 b \sqrt {c \sin (a+b x)}}-\frac {10 c^3 \cos (a+b x) \sqrt {c \sin (a+b x)}}{21 b}-\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]

[In]

Int[(c*Sin[a + b*x])^(7/2),x]

[Out]

(10*c^4*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(21*b*Sqrt[c*Sin[a + b*x]]) - (10*c^3*Cos[a + b*x
]*Sqrt[c*Sin[a + b*x]])/(21*b) - (2*c*Cos[a + b*x]*(c*Sin[a + b*x])^(5/2))/(7*b)

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2720

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac {1}{7} \left (5 c^2\right ) \int (c \sin (a+b x))^{3/2} \, dx \\ & = -\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}+\frac {1}{21} \left (5 c^4\right ) \int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx \\ & = -\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}+\frac {\left (5 c^4 \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{21 \sqrt {c \sin (a+b x)}} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {c^3 \left (-20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+(-23 \cos (a+b x)+3 \cos (3 (a+b x))) \sqrt {\sin (a+b x)}\right ) \sqrt {c \sin (a+b x)}}{42 b \sqrt {\sin (a+b x)}} \]

[In]

Integrate[(c*Sin[a + b*x])^(7/2),x]

[Out]

(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]])*Sq
rt[c*Sin[a + b*x]])/(42*b*Sqrt[Sin[a + b*x]])

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {c^{4} \left (-6 \left (\sin ^{5}\left (b x +a \right )\right )+5 \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) F\left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-4 \left (\sin ^{3}\left (b x +a \right )\right )+10 \sin \left (b x +a \right )\right )}{21 \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\)	\(108\)

[In]

int((c*sin(b*x+a))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/21*c^4*(-6*sin(b*x+a)^5+5*(-sin(b*x+a)+1)^(1/2)*(2*sin(b*x+a)+2)^(1/2)*sin(b*x+a)^(1/2)*EllipticF((-sin(b*x
+a)+1)^(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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int (c \sin (a+b x))^{7/2} \, dx=\frac {5 \, \sqrt {2} \sqrt {-i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 \, \sqrt {2} \sqrt {i \, c} c^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{3} - 8 \, c^{3} \cos \left (b x + a\right )\right )} \sqrt {c \sin \left (b x + a\right )}}{21 \, b} \]

[In]

integrate((c*sin(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

1/21*(5*sqrt(2)*sqrt(-I*c)*c^3*weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*sqrt(2)*sqrt(I*c)*
c^3*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)) + 2*(3*c^3*cos(b*x + a)^3 - 8*c^3*cos(b*x + a))*s
qrt(c*sin(b*x + a)))/b

Sympy [F(-1)]

Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((c*sin(b*x+a))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(7/2), x)

Giac [F]

\[ \int (c \sin (a+b x))^{7/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int (c \sin (a+b x))^{7/2} \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^{7/2} \,d x \]

[In]

int((c*sin(a + b*x))^(7/2),x)

[Out]

int((c*sin(a + b*x))^(7/2), x)