3.17 \(\int \sin ^{\frac{7}{2}}(a+b x) \, dx\)

Optimal. Leaf size=70 \[ \frac{10 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac{10 \sqrt{\sin (a+b x)} \cos (a+b x)}{21 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0287382, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2641} \[ \frac{10 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac{10 \sqrt{\sin (a+b x)} \cos (a+b x)}{21 b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^(7/2),x]

[Out]

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

Rule 2635

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] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \sin ^{\frac{7}{2}}(a+b x) \, dx &=-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{5}{7} \int \sin ^{\frac{3}{2}}(a+b x) \, dx\\ &=-\frac{10 \cos (a+b x) \sqrt{\sin (a+b x)}}{21 b}-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{5}{21} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{10 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{21 b}-\frac{10 \cos (a+b x) \sqrt{\sin (a+b x)}}{21 b}-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.118821, size = 55, normalized size = 0.79 \[ \frac{\sqrt{\sin (a+b x)} (3 \cos (3 (a+b x))-23 \cos (a+b x))-20 F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{42 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^(7/2),x]

[Out]

(-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]])/(42*b)

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 104, normalized size = 1.5 \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) } \left ({\frac{2\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{7}}+{\frac{5}{21}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) }{21}} \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^(7/2),x)

[Out]

(2/7*sin(b*x+a)*cos(b*x+a)^4+5/21*(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))-16/21*cos(b*x+a)^2*sin(b*x+a))/cos(b*x+a)/sin(b*x+a)^(1/2)/b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(cos(b*x + a)^2 - 1)*sin(b*x + a)^(3/2), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(7/2), x)