3.16 \(\int \frac{1}{\sin ^{\frac{7}{2}}(b x)} \, dx\)

Optimal. Leaf size=60 \[ \frac{6 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{5 b}-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}-\frac{6 \cos (b x)}{5 b \sqrt{\sin (b x)}} \]

[Out]

(6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b) - (2*Cos[b*x])/(5*b*Sin[b*x]^(5/2)) - (6*Cos[b*x])/(5*b*Sqrt[Sin[b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0243357, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2639} \[ \frac{6 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{5 b}-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}-\frac{6 \cos (b x)}{5 b \sqrt{\sin (b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[b*x]^(-7/2),x]

[Out]

(6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b) - (2*Cos[b*x])/(5*b*Sin[b*x]^(5/2)) - (6*Cos[b*x])/(5*b*Sqrt[Sin[b*x]])

Rule 2636

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

Rule 2639

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

Rubi steps

\begin{align*} \int \frac{1}{\sin ^{\frac{7}{2}}(b x)} \, dx &=-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}+\frac{3}{5} \int \frac{1}{\sin ^{\frac{3}{2}}(b x)} \, dx\\ &=-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}-\frac{6 \cos (b x)}{5 b \sqrt{\sin (b x)}}-\frac{3}{5} \int \sqrt{\sin (b x)} \, dx\\ &=\frac{6 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{5 b}-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}-\frac{6 \cos (b x)}{5 b \sqrt{\sin (b x)}}\\ \end{align*}

Mathematica [A]  time = 0.049694, size = 51, normalized size = 0.85 \[ \frac{-7 \cos (b x)+3 \cos (3 b x)+12 \sin ^{\frac{5}{2}}(b x) E\left (\left .\frac{1}{4} (\pi -2 b x)\right |2\right )}{10 b \sin ^{\frac{5}{2}}(b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[b*x]^(-7/2),x]

[Out]

(-7*Cos[b*x] + 3*Cos[3*b*x] + 12*EllipticE[(Pi - 2*b*x)/4, 2]*Sin[b*x]^(5/2))/(10*b*Sin[b*x]^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 132, normalized size = 2.2 \begin{align*}{\frac{1}{5\,b\cos \left ( bx \right ) } \left ( 6\,\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) } \left ( \sin \left ( bx \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{\sin \left ( bx \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) } \left ( \sin \left ( bx \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{\sin \left ( bx \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( bx \right ) \right ) ^{4}-4\, \left ( \sin \left ( bx \right ) \right ) ^{2}-2 \right ) \left ( \sin \left ( bx \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x)^(7/2),x)

[Out]

1/5/sin(b*x)^(5/2)*(6*(sin(b*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*sin(b*x)^2*EllipticE((sin(b*x
)+1)^(1/2),1/2*2^(1/2))-3*(sin(b*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*sin(b*x)^2*EllipticF((sin
(b*x)+1)^(1/2),1/2*2^(1/2))+6*sin(b*x)^4-4*sin(b*x)^2-2)/cos(b*x)/b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \left (b x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x)^(-7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\sin \left (b x\right )}}{\cos \left (b x\right )^{4} - 2 \, \cos \left (b x\right )^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(sin(b*x))/(cos(b*x)^4 - 2*cos(b*x)^2 + 1), 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(1/sin(b*x)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \left (b x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(7/2),x, algorithm="giac")

[Out]

integrate(sin(b*x)^(-7/2), x)