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

Optimal. Leaf size=70 \[ -\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0268507, 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 = {2636, 2639} \[ -\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^(-7/2),x]

[Out]

(-6*EllipticE[(a - Pi/2 + b*x)/2, 2])/(5*b) - (2*Cos[a + b*x])/(5*b*Sin[a + b*x]^(5/2)) - (6*Cos[a + b*x])/(5*
b*Sqrt[Sin[a + 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}}(a+b x)} \, dx &=-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}+\frac{3}{5} \int \frac{1}{\sin ^{\frac{3}{2}}(a+b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}}-\frac{3}{5} \int \sqrt{\sin (a+b x)} \, dx\\ &=-\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.284052, size = 55, normalized size = 0.79 \[ \frac{2 \left (3 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )-\frac{\left (3 \sin ^2(a+b x)+1\right ) \cos (a+b x)}{\sin ^{\frac{5}{2}}(a+b x)}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^(-7/2),x]

[Out]

(2*(3*EllipticE[(-2*a + Pi - 2*b*x)/4, 2] - (Cos[a + b*x]*(1 + 3*Sin[a + b*x]^2))/Sin[a + b*x]^(5/2)))/(5*b)

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Maple [A]  time = 0.031, size = 160, normalized size = 2.3 \begin{align*}{\frac{1}{5\,b\cos \left ( bx+a \right ) } \left ( 6\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) } \left ( \sin \left ( bx+a \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) } \left ( \sin \left ( bx+a \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}-4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}-2 \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/sin(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x + a)^(-7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+a)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/sin(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate(sin(b*x + a)^(-7/2), x)