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

Optimal. Leaf size=105 \[ -\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}} \]

[Out]

(-2*Cos[a + b*x])/(5*b*c*(c*Sin[a + b*x])^(5/2)) - (6*Cos[a + b*x])/(5*b*c^3*Sqrt[c*Sin[a + b*x]]) - (6*Ellipt
icE[(a - Pi/2 + b*x)/2, 2]*Sqrt[c*Sin[a + b*x]])/(5*b*c^4*Sqrt[Sin[a + b*x]])

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Rubi [A]  time = 0.0510645, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2640, 2639} \[ -\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(-7/2),x]

[Out]

(-2*Cos[a + b*x])/(5*b*c*(c*Sin[a + b*x])^(5/2)) - (6*Cos[a + b*x])/(5*b*c^3*Sqrt[c*Sin[a + b*x]]) - (6*Ellipt
icE[(a - Pi/2 + b*x)/2, 2]*Sqrt[c*Sin[a + b*x]])/(5*b*c^4*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 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.157936, size = 68, normalized size = 0.65 \[ -\frac{2 \left (\frac{3}{2} \sin (2 (a+b x))+\cot (a+b x)-3 \sin ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{5 b c^2 (c \sin (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(-7/2),x]

[Out]

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

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Maple [A]  time = 0.042, size = 168, normalized size = 1.6 \begin{align*}{\frac{1}{5\,{c}^{3} \left ( \sin \left ( bx+a \right ) \right ) ^{3}\cos \left ( bx+a \right ) b} \left ( 6\,\sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2} \left ( \sin \left ( bx+a \right ) \right ) ^{7/2}{\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} \left ( \sin \left ( bx+a \right ) \right ) ^{7/2}{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}-4\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}-2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/c^3*(6*(-sin(b*x+a)+1)^(1/2)*(2*sin(b*x+a)+2)^(1/2)*sin(b*x+a)^(7/2)*EllipticE((-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)^(7/2)*EllipticF((-sin(b*x+a)+1)^(1/2),1/2*2^
(1/2))+6*sin(b*x+a)^5-4*sin(b*x+a)^3-2*sin(b*x+a))/sin(b*x+a)^3/cos(b*x+a)/(c*sin(b*x+a))^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*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{c \sin \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{4} - 2 \, c^{4} \cos \left (b x + a\right )^{2} + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*sin(b*x + a))/(c^4*cos(b*x + a)^4 - 2*c^4*cos(b*x + a)^2 + c^4), 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/(c*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}{\left (c \sin \left (b x + a\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*sin(b*x + a))^(-7/2), x)