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

Optimal. Leaf size=90 \[ -\frac{2 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{6 \cos (a+b x) \sqrt{\csc (a+b x)}}{5 b}-\frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0367819, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3768, 3771, 2639} \[ -\frac{2 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{6 \cos (a+b x) \sqrt{\csc (a+b x)}}{5 b}-\frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(7/2),x]

[Out]

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

Rule 3768

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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

Mathematica [A]  time = 0.200648, size = 63, normalized size = 0.7 \[ \frac{\csc ^{\frac{5}{2}}(a+b x) \left (-7 \cos (a+b x)+3 \cos (3 (a+b x))+12 \sin ^{\frac{5}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{10 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(7/2),x]

[Out]

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

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Maple [A]  time = 1.171, size = 160, normalized size = 1.8 \begin{align*}{\frac{1}{5\,\cos \left ( bx+a \right ) b} \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(csc(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 \csc \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b x + a\right )^{\frac{7}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csc(b*x + a)^(7/2), 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(csc(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 \csc \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*x + a)^(7/2), x)