3.7 \(\int \csc ^7(a+b x) \, dx\)

Optimal. Leaf size=76 \[ -\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b} \]

[Out]

(-5*ArcTanh[Cos[a + b*x]])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x]^3)/(24
*b) - (Cot[a + b*x]*Csc[a + b*x]^5)/(6*b)

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Rubi [A]  time = 0.0363369, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3770} \[ -\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^7,x]

[Out]

(-5*ArcTanh[Cos[a + b*x]])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x]^3)/(24
*b) - (Cot[a + b*x]*Csc[a + b*x]^5)/(6*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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc ^7(a+b x) \, dx &=-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{6} \int \csc ^5(a+b x) \, dx\\ &=-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{8} \int \csc ^3(a+b x) \, dx\\ &=-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{16} \int \csc (a+b x) \, dx\\ &=-\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.013899, size = 151, normalized size = 1.99 \[ -\frac{\csc ^6\left (\frac{1}{2} (a+b x)\right )}{384 b}-\frac{\csc ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}-\frac{5 \csc ^2\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{\sec ^6\left (\frac{1}{2} (a+b x)\right )}{384 b}+\frac{\sec ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{5 \sec ^2\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{5 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{16 b}-\frac{5 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{16 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^7,x]

[Out]

(-5*Csc[(a + b*x)/2]^2)/(64*b) - Csc[(a + b*x)/2]^4/(64*b) - Csc[(a + b*x)/2]^6/(384*b) - (5*Log[Cos[(a + b*x)
/2]])/(16*b) + (5*Log[Sin[(a + b*x)/2]])/(16*b) + (5*Sec[(a + b*x)/2]^2)/(64*b) + Sec[(a + b*x)/2]^4/(64*b) +
Sec[(a + b*x)/2]^6/(384*b)

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Maple [A]  time = 0.022, size = 78, normalized size = 1. \begin{align*} -{\frac{\cot \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{5}}{6\,b}}-{\frac{5\,\cot \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{3}}{24\,b}}-{\frac{5\,\csc \left ( bx+a \right ) \cot \left ( bx+a \right ) }{16\,b}}+{\frac{5\,\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{16\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^7,x)

[Out]

-1/6*cot(b*x+a)*csc(b*x+a)^5/b-5/24*cot(b*x+a)*csc(b*x+a)^3/b-5/16*cot(b*x+a)*csc(b*x+a)/b+5/16/b*ln(csc(b*x+a
)-cot(b*x+a))

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Maxima [A]  time = 1.04107, size = 123, normalized size = 1.62 \begin{align*} \frac{\frac{2 \,{\left (15 \, \cos \left (b x + a\right )^{5} - 40 \, \cos \left (b x + a\right )^{3} + 33 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{96 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(2*(15*cos(b*x + a)^5 - 40*cos(b*x + a)^3 + 33*cos(b*x + a))/(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b
*x + a)^2 - 1) - 15*log(cos(b*x + a) + 1) + 15*log(cos(b*x + a) - 1))/b

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Fricas [B]  time = 0.497951, size = 424, normalized size = 5.58 \begin{align*} \frac{30 \, \cos \left (b x + a\right )^{5} - 80 \, \cos \left (b x + a\right )^{3} - 15 \,{\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 66 \, \cos \left (b x + a\right )}{96 \,{\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(30*cos(b*x + a)^5 - 80*cos(b*x + a)^3 - 15*(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2 - 1)*lo
g(1/2*cos(b*x + a) + 1/2) + 15*(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2 - 1)*log(-1/2*cos(b*x + a
) + 1/2) + 66*cos(b*x + a))/(b*cos(b*x + a)^6 - 3*b*cos(b*x + a)^4 + 3*b*cos(b*x + a)^2 - b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{7}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**7,x)

[Out]

Integral(csc(a + b*x)**7, x)

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Giac [B]  time = 1.29354, size = 246, normalized size = 3.24 \begin{align*} -\frac{\frac{{\left (\frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{45 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{110 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac{45 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 60 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*((9*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 45*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 110*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 - 1)*(cos(b*x + a) + 1)^3/(cos(b*x + a) - 1)^3 + 45*(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) - 9*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + (cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 60*l
og(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b