3.3 \(\int \csc ^3(a+b x) \, dx\)

Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]

[Out]

-1/2*arctanh(cos(b*x+a))/b-1/2*cot(b*x+a)*csc(b*x+a)/b

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3770} \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3,x]

[Out]

-ArcTanh[Cos[a + b*x]]/(2*b) - (Cot[a + b*x]*Csc[a + b*x])/(2*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 ^3(a+b x) \, dx &=-\frac {\cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int \csc (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 75, normalized size = 2.21 \[ -\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3,x]

[Out]

-1/8*Csc[(a + b*x)/2]^2/b - Log[Cos[(a + b*x)/2]]/(2*b) + Log[Sin[(a + b*x)/2]]/(2*b) + Sec[(a + b*x)/2]^2/(8*
b)

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fricas [B]  time = 0.78, size = 72, normalized size = 2.12 \[ -\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{4 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/4*((cos(b*x + a)^2 - 1)*log(1/2*cos(b*x + a) + 1/2) - (cos(b*x + a)^2 - 1)*log(-1/2*cos(b*x + a) + 1/2) - 2
*cos(b*x + a))/(b*cos(b*x + a)^2 - b)

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giac [B]  time = 0.47, size = 92, normalized size = 2.71 \[ -\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="giac")

[Out]

-1/8*((2*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) + (cos(b*x + a) - 1)
/(cos(b*x + a) + 1) - 2*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b

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maple [A]  time = 0.85, size = 40, normalized size = 1.18 \[ -\frac {\cot \left (b x +a \right ) \csc \left (b x +a \right )}{2 b}+\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3,x)

[Out]

-1/2*cot(b*x+a)*csc(b*x+a)/b+1/2/b*ln(csc(b*x+a)-cot(b*x+a))

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maxima [A]  time = 0.32, size = 46, normalized size = 1.35 \[ \frac {\frac {2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(2*cos(b*x + a)/(cos(b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b

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mupad [B]  time = 0.11, size = 36, normalized size = 1.06 \[ \frac {\cos \left (a+b\,x\right )}{2\,b\,\left ({\cos \left (a+b\,x\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*x)^3,x)

[Out]

cos(a + b*x)/(2*b*(cos(a + b*x)^2 - 1)) - atanh(cos(a + b*x))/(2*b)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{3}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**3,x)

[Out]

Integral(csc(a + b*x)**3, x)

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