3.4.89 \(\int \csc (4 x) \sin ^3(x) \, dx\) [389]

Optimal. Leaf size=26 \[ -\frac {1}{4} \tanh ^{-1}(\sin (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}} \]

[Out]

-1/4*arctanh(sin(x))+1/8*arctanh(sin(x)*2^(1/2))*2^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1144, 213} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[4*x]*Sin[x]^3,x]

[Out]

-1/4*ArcTanh[Sin[x]] + ArcTanh[Sqrt[2]*Sin[x]]/(4*Sqrt[2])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rubi steps

\begin {align*} \int \csc (4 x) \sin ^3(x) \, dx &=\text {Subst}\left (\int \frac {x^2}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sin (x)\right )-\text {Subst}\left (\int \frac {1}{-4+8 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{4} \tanh ^{-1}(\sin (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 218, normalized size = 8.38 \begin {gather*} \frac {-2 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\left (-1+\sqrt {2}\right ) \sin \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )-2 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sin \left (\frac {x}{2}\right )}{\left (-1+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )+4 \sqrt {2} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-4 \sqrt {2} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+2 \log \left (\sqrt {2}+2 \sin (x)\right )-\log \left (2-\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )-\log \left (2+\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )}{16 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[4*x]*Sin[x]^3,x]

[Out]

((-2*I)*ArcTan[(Cos[x/2] - (-1 + Sqrt[2])*Sin[x/2])/((1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] - (2*I)*ArcTan[(Cos[x
/2] - (1 + Sqrt[2])*Sin[x/2])/((-1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] + 4*Sqrt[2]*Log[Cos[x/2] - Sin[x/2]] - 4*S
qrt[2]*Log[Cos[x/2] + Sin[x/2]] + 2*Log[Sqrt[2] + 2*Sin[x]] - Log[2 - Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]] - Log[2
+ Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/(16*Sqrt[2])

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Maple [A]
time = 0.10, size = 28, normalized size = 1.08

method	result	size
default	\(\frac {\arctanh \left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{8}+\frac {\ln \left (-1+\sin \left (x \right )\right )}{8}-\frac {\ln \left (\sin \left (x \right )+1\right )}{8}\)	\(28\)
risch	\(-\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{4}+\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/sin(4*x),x,method=_RETURNVERBOSE)

[Out]

1/8*arctanh(sin(x)*2^(1/2))*2^(1/2)+1/8*ln(-1+sin(x))-1/8*ln(sin(x)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (18) = 36\).
time = 2.45, size = 171, normalized size = 6.58 \begin {gather*} \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/sin(4*x),x, algorithm="maxima")

[Out]

1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 2*sqrt(2)*sin(x) + 2) - 1/32*sqrt(2)*log(2*cos(x
)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) - 2*sqrt(2)*sin(x) + 2) + 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqr
t(2)*cos(x) + 2*sqrt(2)*sin(x) + 2) - 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) - 2*sqrt(2)*
sin(x) + 2) - 1/8*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + 1/8*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
time = 0.99, size = 50, normalized size = 1.92 \begin {gather*} \frac {1}{16} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/sin(4*x),x, algorithm="fricas")

[Out]

1/16*sqrt(2)*log(-(2*cos(x)^2 - 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x
) + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (22) = 44\).
time = 8.02, size = 294, normalized size = 11.31 \begin {gather*} \frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/sin(4*x),x)

[Out]

4093147632754948*log(tan(x/2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) + 2894292447518688*sqrt(2)*
log(tan(x/2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 4093147632754948*log(tan(x/2) + 1)/(163725
90531019792 + 11577169790074752*sqrt(2)) - 2894292447518688*sqrt(2)*log(tan(x/2) + 1)/(16372590531019792 + 115
77169790074752*sqrt(2)) + 1447146223759344*log(tan(x/2) - 1 + sqrt(2))/(16372590531019792 + 11577169790074752*
sqrt(2)) + 1023286908188737*sqrt(2)*log(tan(x/2) - 1 + sqrt(2))/(16372590531019792 + 11577169790074752*sqrt(2)
) + 1447146223759344*log(tan(x/2) + 1 + sqrt(2))/(16372590531019792 + 11577169790074752*sqrt(2)) + 10232869081
88737*sqrt(2)*log(tan(x/2) + 1 + sqrt(2))/(16372590531019792 + 11577169790074752*sqrt(2)) - 1447146223759344*l
og(tan(x/2) - sqrt(2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 1023286908188737*sqrt(2)*log(tan(
x/2) - sqrt(2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 1447146223759344*log(tan(x/2) - sqrt(2)
+ 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 1023286908188737*sqrt(2)*log(tan(x/2) - sqrt(2) + 1)/(1
6372590531019792 + 11577169790074752*sqrt(2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
time = 1.11, size = 48, normalized size = 1.85 \begin {gather*} -\frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/sin(4*x),x, algorithm="giac")

[Out]

-1/16*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x
) + 1)

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Mupad [B]
time = 0.29, size = 27, normalized size = 1.04 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{8}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/sin(4*x),x)

[Out]

(2^(1/2)*atanh(2^(1/2)*sin(x)))/8 - atanh(sin(x/2)/cos(x/2))/2

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