3.488 \(\int x \sec (x) \tan ^3(x) \, dx\)
Optimal. Leaf size=30 \[ \frac{1}{3} x \sec ^3(x)-x \sec (x)+\frac{5}{6} \tanh ^{-1}(\sin (x))-\frac{1}{6} \tan (x) \sec (x) \]
[Out]
(5*ArcTanh[Sin[x]])/6 - x*Sec[x] + (x*Sec[x]^3)/3 - (Sec[x]*Tan[x])/6
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Rubi [A] time = 0.061406, antiderivative size = 30, normalized size of antiderivative =
1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5
\[ \frac{1}{3} x \sec ^3(x)-x \sec (x)+\frac{5}{6} \tanh ^{-1}(\sin (x))-\frac{1}{6} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
[In] Int[x*Sec[x]*Tan[x]^3,x]
[Out]
(5*ArcTanh[Sin[x]])/6 - x*Sec[x] + (x*Sec[x]^3)/3 - (Sec[x]*Tan[x])/6
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sin ^{3}{\left (x \right )}}{\cos ^{4}{\left (x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*sin(x)**3/cos(x)**4,x)
[Out]
Integral(x*sin(x)**3/cos(x)**4, x)
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Mathematica [B] time = 0.182542, size = 104, normalized size = 3.47 \[ -\frac{1}{24} \sec ^3(x) \left (4 x+2 \sin (2 x)+12 x \cos (2 x)+5 \cos (3 x) \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+15 \cos (x) \left (\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )-5 \cos (3 x) \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*Sec[x]*Tan[x]^3,x]
[Out]
-(Sec[x]^3*(4*x + 12*x*Cos[2*x] + 5*Cos[3*x]*Log[Cos[x/2] - Sin[x/2]] + 15*Cos[x
]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) - 5*Cos[3*x]*Log[Cos[x/2
] + Sin[x/2]] + 2*Sin[2*x]))/24
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Maple [A] time = 0.174, size = 30, normalized size = 1. \[ -{\frac{x}{\cos \left ( x \right ) }}+{\frac{5\,\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{6}}+{\frac{x}{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}}-{\frac{\sec \left ( x \right ) \tan \left ( x \right ) }{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*sin(x)^3/cos(x)^4,x)
[Out]
-x/cos(x)+5/6*ln(sec(x)+tan(x))+1/3*x/cos(x)^3-1/6*sec(x)*tan(x)
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Maxima [A] time = 1.58442, size = 836, normalized size = 27.87 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)^3/cos(x)^4,x, algorithm="maxima")
[Out]
-1/12*(48*x*sin(3*x)*sin(2*x) + 4*(6*x*cos(5*x) + 4*x*cos(3*x) + 6*x*cos(x) + si
n(5*x) - sin(x))*cos(6*x) + 12*(6*x*cos(4*x) + 6*x*cos(2*x) + 2*x - sin(4*x) - s
in(2*x))*cos(5*x) + 12*(4*x*cos(3*x) + 6*x*cos(x) - sin(x))*cos(4*x) + 16*(3*x*c
os(2*x) + x)*cos(3*x) + 12*(6*x*cos(x) - sin(x))*cos(2*x) + 24*x*cos(x) - 5*(2*(
3*cos(4*x) + 3*cos(2*x) + 1)*cos(6*x) + cos(6*x)^2 + 6*(3*cos(2*x) + 1)*cos(4*x)
+ 9*cos(4*x)^2 + 9*cos(2*x)^2 + 6*(sin(4*x) + sin(2*x))*sin(6*x) + sin(6*x)^2 +
9*sin(4*x)^2 + 18*sin(4*x)*sin(2*x) + 9*sin(2*x)^2 + 6*cos(2*x) + 1)*log(cos(x)
^2 + sin(x)^2 + 2*sin(x) + 1) + 5*(2*(3*cos(4*x) + 3*cos(2*x) + 1)*cos(6*x) + co
s(6*x)^2 + 6*(3*cos(2*x) + 1)*cos(4*x) + 9*cos(4*x)^2 + 9*cos(2*x)^2 + 6*(sin(4*
x) + sin(2*x))*sin(6*x) + sin(6*x)^2 + 9*sin(4*x)^2 + 18*sin(4*x)*sin(2*x) + 9*s
in(2*x)^2 + 6*cos(2*x) + 1)*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) + 4*(6*x*sin
(5*x) + 4*x*sin(3*x) + 6*x*sin(x) - cos(5*x) + cos(x))*sin(6*x) + 4*(18*x*sin(4*
x) + 18*x*sin(2*x) + 3*cos(4*x) + 3*cos(2*x) + 1)*sin(5*x) + 12*(4*x*sin(3*x) +
6*x*sin(x) + cos(x))*sin(4*x) + 12*(6*x*sin(x) + cos(x))*sin(2*x) - 4*sin(x))/(2
*(3*cos(4*x) + 3*cos(2*x) + 1)*cos(6*x) + cos(6*x)^2 + 6*(3*cos(2*x) + 1)*cos(4*
x) + 9*cos(4*x)^2 + 9*cos(2*x)^2 + 6*(sin(4*x) + sin(2*x))*sin(6*x) + sin(6*x)^2
+ 9*sin(4*x)^2 + 18*sin(4*x)*sin(2*x) + 9*sin(2*x)^2 + 6*cos(2*x) + 1)
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Fricas [A] time = 0.228565, size = 63, normalized size = 2.1 \[ \frac{5 \, \cos \left (x\right )^{3} \log \left (\sin \left (x\right ) + 1\right ) - 5 \, \cos \left (x\right )^{3} \log \left (-\sin \left (x\right ) + 1\right ) - 12 \, x \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 4 \, x}{12 \, \cos \left (x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)^3/cos(x)^4,x, algorithm="fricas")
[Out]
1/12*(5*cos(x)^3*log(sin(x) + 1) - 5*cos(x)^3*log(-sin(x) + 1) - 12*x*cos(x)^2 -
2*cos(x)*sin(x) + 4*x)/cos(x)^3
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Sympy [A] time = 3.74601, size = 551, normalized size = 18.37 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)**3/cos(x)**4,x)
[Out]
4*x*tan(x/2)**6/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 12*x*tan
(x/2)**4/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 12*x*tan(x/2)**
2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 4*x/(6*tan(x/2)**6 - 1
8*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 5*log(tan(x/2) - 1)*tan(x/2)**6/(6*tan(x/2
)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 15*log(tan(x/2) - 1)*tan(x/2)**4/(
6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 15*log(tan(x/2) - 1)*tan(
x/2)**2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 5*log(tan(x/2) -
1)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 5*log(tan(x/2) + 1)*
tan(x/2)**6/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) - 15*log(tan(x
/2) + 1)*tan(x/2)**4/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 15*
log(tan(x/2) + 1)*tan(x/2)**2/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 -
6) - 5*log(tan(x/2) + 1)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6)
- 2*tan(x/2)**5/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6) + 2*tan(x/
2)/(6*tan(x/2)**6 - 18*tan(x/2)**4 + 18*tan(x/2)**2 - 6)
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GIAC/XCAS [A] time = 0.436535, size = 460, normalized size = 15.33 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)^3/cos(x)^4,x, algorithm="giac")
[Out]
1/12*(8*x*tan(1/2*x)^6 + 5*ln(2*(tan(1/2*x)^2 + 2*tan(1/2*x) + 1)/(tan(1/2*x)^2
+ 1))*tan(1/2*x)^6 - 5*ln(2*(tan(1/2*x)^2 - 2*tan(1/2*x) + 1)/(tan(1/2*x)^2 + 1)
)*tan(1/2*x)^6 - 24*x*tan(1/2*x)^4 - 15*ln(2*(tan(1/2*x)^2 + 2*tan(1/2*x) + 1)/(
tan(1/2*x)^2 + 1))*tan(1/2*x)^4 + 15*ln(2*(tan(1/2*x)^2 - 2*tan(1/2*x) + 1)/(tan
(1/2*x)^2 + 1))*tan(1/2*x)^4 - 4*tan(1/2*x)^5 - 24*x*tan(1/2*x)^2 + 15*ln(2*(tan
(1/2*x)^2 + 2*tan(1/2*x) + 1)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^2 - 15*ln(2*(tan(1/
2*x)^2 - 2*tan(1/2*x) + 1)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^2 + 8*x - 5*ln(2*(tan(
1/2*x)^2 + 2*tan(1/2*x) + 1)/(tan(1/2*x)^2 + 1)) + 5*ln(2*(tan(1/2*x)^2 - 2*tan(
1/2*x) + 1)/(tan(1/2*x)^2 + 1)) + 4*tan(1/2*x))/(tan(1/2*x)^6 - 3*tan(1/2*x)^4 +
3*tan(1/2*x)^2 - 1)