3.4.19 \(\int \frac {(-4+x^4) (-1+x^4)^{3/4}}{x^{12}} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (x^4-1\right )^{3/4} \left (-5 x^8-23 x^4+28\right )}{77 x^{11}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} -\frac {4 \left (x^4-1\right )^{7/4}}{11 x^{11}}-\frac {5 \left (x^4-1\right )^{7/4}}{77 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4 + x^4)*(-1 + x^4)^(3/4))/x^12,x]

[Out]

(-4*(-1 + x^4)^(7/4))/(11*x^11) - (5*(-1 + x^4)^(7/4))/(77*x^7)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (-4+x^4\right ) \left (-1+x^4\right )^{3/4}}{x^{12}} \, dx &=-\frac {4 \left (-1+x^4\right )^{7/4}}{11 x^{11}}-\frac {5}{11} \int \frac {\left (-1+x^4\right )^{3/4}}{x^8} \, dx\\ &=-\frac {4 \left (-1+x^4\right )^{7/4}}{11 x^{11}}-\frac {5 \left (-1+x^4\right )^{7/4}}{77 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} -\frac {\left (x^4-1\right )^{7/4} \left (5 x^4+28\right )}{77 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + x^4)*(-1 + x^4)^(3/4))/x^12,x]

[Out]

-1/77*((-1 + x^4)^(7/4)*(28 + 5*x^4))/x^11

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IntegrateAlgebraic [A]  time = 0.14, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (-28-5 x^4\right ) \left (-1+x^4\right )^{7/4}}{77 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4 + x^4)*(-1 + x^4)^(3/4))/x^12,x]

[Out]

((-28 - 5*x^4)*(-1 + x^4)^(7/4))/(77*x^11)

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fricas [A]  time = 0.43, size = 24, normalized size = 0.86 \begin {gather*} -\frac {{\left (5 \, x^{8} + 23 \, x^{4} - 28\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4-1)^(3/4)/x^12,x, algorithm="fricas")

[Out]

-1/77*(5*x^8 + 23*x^4 - 28)*(x^4 - 1)^(3/4)/x^11

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )}}{x^{12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4-1)^(3/4)/x^12,x, algorithm="giac")

[Out]

integrate((x^4 - 1)^(3/4)*(x^4 - 4)/x^12, x)

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maple [A]  time = 0.10, size = 25, normalized size = 0.89

method	result	size
trager	\(-\frac {\left (5 x^{8}+23 x^{4}-28\right ) \left (x^{4}-1\right )^{\frac {3}{4}}}{77 x^{11}}\)	\(25\)
risch	\(-\frac {5 x^{12}+18 x^{8}-51 x^{4}+28}{77 x^{11} \left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(30\)
gosper	\(-\frac {\left (x^{2}+1\right ) \left (1+x \right ) \left (-1+x \right ) \left (5 x^{4}+28\right ) \left (x^{4}-1\right )^{\frac {3}{4}}}{77 x^{11}}\)	\(31\)
meijerg	\(\frac {4 \mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}} \left (-\frac {4}{7} x^{8}-\frac {3}{7} x^{4}+1\right ) \left (-x^{4}+1\right )^{\frac {3}{4}}}{11 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} x^{11}}-\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}} \left (-x^{4}+1\right )^{\frac {7}{4}}}{7 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} x^{7}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-4)*(x^4-1)^(3/4)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/77*(5*x^8+23*x^4-28)/x^11*(x^4-1)^(3/4)

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maxima [A]  time = 0.32, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {11}{4}}}{11 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4-1)^(3/4)/x^12,x, algorithm="maxima")

[Out]

-3/7*(x^4 - 1)^(7/4)/x^7 + 4/11*(x^4 - 1)^(11/4)/x^11

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mupad [B]  time = 0.36, size = 39, normalized size = 1.39 \begin {gather*} -\frac {23\,x^4\,{\left (x^4-1\right )}^{3/4}-28\,{\left (x^4-1\right )}^{3/4}+5\,x^8\,{\left (x^4-1\right )}^{3/4}}{77\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)^(3/4)*(x^4 - 4))/x^12,x)

[Out]

-(23*x^4*(x^4 - 1)^(3/4) - 28*(x^4 - 1)^(3/4) + 5*x^8*(x^4 - 1)^(3/4))/(77*x^11)

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sympy [C]  time = 3.61, size = 420, normalized size = 15.00 \begin {gather*} \begin {cases} \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} - \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {otherwise} \end {cases} - 4 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {3 \left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{4} \Gamma \left (- \frac {3}{4}\right )} - \frac {7 \left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {4 x^{8} \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} - \frac {x^{4} \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {7 \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{12} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{8} \Gamma \left (- \frac {3}{4}\right )} - \frac {10 \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-4)*(x**4-1)**(3/4)/x**12,x)

[Out]

Piecewise(((-1 + x**(-4))**(3/4)*exp(-I*pi/4)*gamma(-7/4)/(4*gamma(-3/4)) - (-1 + x**(-4))**(3/4)*exp(-I*pi/4)
*gamma(-7/4)/(4*x**4*gamma(-3/4)), 1/Abs(x**4) > 1), (-(1 - 1/x**4)**(3/4)*gamma(-7/4)/(4*gamma(-3/4)) + (1 -
1/x**4)**(3/4)*gamma(-7/4)/(4*x**4*gamma(-3/4)), True)) - 4*Piecewise(((-1 + x**(-4))**(3/4)*exp(3*I*pi/4)*gam
ma(-11/4)/(4*gamma(-3/4)) + 3*(-1 + x**(-4))**(3/4)*exp(3*I*pi/4)*gamma(-11/4)/(16*x**4*gamma(-3/4)) - 7*(-1 +
x**(-4))**(3/4)*exp(3*I*pi/4)*gamma(-11/4)/(16*x**8*gamma(-3/4)), 1/Abs(x**4) > 1), (4*x**8*(1 - 1/x**4)**(3/
4)*gamma(-11/4)/(16*x**8*gamma(-3/4) - 16*x**4*gamma(-3/4)) - x**4*(1 - 1/x**4)**(3/4)*gamma(-11/4)/(16*x**8*g
amma(-3/4) - 16*x**4*gamma(-3/4)) + 7*(1 - 1/x**4)**(3/4)*gamma(-11/4)/(16*x**12*gamma(-3/4) - 16*x**8*gamma(-
3/4)) - 10*(1 - 1/x**4)**(3/4)*gamma(-11/4)/(16*x**8*gamma(-3/4) - 16*x**4*gamma(-3/4)), True))

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