3.34 \(\int (a \tanh ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=86 \[ -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {a \tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {a \sqrt {a \tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)} \]

[Out]

-2/3*a*(a*tanh(x)^3)^(1/2)-a*arctan(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)+a*arctanh(tanh(x)^(1/2))*
(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)-2/7*a*(a*tanh(x)^3)^(1/2)*tanh(x)^2

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Rubi [A]  time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3658, 3473, 3476, 329, 298, 203, 206} \[ -\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}-\frac {2}{3} a \sqrt {a \tanh ^3(x)}+\frac {a \tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {a \sqrt {a \tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Tanh[x]^3)^(3/2),x]

[Out]

(-2*a*Sqrt[a*Tanh[x]^3])/3 - (a*ArcTan[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) + (a*ArcTanh[Sqrt[Tanh[
x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) - (2*a*Tanh[x]^2*Sqrt[a*Tanh[x]^3])/7

Rule 203

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

Rule 206

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

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \left (a \tanh ^3(x)\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \tanh ^{\frac {9}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \tanh ^{\frac {5}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \sqrt {\tanh (x)} \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}-\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}-\frac {\left (2 a \sqrt {a \tanh ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {a \tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {a \tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 55, normalized size = 0.64 \[ -\frac {\left (a \tanh ^3(x)\right )^{3/2} \left (-21 \tanh ^{-1}\left (\sqrt {\tanh (x)}\right )+6 \tanh ^{\frac {7}{2}}(x)+14 \tanh ^{\frac {3}{2}}(x)+21 \tan ^{-1}\left (\sqrt {\tanh (x)}\right )\right )}{21 \tanh ^{\frac {9}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Tanh[x]^3)^(3/2),x]

[Out]

-1/21*((a*Tanh[x]^3)^(3/2)*(21*ArcTan[Sqrt[Tanh[x]]] - 21*ArcTanh[Sqrt[Tanh[x]]] + 14*Tanh[x]^(3/2) + 6*Tanh[x
]^(7/2)))/Tanh[x]^(9/2)

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fricas [B]  time = 0.83, size = 1269, normalized size = 14.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^3)^(3/2),x, algorithm="fricas")

[Out]

[-1/84*(42*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^
4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^
2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(-a)*arctan((cosh(x)^2 + 2*cosh(x)*sinh(x) +
sinh(x)^2)*sqrt(-a)*sqrt(a*sinh(x)/cosh(x))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)) - 21*(a*cos
h(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x
)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^
5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(-a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)^3*sinh(x) + 6*a*cosh(x)^2
*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 - 2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*
sqrt(a*sinh(x)/cosh(x)) - 2*a)/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3
+ sinh(x)^4)) + 16*(5*a*cosh(x)^6 + 30*a*cosh(x)*sinh(x)^5 + 5*a*sinh(x)^6 - a*cosh(x)^4 + (75*a*cosh(x)^2 - a
)*sinh(x)^4 + 4*(25*a*cosh(x)^3 - a*cosh(x))*sinh(x)^3 + a*cosh(x)^2 + (75*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*si
nh(x)^2 + 2*(15*a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - 5*a)*sqrt(a*sinh(x)/cosh(x)))/(cosh(x)^6 +
6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*si
nh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*si
nh(x) + 1), -1/84*(42*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 +
a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 +
a)*sinh(x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(a)*arctan((cosh(x)^2 + 2*cosh(x)*
sinh(x) + sinh(x)^2 + 1)*sqrt(a*sinh(x)/cosh(x))/sqrt(a)) - 21*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x
)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x
)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) +
a)*sqrt(a)*log(2*a*cosh(x)^4 + 8*a*cosh(x)^3*sinh(x) + 12*a*cosh(x)^2*sinh(x)^2 + 8*a*cosh(x)*sinh(x)^3 + 2*a*
sinh(x)^4 + 2*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(2*co
sh(x)^3 + cosh(x))*sinh(x))*sqrt(a)*sqrt(a*sinh(x)/cosh(x)) - a) + 16*(5*a*cosh(x)^6 + 30*a*cosh(x)*sinh(x)^5
+ 5*a*sinh(x)^6 - a*cosh(x)^4 + (75*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(25*a*cosh(x)^3 - a*cosh(x))*sinh(x)^3 + a*
cosh(x)^2 + (75*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 2*(15*a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*si
nh(x) - 5*a)*sqrt(a*sinh(x)/cosh(x)))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(
x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*c
osh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)]

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giac [B]  time = 0.27, size = 342, normalized size = 3.98 \[ -\frac {1}{42} \, {\left (42 \, \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 21 \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \frac {16 \, {\left (21 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{6} a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 42 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{5} a^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 119 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{4} a^{2} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 56 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{3} a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 63 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{2} a^{3} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 14 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )} a^{\frac {7}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 5 \, a^{4} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}\right )}^{7}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^3)^(3/2),x, algorithm="giac")

[Out]

-1/42*(42*sqrt(a)*arctan(-(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))/sqrt(a))*sgn(e^(4*x) - 1) + 21*sqrt(a)*log(a
bs(-sqrt(a)*e^(2*x) + sqrt(a*e^(4*x) - a)))*sgn(e^(4*x) - 1) + 16*(21*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^
6*a*sgn(e^(4*x) - 1) + 42*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^5*a^(3/2)*sgn(e^(4*x) - 1) + 119*(sqrt(a)*e^
(2*x) - sqrt(a*e^(4*x) - a))^4*a^2*sgn(e^(4*x) - 1) + 56*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^3*a^(5/2)*sgn
(e^(4*x) - 1) + 63*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^2*a^3*sgn(e^(4*x) - 1) + 14*(sqrt(a)*e^(2*x) - sqrt
(a*e^(4*x) - a))*a^(7/2)*sgn(e^(4*x) - 1) + 5*a^4*sgn(e^(4*x) - 1))/(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a) + s
qrt(a))^7)*a

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maple [A]  time = 0.10, size = 76, normalized size = 0.88 \[ -\frac {\left (a \left (\tanh ^{3}\relax (x )\right )\right )^{\frac {3}{2}} \left (21 a^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a \tanh \relax (x )}}{\sqrt {a}}\right )-21 a^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {a \tanh \relax (x )}}{\sqrt {a}}\right )+6 \left (a \tanh \relax (x )\right )^{\frac {7}{2}}+14 a^{2} \left (a \tanh \relax (x )\right )^{\frac {3}{2}}\right )}{21 \tanh \relax (x )^{3} \left (a \tanh \relax (x )\right )^{\frac {3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^3)^(3/2),x)

[Out]

-1/21*(a*tanh(x)^3)^(3/2)*(21*a^(7/2)*arctan((a*tanh(x))^(1/2)/a^(1/2))-21*a^(7/2)*arctanh((a*tanh(x))^(1/2)/a
^(1/2))+6*(a*tanh(x))^(7/2)+14*a^2*(a*tanh(x))^(3/2))/tanh(x)^3/(a*tanh(x))^(3/2)/a^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \tanh \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*tanh(x)^3)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {tanh}\relax (x)}^3\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^3)^(3/2),x)

[Out]

int((a*tanh(x)^3)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \tanh ^{3}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)**3)**(3/2),x)

[Out]

Integral((a*tanh(x)**3)**(3/2), x)

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