### 3.210 $$\int \tanh ^3(x) \sqrt{a+b \tanh ^2(x)} \, dx$$

Optimal. Leaf size=63 $-\frac{\left (a+b \tanh ^2(x)\right )^{3/2}}{3 b}-\sqrt{a+b \tanh ^2(x)}+\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right )$

[Out]

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

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Rubi [A]  time = 0.119459, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.353, Rules used = {3670, 446, 80, 50, 63, 208} $-\frac{\left (a+b \tanh ^2(x)\right )^{3/2}}{3 b}-\sqrt{a+b \tanh ^2(x)}+\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[x]^3*Sqrt[a + b*Tanh[x]^2],x]

[Out]

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

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.160509, size = 60, normalized size = 0.95 $\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right )-\frac{\sqrt{a+b \tanh ^2(x)} \left (a+b \tanh ^2(x)+3 b\right )}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[x]^3*Sqrt[a + b*Tanh[x]^2],x]

[Out]

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

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Maple [B]  time = 0.045, size = 253, normalized size = 4. \begin{align*} -{\frac{1}{3\,b} \left ( a+b \left ( \tanh \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}b-2\, \left ( 1+\tanh \left ( x \right ) \right ) b+a+b}}+{\frac{1}{2}\sqrt{b}\ln \left ({( \left ( 1+\tanh \left ( x \right ) \right ) b-b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}b-2\, \left ( 1+\tanh \left ( x \right ) \right ) b+a+b} \right ) }+{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{1+\tanh \left ( x \right ) } \left ( 2\,a+2\,b-2\, \left ( 1+\tanh \left ( x \right ) \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}b-2\, \left ( 1+\tanh \left ( x \right ) \right ) b+a+b} \right ) } \right ) }-{\frac{1}{2}\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}b+2\, \left ( \tanh \left ( x \right ) -1 \right ) b+a+b}}-{\frac{1}{2}\sqrt{b}\ln \left ({( \left ( \tanh \left ( x \right ) -1 \right ) b+b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}b+2\, \left ( \tanh \left ( x \right ) -1 \right ) b+a+b} \right ) }+{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{\tanh \left ( x \right ) -1} \left ( 2\,a+2\,b+2\, \left ( \tanh \left ( x \right ) -1 \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}b+2\, \left ( \tanh \left ( x \right ) -1 \right ) b+a+b} \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tanh(x)^2)^(1/2)*tanh(x)^3,x)

[Out]

-1/3*(a+b*tanh(x)^2)^(3/2)/b-1/2*((1+tanh(x))^2*b-2*(1+tanh(x))*b+a+b)^(1/2)+1/2*b^(1/2)*ln(((1+tanh(x))*b-b)/
b^(1/2)+((1+tanh(x))^2*b-2*(1+tanh(x))*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+tanh(x))*b+2*(a+b)^(1/2)
*((1+tanh(x))^2*b-2*(1+tanh(x))*b+a+b)^(1/2))/(1+tanh(x)))-1/2*((tanh(x)-1)^2*b+2*(tanh(x)-1)*b+a+b)^(1/2)-1/2
*b^(1/2)*ln(((tanh(x)-1)*b+b)/b^(1/2)+((tanh(x)-1)^2*b+2*(tanh(x)-1)*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b
+2*(tanh(x)-1)*b+2*(a+b)^(1/2)*((tanh(x)-1)^2*b+2*(tanh(x)-1)*b+a+b)^(1/2))/(tanh(x)-1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tanh \left (x\right )^{2} + a} \tanh \left (x\right )^{3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*tanh(x)^2 + a)*tanh(x)^3, x)

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Fricas [B]  time = 3.91679, size = 6839, normalized size = 108.56 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^3,x, algorithm="fricas")

[Out]

[1/12*(3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 + 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4
+ 4*(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2
+ 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(a + b)*log(((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 +
a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh(x)^8 + 2*(2*a^3 + a^2*b)*cosh(x)^6 + 2*(2*a^3 + a^2*b + 14*(a^3
+ a^2*b)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*cosh(x)^3 + 3*(2*a^3 + a^2*b)*cosh(x))*sinh(x)^5 + (6*a^3
+ 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (70*(a^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 + 30*(2*a^3
+ a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*cosh(x)^5 + 10*(2*a^3 + a^2*b)*cosh(x)^3 + (6*a^3 + 4*a^2*
b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2*
(14*(a^3 + a^2*b)*cosh(x)^6 + 15*(2*a^3 + a^2*b)*cosh(x)^4 + 2*a^3 + 3*a^2*b - b^3 + 3*(6*a^3 + 4*a^2*b - a*b^
2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 + 3*a^2*cosh(
x)^4 + 3*(5*a^2*cosh(x)^2 + a^2)*sinh(x)^4 + 4*(5*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^3 + (3*a^2 + 2*a*b -
b^2)*cosh(x)^2 + (15*a^2*cosh(x)^4 + 18*a^2*cosh(x)^2 + 3*a^2 + 2*a*b - b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2
*(3*a^2*cosh(x)^5 + 6*a^2*cosh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x))*sinh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x
)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*(a^3 + a^2*b)*cosh(x)^7 +
3*(2*a^3 + a^2*b)*cosh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^3 + (2*a^3 + 3*a^2*b - b^3)*cosh(x))*si
nh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(
x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 + 3*b*cosh(x)^
4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(
x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(a + b)*log
(-((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*b*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^
2 - b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2
+ (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + b)*cosh(x)^3 - b*cosh(x))
*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((a + 4*b)*cosh(x)^4 + 4*(a + 4*b)*
cosh(x)*sinh(x)^3 + (a + 4*b)*sinh(x)^4 + 2*(a + 2*b)*cosh(x)^2 + 2*(3*(a + 4*b)*cosh(x)^2 + a + 2*b)*sinh(x)^
2 + 4*((a + 4*b)*cosh(x)^3 + (a + 2*b)*cosh(x))*sinh(x) + a + 4*b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2
+ a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 + 3
*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3
*(5*b*cosh(x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) + b), -1/
6*(3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 + 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 4*
(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*
(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(-a - b)*arctan(sqrt(2)*(a*cosh(x)^2 + 2*a*cosh(x)*
sinh(x) + a*sinh(x)^2 + a + b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 -
2*cosh(x)*sinh(x) + sinh(x)^2))/((a^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3 + (a^2 + a*b)*sinh(x)
^4 + (2*a^2 + a*b - b^2)*cosh(x)^2 + (6*(a^2 + a*b)*cosh(x)^2 + 2*a^2 + a*b - b^2)*sinh(x)^2 + a^2 + 2*a*b + b
^2 + 2*(2*(a^2 + a*b)*cosh(x)^3 + (2*a^2 + a*b - b^2)*cosh(x))*sinh(x))) + 3*(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x
)^5 + b*sinh(x)^6 + 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^
3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(
x))*sinh(x) + b)*sqrt(-a - b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a - b)*sqrt
(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*cosh(x)
^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*s
inh(x)^2 + 4*((a + b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)) + 2*sqrt(2)*((a + 4*b)*cosh(x)^4 + 4*(a +
4*b)*cosh(x)*sinh(x)^3 + (a + 4*b)*sinh(x)^4 + 2*(a + 2*b)*cosh(x)^2 + 2*(3*(a + 4*b)*cosh(x)^2 + a + 2*b)*si
nh(x)^2 + 4*((a + 4*b)*cosh(x)^3 + (a + 2*b)*cosh(x))*sinh(x) + a + 4*b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sin
h(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)
^6 + 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 4*(5*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 3*b*cosh(x)
^2 + 3*(5*b*cosh(x)^4 + 6*b*cosh(x)^2 + b)*sinh(x)^2 + 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + b*cosh(x))*sinh(x) + b
)]

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Sympy [A]  time = 4.42507, size = 71, normalized size = 1.13 \begin{align*} - \frac{2 \left (\frac{b^{2} \sqrt{a + b \tanh ^{2}{\left (x \right )}}}{2} + \frac{b^{2} \left (a + b\right ) \operatorname{atan}{\left (\frac{\sqrt{a + b \tanh ^{2}{\left (x \right )}}}{\sqrt{- a - b}} \right )}}{2 \sqrt{- a - b}} + \frac{b \left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac{3}{2}}}{6}\right )}{b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(x)**2)**(1/2)*tanh(x)**3,x)

[Out]

-2*(b**2*sqrt(a + b*tanh(x)**2)/2 + b**2*(a + b)*atan(sqrt(a + b*tanh(x)**2)/sqrt(-a - b))/(2*sqrt(-a - b)) +
b*(a + b*tanh(x)**2)**(3/2)/6)/b**2

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError