3.472 \(\int \coth ^3(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)
Optimal. Leaf size=140 \[ \frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}+\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f} \]
[Out]
-(Sqrt[a]*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*f) + ((2*a + 3*b)*Sqrt[a + b*Sinh[e + f
*x]^2])/(2*f) + ((2*a + 3*b)*(a + b*Sinh[e + f*x]^2)^(3/2))/(6*a*f) - (Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)
^(5/2))/(2*a*f)
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Rubi [A] time = 0.147753, antiderivative size = 140, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{3194, 78, 50, 63, 208} \[ \frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}+\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f} \]
Antiderivative was successfully verified.
[In]
Int[Coth[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-(Sqrt[a]*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*f) + ((2*a + 3*b)*Sqrt[a + b*Sinh[e + f
*x]^2])/(2*f) + ((2*a + 3*b)*(a + b*Sinh[e + f*x]^2)^(3/2))/(6*a*f) - (Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)
^(5/2))/(2*a*f)
Rule 3194
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
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 \coth ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1+x) (a+b x)^{3/2}}{x^2} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f}+\frac{(2 a+3 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=\frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f}+\frac{(2 a+3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sinh ^2(e+f x)\right )}{4 f}\\ &=\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}+\frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f}+\frac{(a (2 a+3 b)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 f}\\ &=\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}+\frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f}+\frac{(a (2 a+3 b)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^2(e+f x)}\right )}{2 b f}\\ &=-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}+\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}+\frac{(2 a+3 b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{6 a f}-\frac{\text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{2 a f}\\ \end{align*}
Mathematica [A] time = 0.44647, size = 90, normalized size = 0.64 \[ \frac{\sqrt{a+b \sinh ^2(e+f x)} \left (-3 a \text{csch}^2(e+f x)+8 a+b \cosh (2 (e+f x))+5 b\right )-3 \sqrt{a} (2 a+3 b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{6 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
(-3*Sqrt[a]*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + (8*a + 5*b + b*Cosh[2*(e + f*x)] - 3*a*
Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(6*f)
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Maple [C] time = 0.124, size = 84, normalized size = 0.6 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({ \left ({b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{3}+ \left ( 2\,ab+{b}^{2} \right ) \sinh \left ( fx+e \right ) +{\frac{{a}^{2}+2\,ab}{\sinh \left ( fx+e \right ) }}+{\frac{{a}^{2}}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{3}}} \right ){\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x)
[Out]
`int/indef0`((b^2*sinh(f*x+e)^3+(2*a*b+b^2)*sinh(f*x+e)+(a^2+2*a*b)/sinh(f*x+e)+a^2/sinh(f*x+e)^3)/(a+b*sinh(f
*x+e)^2)^(1/2),sinh(f*x+e))/f
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \coth \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*coth(f*x + e)^3, x)
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Fricas [B] time = 5.47209, size = 6418, normalized size = 45.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/24*(6*((2*a + 3*b)*cosh(f*x + e)^7 + 7*(2*a + 3*b)*cosh(f*x + e)*sinh(f*x + e)^6 + (2*a + 3*b)*sinh(f*x + e
)^7 - 2*(2*a + 3*b)*cosh(f*x + e)^5 + (21*(2*a + 3*b)*cosh(f*x + e)^2 - 4*a - 6*b)*sinh(f*x + e)^5 + 5*(7*(2*a
+ 3*b)*cosh(f*x + e)^3 - 2*(2*a + 3*b)*cosh(f*x + e))*sinh(f*x + e)^4 + (2*a + 3*b)*cosh(f*x + e)^3 + (35*(2*
a + 3*b)*cosh(f*x + e)^4 - 20*(2*a + 3*b)*cosh(f*x + e)^2 + 2*a + 3*b)*sinh(f*x + e)^3 + (21*(2*a + 3*b)*cosh(
f*x + e)^5 - 20*(2*a + 3*b)*cosh(f*x + e)^3 + 3*(2*a + 3*b)*cosh(f*x + e))*sinh(f*x + e)^2 + (7*(2*a + 3*b)*co
sh(f*x + e)^6 - 10*(2*a + 3*b)*cosh(f*x + e)^4 + 3*(2*a + 3*b)*cosh(f*x + e)^2)*sinh(f*x + e))*sqrt(a)*log((b*
cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(4*a - b)*cosh(f*x + e)^2 + 2*(3*b
*cosh(f*x + e)^2 + 4*a - b)*sinh(f*x + e)^2 - 4*sqrt(2)*sqrt(a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 +
2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))*(cosh(f*x + e) + sinh(f*x + e))
+ 4*(b*cosh(f*x + e)^3 + (4*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f
*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 - 1)*sinh(f*x + e)^2 - 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)
^3 - cosh(f*x + e))*sinh(f*x + e) + 1)) + sqrt(2)*(b*cosh(f*x + e)^8 + 8*b*cosh(f*x + e)*sinh(f*x + e)^7 + b*s
inh(f*x + e)^8 + 8*(2*a + b)*cosh(f*x + e)^6 + 4*(7*b*cosh(f*x + e)^2 + 4*a + 2*b)*sinh(f*x + e)^6 + 8*(7*b*co
sh(f*x + e)^3 + 6*(2*a + b)*cosh(f*x + e))*sinh(f*x + e)^5 - 2*(28*a + 9*b)*cosh(f*x + e)^4 + 2*(35*b*cosh(f*x
+ e)^4 + 60*(2*a + b)*cosh(f*x + e)^2 - 28*a - 9*b)*sinh(f*x + e)^4 + 8*(7*b*cosh(f*x + e)^5 + 20*(2*a + b)*c
osh(f*x + e)^3 - (28*a + 9*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 8*(2*a + b)*cosh(f*x + e)^2 + 4*(7*b*cosh(f*x +
e)^6 + 30*(2*a + b)*cosh(f*x + e)^4 - 3*(28*a + 9*b)*cosh(f*x + e)^2 + 4*a + 2*b)*sinh(f*x + e)^2 + 8*(b*cosh
(f*x + e)^7 + 6*(2*a + b)*cosh(f*x + e)^5 - (28*a + 9*b)*cosh(f*x + e)^3 + 2*(2*a + b)*cosh(f*x + e))*sinh(f*x
+ e) + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x
+ e) + sinh(f*x + e)^2)))/(f*cosh(f*x + e)^7 + 7*f*cosh(f*x + e)*sinh(f*x + e)^6 + f*sinh(f*x + e)^7 - 2*f*cos
h(f*x + e)^5 + (21*f*cosh(f*x + e)^2 - 2*f)*sinh(f*x + e)^5 + 5*(7*f*cosh(f*x + e)^3 - 2*f*cosh(f*x + e))*sinh
(f*x + e)^4 + f*cosh(f*x + e)^3 + (35*f*cosh(f*x + e)^4 - 20*f*cosh(f*x + e)^2 + f)*sinh(f*x + e)^3 + (21*f*co
sh(f*x + e)^5 - 20*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e))*sinh(f*x + e)^2 + (7*f*cosh(f*x + e)^6 - 10*f*cosh(f
*x + e)^4 + 3*f*cosh(f*x + e)^2)*sinh(f*x + e)), 1/24*(12*((2*a + 3*b)*cosh(f*x + e)^7 + 7*(2*a + 3*b)*cosh(f*
x + e)*sinh(f*x + e)^6 + (2*a + 3*b)*sinh(f*x + e)^7 - 2*(2*a + 3*b)*cosh(f*x + e)^5 + (21*(2*a + 3*b)*cosh(f*
x + e)^2 - 4*a - 6*b)*sinh(f*x + e)^5 + 5*(7*(2*a + 3*b)*cosh(f*x + e)^3 - 2*(2*a + 3*b)*cosh(f*x + e))*sinh(f
*x + e)^4 + (2*a + 3*b)*cosh(f*x + e)^3 + (35*(2*a + 3*b)*cosh(f*x + e)^4 - 20*(2*a + 3*b)*cosh(f*x + e)^2 + 2
*a + 3*b)*sinh(f*x + e)^3 + (21*(2*a + 3*b)*cosh(f*x + e)^5 - 20*(2*a + 3*b)*cosh(f*x + e)^3 + 3*(2*a + 3*b)*c
osh(f*x + e))*sinh(f*x + e)^2 + (7*(2*a + 3*b)*cosh(f*x + e)^6 - 10*(2*a + 3*b)*cosh(f*x + e)^4 + 3*(2*a + 3*b
)*cosh(f*x + e)^2)*sinh(f*x + e))*sqrt(-a)*arctan(1/2*sqrt(2)*sqrt(-a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x +
e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/(a*cosh(f*x + e) + a*sinh
(f*x + e))) + sqrt(2)*(b*cosh(f*x + e)^8 + 8*b*cosh(f*x + e)*sinh(f*x + e)^7 + b*sinh(f*x + e)^8 + 8*(2*a + b)
*cosh(f*x + e)^6 + 4*(7*b*cosh(f*x + e)^2 + 4*a + 2*b)*sinh(f*x + e)^6 + 8*(7*b*cosh(f*x + e)^3 + 6*(2*a + b)*
cosh(f*x + e))*sinh(f*x + e)^5 - 2*(28*a + 9*b)*cosh(f*x + e)^4 + 2*(35*b*cosh(f*x + e)^4 + 60*(2*a + b)*cosh(
f*x + e)^2 - 28*a - 9*b)*sinh(f*x + e)^4 + 8*(7*b*cosh(f*x + e)^5 + 20*(2*a + b)*cosh(f*x + e)^3 - (28*a + 9*b
)*cosh(f*x + e))*sinh(f*x + e)^3 + 8*(2*a + b)*cosh(f*x + e)^2 + 4*(7*b*cosh(f*x + e)^6 + 30*(2*a + b)*cosh(f*
x + e)^4 - 3*(28*a + 9*b)*cosh(f*x + e)^2 + 4*a + 2*b)*sinh(f*x + e)^2 + 8*(b*cosh(f*x + e)^7 + 6*(2*a + b)*co
sh(f*x + e)^5 - (28*a + 9*b)*cosh(f*x + e)^3 + 2*(2*a + b)*cosh(f*x + e))*sinh(f*x + e) + b)*sqrt((b*cosh(f*x
+ e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(f
*cosh(f*x + e)^7 + 7*f*cosh(f*x + e)*sinh(f*x + e)^6 + f*sinh(f*x + e)^7 - 2*f*cosh(f*x + e)^5 + (21*f*cosh(f*
x + e)^2 - 2*f)*sinh(f*x + e)^5 + 5*(7*f*cosh(f*x + e)^3 - 2*f*cosh(f*x + e))*sinh(f*x + e)^4 + f*cosh(f*x + e
)^3 + (35*f*cosh(f*x + e)^4 - 20*f*cosh(f*x + e)^2 + f)*sinh(f*x + e)^3 + (21*f*cosh(f*x + e)^5 - 20*f*cosh(f*
x + e)^3 + 3*f*cosh(f*x + e))*sinh(f*x + e)^2 + (7*f*cosh(f*x + e)^6 - 10*f*cosh(f*x + e)^4 + 3*f*cosh(f*x + e
)^2)*sinh(f*x + e))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(3/2),x)
[Out]
Timed out
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError