3.1.80 \(\int \text {csch}^5(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [80]

Optimal. Leaf size=135 \[ -\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 (a-b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 f} \]

[Out]

-1/4*(a-b+b*cosh(f*x+e)^2)^(3/2)*coth(f*x+e)*csch(f*x+e)^3/f-3/8*(a-b)^2*arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*co
sh(f*x+e)^2)^(1/2))/f/a^(1/2)+3/8*(a-b)*coth(f*x+e)*csch(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(1/2)/f

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Rubi [A]
time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3265, 386, 385, 212} \begin {gather*} -\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{8 \sqrt {a} f}-\frac {\coth (e+f x) \text {csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 f}+\frac {3 (a-b) \coth (e+f x) \text {csch}(e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(-3*(a - b)^2*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqr
t[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e +
f*x]*Csch[e + f*x]^3)/(4*f)

Rule 212

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

Rule 385

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

Rule 386

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

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \text {csch}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 f}-\frac {(3 (a-b)) \text {Subst}\left (\int \frac {\sqrt {a-b+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{4 f}\\ &=\frac {3 (a-b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 f}-\frac {\left (3 (a-b)^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{8 f}\\ &=\frac {3 (a-b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 f}-\frac {\left (3 (a-b)^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 f}\\ &=-\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 (a-b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 f}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 123, normalized size = 0.91 \begin {gather*} \frac {-6 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )+\sqrt {2} \sqrt {a} \sqrt {2 a-b+b \cosh (2 (e+f x))} \coth (e+f x) \text {csch}(e+f x) \left (3 a-5 b-2 a \text {csch}^2(e+f x)\right )}{16 \sqrt {a} f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(-6*(a - b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + Sqrt[2]*Sqrt[a]*S
qrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x]*(3*a - 5*b - 2*a*Csch[e + f*x]^2))/(16*Sqrt[a]*
f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs. \(2(119)=238\).
time = 1.39, size = 379, normalized size = 2.81

method	result	size
default	\(\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-3 a^{2} \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+6 a b \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )-3 b^{2} \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+6 a^{\frac {3}{2}} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (\sinh ^{2}\left (f x +e \right )\right )-10 b \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \sqrt {a}\, \left (\sinh ^{2}\left (f x +e \right )\right )-4 a^{\frac {3}{2}} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\right )}{16 \sqrt {a}\, \sinh \left (f x +e \right )^{4} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-3*a^2*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b
)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4+6*a*b*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e
)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4-3*b^2*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*co
sh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4+6*a^(3/2)*((a+b*sinh(f*x+e)^2)*cosh(f
*x+e)^2)^(1/2)*sinh(f*x+e)^2-10*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*a^(1/2)*sinh(f*x+e)^2-4*a^(3/2)*((
a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/a^(1/2)/sinh(f*x+e)^4/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*csch(f*x + e)^5, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1515 vs. \(2 (119) = 238\).
time = 0.61, size = 3133, normalized size = 23.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(3*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^8 + 8*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2 - 2
*a*b + b^2)*sinh(f*x + e)^8 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2
- a^2 + 2*a*b - b^2)*sinh(f*x + e)^6 + 8*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(
f*x + e))*sinh(f*x + e)^5 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 2*(35*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4
- 30*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 3*a^2 - 6*a*b + 3*b^2)*sinh(f*x + e)^4 + 8*(7*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^5 - 10*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3
- 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 - 15*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^4 + 9*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^2 + a^2 - 2*a*b + b^
2 + 8*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^7 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 + 3*(a^2 - 2*a*b + b^2)*cos
h(f*x + e)^3 - (a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log(-((a + b)*cosh(f*x + e)^4 + 4*(a
+ b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a + b)*sinh(f*x + e)^4 + 2*(3*a - b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh
(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x
+ e)^2 + 1)*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)) + 4*((a + b)*cosh(f*x + e)^3 + (3*a - b)*cosh(f*x + e))*sinh(f*x + e) + a +
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)) + 2*sqrt(2)*((3*a^2 - 5*a
*b)*cosh(f*x + e)^6 + 6*(3*a^2 - 5*a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2 - 5*a*b)*sinh(f*x + e)^6 - (11*
a^2 - 5*a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2 + 5*a*b)*sinh(f*x + e)^4 + 4*(5*(3
*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x +
e)^2 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2 + 5*a*b)*sinh(f*x + e
)^2 + 3*a^2 - 5*a*b + 2*(3*(3*a^2 - 5*a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*
a*b)*cosh(f*x + e))*sinh(f*x + e))*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*f*cosh(f*x + e)^8 + 8*a*f*cosh(f*x + e)*sinh(f*x + e)^7 +
a*f*sinh(f*x + e)^8 - 4*a*f*cosh(f*x + e)^6 + 4*(7*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^6 + 6*a*f*cosh(f*
x + e)^4 + 8*(7*a*f*cosh(f*x + e)^3 - 3*a*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a*f*cosh(f*x + e)^4 - 30*a*
f*cosh(f*x + e)^2 + 3*a*f)*sinh(f*x + e)^4 - 4*a*f*cosh(f*x + e)^2 + 8*(7*a*f*cosh(f*x + e)^5 - 10*a*f*cosh(f*
x + e)^3 + 3*a*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a*f*cosh(f*x + e)^6 - 15*a*f*cosh(f*x + e)^4 + 9*a*f*co
sh(f*x + e)^2 - a*f)*sinh(f*x + e)^2 + a*f + 8*(a*f*cosh(f*x + e)^7 - 3*a*f*cosh(f*x + e)^5 + 3*a*f*cosh(f*x +
e)^3 - a*f*cosh(f*x + e))*sinh(f*x + e)), 1/8*(3*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^8 + 8*(a^2 - 2*a*b + b^2)
*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^8 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 +
4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^6 + 8*(7*(a^2 - 2*a*b + b^2)*cosh
(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 2
*(35*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 - 30*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 3*a^2 - 6*a*b + 3*b^2)*sin
h(f*x + e)^4 + 8*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 - 10*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 + 3*(a^2 - 2*
a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 4*(7*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^6 - 15*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 9*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b
- b^2)*sinh(f*x + e)^2 + a^2 - 2*a*b + b^2 + 8*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^7 - 3*(a^2 - 2*a*b + b^2)*co
sh(f*x + e)^5 + 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - (a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt
(-a)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-a)*sqrt((b*c
osh(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))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2
+ 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x
+ e) + b)) + sqrt(2)*((3*a^2 - 5*a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - 5*a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*
a^2 - 5*a*b)*sinh(f*x + e)^6 - (11*a^2 - 5*a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2
+ 5*a*b)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e))*sinh(f*x +
e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e)^2 + (15*(...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)**5*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2145 vs. \(2 (119) = 238\).
time = 0.90, size = 2145, normalized size = 15.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*(3*(a^2 - 2*a*b + b^2)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e)
- 2*b*e^(2*f*x + 2*e) + b) - sqrt(b))/sqrt(-a))/sqrt(-a) - 2*(3*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x +
4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a^2 - 6*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x +
4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a*b - 5*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x +
4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*b^2 - 21*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x +
4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a^2*sqrt(b) - 22*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e
^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a*b^(3/2) + 35*(sqrt(b)*e^(2*f*x + 2*e) - s
qrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*b^(5/2) - 44*(sqrt(b)*e^(2*f*x + 2*e
) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^3 - 105*(sqrt(b)*e^(2*f*x + 2
*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b + 246*(sqrt(b)*e^(2*f*x
+ 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a*b^2 - 105*(sqrt(b)*e^(2
*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b^3 - 292*(sqrt(b)*e^
(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^3*sqrt(b) + 735*(
sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^2*b^(3/
2) - 690*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4
*a*b^(5/2) + 175*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e)
+ b))^4*b^(7/2) - 176*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x
+ 2*e) + b))^3*a^4 + 936*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*
x + 2*e) + b))^3*a^3*b - 1623*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^
(2*f*x + 2*e) + b))^3*a^2*b^2 + 950*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) -
2*b*e^(2*f*x + 2*e) + b))^3*a*b^3 - 175*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e
) - 2*b*e^(2*f*x + 2*e) + b))^3*b^4 + 528*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2
*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^4*sqrt(b) - 1640*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e
^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^3*b^(3/2) + 1761*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4
*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^2*b^(5/2) - 714*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^
(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a*b^(7/2) + 105*(sqrt(b)*e^(2*f*x + 2*e) - s
qrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*b^(9/2) + 192*(sqrt(b)*e^(2*f*x + 2*
e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^5 - 912*(sqrt(b)*e^(2*f*x + 2*
e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^4*b + 1540*(sqrt(b)*e^(2*f*x +
2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^3*b^2 - 963*(sqrt(b)*e^(2*f
*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^2*b^3 + 282*(sqrt(b)*e^
(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a*b^4 - 35*(sqrt(b)*e
^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*b^5 - 192*a^5*sqrt(b
) + 560*a^4*b^(3/2) - 500*a^3*b^(5/2) + 213*a^2*b^(7/2) - 46*a*b^(9/2) + 5*b^(11/2))/((sqrt(b)*e^(2*f*x + 2*e)
- sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2 - 2*(sqrt(b)*e^(2*f*x + 2*e) - s
qrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*sqrt(b) - 4*a + b)^4)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {sinh}\left (e+f\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(e + f*x)^2)^(3/2)/sinh(e + f*x)^5,x)

[Out]

int((a + b*sinh(e + f*x)^2)^(3/2)/sinh(e + f*x)^5, x)

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