3.4.72 \(\int \text {sech}^4(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [372]

Optimal. Leaf size=193 \[ \frac {2 (a+b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-b) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]

[Out]

2/3*(a+b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1
-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*b*(1/(1+s
inh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech
(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(a-b)*sech(f*x+e)^2*(a+b*s
inh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f

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Rubi [A]
time = 0.13, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3271, 424, 539, 429, 422} \begin {gather*} -\frac {b \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (a+b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-b) \tanh (e+f x) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(2*(a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sec
h[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a
+ b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a - b)*Sech[e + f*x]^2*Sqrt[
a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 424

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

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

Rule 3271

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.46, size = 197, normalized size = 1.02 \begin {gather*} \frac {4 i a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (2 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\frac {\left (8 a^2-3 a b+b^2+\left (4 a^2+6 a b-2 b^2\right ) \cosh (2 (e+f x))+b (a+b) \cosh (4 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{\sqrt {2}}}{6 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

((4*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*a*(2*a + b)*Sqrt[
(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + ((8*a^2 - 3*a*b + b^2 + (4*a^2 + 6*a*b - 2*b^
2)*Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*f*Sqrt[2*a - b
+ b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 1.70, size = 324, normalized size = 1.68

method	result	size
default	\(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (2 \sqrt {-\frac {b}{a}}\, a^{2}+\sqrt {-\frac {b}{a}}\, a b -3 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (a \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 a \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 b \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \cosh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(324\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((2*(-1/a*b)^(1/2)*a*b+2*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(2*(-1/a*b)^(1/2)*a^2+(-1/a*b)^(1/2
)*a*b-3*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*
b*(a*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))+2*b*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-2
*a*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-2*b*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2)))*co
sh(f*x+e)^2+((-1/a*b)^(1/2)*a^2-2*(-1/a*b)^(1/2)*a*b+(-1/a*b)^(1/2)*b^2)*sinh(f*x+e))/cosh(f*x+e)^3/(-1/a*b)^(
1/2)/(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(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^4, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2071 vs. \(2 (205) = 410\).
time = 0.15, size = 2071, normalized size = 10.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(((2*a^2 + a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a^2 + a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 + a*
b - b^2)*sinh(f*x + e)^6 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 +
2*a^2 + a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x
+ e))*sinh(f*x + e)^3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 6*
(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b - b^2)*sinh(f*x + e)^2 + 2*a^2 + a*b - b^2 + 6*((2*a^2 + a*b
- b^2)*cosh(f*x + e)^5 + 2*(2*a^2 + a*b - b^2)*cosh(f*x + e)^3 + (2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x
+ e) - 2*((a*b + b^2)*cosh(f*x + e)^6 + 6*(a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b + b^2)*sinh(f*x + e
)^6 + 3*(a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(a*b + b^2)*cosh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(a*
b + b^2)*cosh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e)^2 + 3*(5
*(a*b + b^2)*cosh(f*x + e)^4 + 6*(a*b + b^2)*cosh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^2 + a*b + b^2 + 6*((a*
b + b^2)*cosh(f*x + e)^5 + 2*(a*b + b^2)*cosh(f*x + e)^3 + (a*b + b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2
- a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*
b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b
)/b^2))/b^2) - ((2*a^2 - a*b)*cosh(f*x + e)^6 + 6*(2*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 - a*b)*
sinh(f*x + e)^6 + 3*(2*a^2 - a*b)*cosh(f*x + e)^4 + 3*(5*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x
+ e)^4 + 4*(5*(2*a^2 - a*b)*cosh(f*x + e)^3 + 3*(2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 - a*b
)*cosh(f*x + e)^2 + 3*(5*(2*a^2 - a*b)*cosh(f*x + e)^4 + 6*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f
*x + e)^2 + 2*a^2 - a*b + 6*((2*a^2 - a*b)*cosh(f*x + e)^5 + 2*(2*a^2 - a*b)*cosh(f*x + e)^3 + (2*a^2 - a*b)*c
osh(f*x + e))*sinh(f*x + e) - 2*((a*b + 2*b^2)*cosh(f*x + e)^6 + 6*(a*b + 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5
+ (a*b + 2*b^2)*sinh(f*x + e)^6 + 3*(a*b + 2*b^2)*cosh(f*x + e)^4 + 3*(5*(a*b + 2*b^2)*cosh(f*x + e)^2 + a*b
+ 2*b^2)*sinh(f*x + e)^4 + 4*(5*(a*b + 2*b^2)*cosh(f*x + e)^3 + 3*(a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3
+ 3*(a*b + 2*b^2)*cosh(f*x + e)^2 + 3*(5*(a*b + 2*b^2)*cosh(f*x + e)^4 + 6*(a*b + 2*b^2)*cosh(f*x + e)^2 + a*
b + 2*b^2)*sinh(f*x + e)^2 + a*b + 2*b^2 + 6*((a*b + 2*b^2)*cosh(f*x + e)^5 + 2*(a*b + 2*b^2)*cosh(f*x + e)^3
+ (a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) -
2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))),
(8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - sqrt(2)*((a*b + b^2)*cosh(f*x + e)^5 + 5
*(a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^4 + (a*b + b^2)*sinh(f*x + e)^5 + (3*a*b + b^2)*cosh(f*x + e)^3 + (10
*(a*b + b^2)*cosh(f*x + e)^2 + 3*a*b + b^2)*sinh(f*x + e)^3 + 2*b^2*cosh(f*x + e) + (10*(a*b + b^2)*cosh(f*x +
e)^3 + 3*(3*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (5*(a*b + b^2)*cosh(f*x + e)^4 + 3*(3*a*b + b^2)*cosh
(f*x + e)^2 + 2*b^2)*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)))/(b*f*cosh(f*x + e)^6 + 6*b*f*cosh(f*x + e)*sinh(f*x + e)^5
+ b*f*sinh(f*x + e)^6 + 3*b*f*cosh(f*x + e)^4 + 3*(5*b*f*cosh(f*x + e)^2 + b*f)*sinh(f*x + e)^4 + 3*b*f*cosh(f
*x + e)^2 + 4*(5*b*f*cosh(f*x + e)^3 + 3*b*f*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(5*b*f*cosh(f*x + e)^4 + 6*b*f
*cosh(f*x + e)^2 + b*f)*sinh(f*x + e)^2 + b*f + 6*(b*f*cosh(f*x + e)^5 + 2*b*f*cosh(f*x + e)^3 + b*f*cosh(f*x
+ e))*sinh(f*x + e))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8010 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1574 vs. \(2 (205) = 410\).
time = 4.65, size = 1574, normalized size = 8.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

1/3*(6*b^2*arctan(-(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))*e^e/sqrt(-b) + 3*(a^2*e^e + a*b*e^e - 2*b^2*e^e)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqr
t(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 - b))/sqrt(a - b) - 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))^5*a^2*e^
e + 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))^5*a*
b*e^e - 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))^
5*b^2*e^e + 15*(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*sqrt(b)*e^e - 9*(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^(3/2)*e^e - 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))^4*b^(5/2)*e^e + 32*(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*e^e - 66*(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*e^e + 30*(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^2*e^e + 4*(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^3*e^e - 96*(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*sqrt(b)*e^e + 222*(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^(3/2)*e^e - 210*(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^(5/2)*e^e + 84*(s
qrt(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*b^(7/2)*e^
e - 48*(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
*e^e + 144*(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*e^e - 321*(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^2*e^e + 351*(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^3*e^e - 126*(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^4*e^e - 48*a^4*sqrt(b)*e^e + 16*a^3*b^(3/2)*e^e + 147*a^2*b^(5/2)*e^e - 165*a*b^(7/2)*e
^e + 50*b^(9/2)*e^e)/((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) - 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) + 4*a - 3*b)^3)/f^2

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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 {cosh}\left (e+f\,x\right )}^4} \,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)/cosh(e + f*x)^4,x)

[Out]

int((a + b*sinh(e + f*x)^2)^(3/2)/cosh(e + f*x)^4, x)

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