3.391 \(\int \frac{\cosh ^5(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\)
Optimal. Leaf size=134 \[ -\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{b^{5/2} f}-\frac{(a-b) \sinh (e+f x) \cosh ^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
[Out]
ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(b^(5/2)*f) - ((a - b)*Cosh[e + f*x]^2*Sinh[e + f
*x])/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((a - b)*(3*a + 2*b)*Sinh[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin
h[e + f*x]^2])
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Rubi [A] time = 0.141492, antiderivative size = 134, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{3190, 413, 385, 217, 206} \[ -\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{b^{5/2} f}-\frac{(a-b) \sinh (e+f x) \cosh ^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(b^(5/2)*f) - ((a - b)*Cosh[e + f*x]^2*Sinh[e + f
*x])/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((a - b)*(3*a + 2*b)*Sinh[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin
h[e + f*x]^2])
Rule 3190
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, 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 - 1)/2]
Rule 413
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 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 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])
Rubi steps
\begin{align*} \int \frac{\cosh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{(a-b) \cosh ^2(e+f x) \sinh (e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{a+2 b+3 a x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a b f}\\ &=-\frac{(a-b) \cosh ^2(e+f x) \sinh (e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{b^2 f}\\ &=-\frac{(a-b) \cosh ^2(e+f x) \sinh (e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{b^2 f}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{b^{5/2} f}-\frac{(a-b) \cosh ^2(e+f x) \sinh (e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.863662, size = 126, normalized size = 0.94 \[ \frac{\frac{2 \sqrt{2} (b-a) \sinh (e+f x) \left (3 a^2+b (2 a+b) \cosh (2 (e+f x))+a b-b^2\right )}{3 a^2 b^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right )}{b^{5/2}}}{f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]
[Out]
(ArcTanh[(Sqrt[2]*Sqrt[b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/b^(5/2) + (2*Sqrt[2]*(-a + b)*(3
*a^2 + a*b - b^2 + b*(2*a + b)*Cosh[2*(e + f*x)])*Sinh[e + f*x])/(3*a^2*b^2*(2*a - b + b*Cosh[2*(e + f*x)])^(3
/2)))/f
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Maple [C] time = 0.082, size = 65, normalized size = 0.5 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{4}}{{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+2\,ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+{a}^{2}}{\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(cosh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),x)
[Out]
`int/indef0`(cosh(f*x+e)^4/(b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)/(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 \frac{\cosh \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(cosh(f*x + e)^5/(b*sinh(f*x + e)^2 + a)^(5/2), x)
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Fricas [B] time = 5.85418, size = 15815, normalized size = 118.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*(a^2*b^2*cosh(f*x + e)^8 + 8*a^2*b^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a
^3*b - a^2*b^2)*cosh(f*x + e)^6 + 4*(7*a^2*b^2*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2
*b^2*cosh(f*x + e)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*
cosh(f*x + e)^4 + 2*(35*a^2*b^2*cosh(f*x + e)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*
x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 +
(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^
2*b^2*cosh(f*x + e)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^
2*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*(a^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 +
(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^3 + (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*log
(-((a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^8 + 8*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b
- 2*a*b^2 + b^3)*sinh(f*x + e)^8 + 2*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^6 + 2*(a^3 - 4*a^2*b + 5*
a*b^2 - 2*b^3 + 14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(14*(a^2*b - 2*a*b^2 + b^3)*co
sh(f*x + e)^3 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b
^3)*cosh(f*x + e)^4 + (70*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^4 + 9*a^2*b - 14*a*b^2 + 6*b^3 + 30*(a^3 - 4*a
^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^5 + 10*
(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e))*sinh(f*x + e)^
3 + b^3 + 2*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + 2*(14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^6 + 15*(a^3 - 4*a^
2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + 3*a*b^2 - 2*b^3 + 3*(9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^2)*sin
h(f*x + e)^2 + sqrt(2)*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e
)^5 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^6 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(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*b - 3*b^2)*cosh(f*x + e)^2 + (15*(a^2 - 2*a*b + b^2)*cosh(f*x +
e)^4 - 18*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - 4*a*b + 3*b^2)*sinh(f*x + e)^2 - b^2 + 2*(3*(a^2 - 2*a*b + b^
2)*cosh(f*x + e)^5 - 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - (4*a*b - 3*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqr
t(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)) + 4*(2*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^7 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f
*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^3 + (3*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e))/(co
sh(f*x + e)^6 + 6*cosh(f*x + e)^5*sinh(f*x + e) + 15*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*cosh(f*x + e)^3*sinh
(f*x + e)^3 + 15*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*cosh(f*x + e)*sinh(f*x + e)^5 + sinh(f*x + e)^6)) + 3*(a^
2*b^2*cosh(f*x + e)^8 + 8*a^2*b^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a^3*b - a^2*b
^2)*cosh(f*x + e)^6 + 4*(7*a^2*b^2*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2*b^2*cosh(f*
x + e)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e
)^4 + 2*(35*a^2*b^2*cosh(f*x + e)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2)*si
nh(f*x + e)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 + (8*a^4 - 8*a
^3*b + 3*a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^2*b^2*cosh(f
*x + e)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(
f*x + e)^2)*sinh(f*x + e)^2 + 8*(a^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 + (8*a^4 - 8*
a^3*b + 3*a^2*b^2)*cosh(f*x + e)^3 + (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*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*a*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2
+ a)*sinh(f*x + e)^2 + sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(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) + si
nh(f*x + e)^2)) + 4*(b*cosh(f*x + e)^3 + a*cosh(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x + e)^2 + 2*cosh(f*x + e
)*sinh(f*x + e) + sinh(f*x + e)^2)) - 8*sqrt(2)*((2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^6 + 6*(2*a^2*b^2 - a*
b^3 - b^4)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2*b^2 - a*b^3 - b^4)*sinh(f*x + e)^6 + 3*(2*a^3*b - 2*a^2*b^2
- a*b^3 + b^4)*cosh(f*x + e)^4 + 3*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4 + 5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x +
e)^2)*sinh(f*x + e)^4 - 2*a^2*b^2 + a*b^3 + b^4 + 4*(5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b
- 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e))*sinh(f*x + e)^3 - 3*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x +
e)^2 + 3*(5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^4 - 2*a^3*b + 2*a^2*b^2 + a*b^3 - b^4 + 6*(2*a^3*b - 2*a^2
*b^2 - a*b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^5 + 2*(2*a^3
*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e)^3 - (2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*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^2*b^5*f*cosh(f*x + e)^8 + 8*a^2*b^5*f*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^5*f*sinh(f*x
+ e)^8 + a^2*b^5*f + 4*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^6 + 4*(7*a^2*b^5*f*cosh(f*x + e)^2 + (2*a^3*b^4
- a^2*b^5)*f)*sinh(f*x + e)^6 + 2*(8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e)^4 + 8*(7*a^2*b^5*f*cosh(
f*x + e)^3 + 3*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a^2*b^5*f*cosh(f*x + e)^4 + 30*(
2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^2 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f)*sinh(f*x + e)^4 + 4*(2*a^3*b^4
- a^2*b^5)*f*cosh(f*x + e)^2 + 8*(7*a^2*b^5*f*cosh(f*x + e)^5 + 10*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^3 +
(8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a^2*b^5*f*cosh(f*x + e)^6 + 15*(2*
a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^4 + 3*(8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e)^2 + (2*a^3*b^4 -
a^2*b^5)*f)*sinh(f*x + e)^2 + 8*(a^2*b^5*f*cosh(f*x + e)^7 + 3*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^5 + (8*a^
4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e)^3 + (2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)), -1/6
*(3*(a^2*b^2*cosh(f*x + e)^8 + 8*a^2*b^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a^3*b
- a^2*b^2)*cosh(f*x + e)^6 + 4*(7*a^2*b^2*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2*b^2*
cosh(f*x + e)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(
f*x + e)^4 + 2*(35*a^2*b^2*cosh(f*x + e)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*x + e
)^2)*sinh(f*x + e)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 + (8*a^
4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^2*b^2
*cosh(f*x + e)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^2*b^2
)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*(a^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 + (8*a
^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^3 + (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-b)*arctan(
sqrt(2)*((a - b)*cosh(f*x + e)^2 + 2*(a - b)*cosh(f*x + e)*sinh(f*x + e) + (a - b)*sinh(f*x + e)^2 + b)*sqrt(-
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) + s
inh(f*x + e)^2))/((a*b - b^2)*cosh(f*x + e)^4 + 4*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*sinh
(f*x + e)^4 - (3*a*b - 2*b^2)*cosh(f*x + e)^2 + (6*(a*b - b^2)*cosh(f*x + e)^2 - 3*a*b + 2*b^2)*sinh(f*x + e)^
2 - b^2 + 2*(2*(a*b - b^2)*cosh(f*x + e)^3 - (3*a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e))) + 3*(a^2*b^2*cosh(
f*x + e)^8 + 8*a^2*b^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a^3*b - a^2*b^2)*cosh(f*
x + e)^6 + 4*(7*a^2*b^2*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2*b^2*cosh(f*x + e)^3 +
3*(2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^4 + 2*(35
*a^2*b^2*cosh(f*x + e)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)
^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 + (8*a^4 - 8*a^3*b + 3*a^
2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^2*b^2*cosh(f*x + e)^6 +
15*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^2)
*sinh(f*x + e)^2 + 8*(a^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 + (8*a^4 - 8*a^3*b + 3*a
^2*b^2)*cosh(f*x + e)^3 + (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-b)*arctan(sqrt(2)*(cosh(f*x
+ e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-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))/(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)) + 4*sqrt(2)*((2*a^
2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^6 + 6*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2*b^2
- a*b^3 - b^4)*sinh(f*x + e)^6 + 3*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e)^4 + 3*(2*a^3*b - 2*a^2*b^
2 - a*b^3 + b^4 + 5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 - 2*a^2*b^2 + a*b^3 + b^4 + 4*(
5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e))*sinh(f*x +
e)^3 - 3*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e)^2 + 3*(5*(2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^4
- 2*a^3*b + 2*a^2*b^2 + a*b^3 - b^4 + 6*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 +
6*((2*a^2*b^2 - a*b^3 - b^4)*cosh(f*x + e)^5 + 2*(2*a^3*b - 2*a^2*b^2 - a*b^3 + b^4)*cosh(f*x + e)^3 - (2*a^3
*b - 2*a^2*b^2 - a*b^3 + b^4)*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^2*b^5*f*cosh(f*x + e)^8 + 8*a^2*
b^5*f*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^5*f*sinh(f*x + e)^8 + a^2*b^5*f + 4*(2*a^3*b^4 - a^2*b^5)*f*cosh(f
*x + e)^6 + 4*(7*a^2*b^5*f*cosh(f*x + e)^2 + (2*a^3*b^4 - a^2*b^5)*f)*sinh(f*x + e)^6 + 2*(8*a^4*b^3 - 8*a^3*b
^4 + 3*a^2*b^5)*f*cosh(f*x + e)^4 + 8*(7*a^2*b^5*f*cosh(f*x + e)^3 + 3*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e))*
sinh(f*x + e)^5 + 2*(35*a^2*b^5*f*cosh(f*x + e)^4 + 30*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^2 + (8*a^4*b^3 -
8*a^3*b^4 + 3*a^2*b^5)*f)*sinh(f*x + e)^4 + 4*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^2 + 8*(7*a^2*b^5*f*cosh(f*
x + e)^5 + 10*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^3 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e))*s
inh(f*x + e)^3 + 4*(7*a^2*b^5*f*cosh(f*x + e)^6 + 15*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^4 + 3*(8*a^4*b^3 -
8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e)^2 + (2*a^3*b^4 - a^2*b^5)*f)*sinh(f*x + e)^2 + 8*(a^2*b^5*f*cosh(f*x +
e)^7 + 3*(2*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^5 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*f*cosh(f*x + e)^3 + (2*
a^3*b^4 - a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e))]
________________________________________________________________________________________
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(cosh(f*x+e)**5/(a+b*sinh(f*x+e)**2)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")
[Out]
integrate(cosh(f*x + e)^5/(b*sinh(f*x + e)^2 + a)^(5/2), x)