3.352 \(\int \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx\)
Optimal. Leaf size=117 \[ -\frac{a (a-4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 b^{3/2} f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f} \]
[Out]
-(a*(a - 4*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a - 4*b)*Sinh[e
+ f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(8*b*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*b*f)
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Rubi [A] time = 0.113507, antiderivative size = 117, 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, 388, 195, 217, 206} \[ -\frac{a (a-4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 b^{3/2} f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
-(a*(a - 4*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a - 4*b)*Sinh[e
+ f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(8*b*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*b*f)
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 388
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Rule 195
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])
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 \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \sqrt{a+b x^2} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a-4 b) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\sinh (e+f x)\right )}{4 b f}\\ &=-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a (a-4 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{8 b f}\\ &=-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a (a-4 b)) \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 )}{8 b f}\\ &=-\frac{a (a-4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 b^{3/2} f}-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 b f}\\ \end{align*}
Mathematica [A] time = 0.663081, size = 124, normalized size = 1.06 \[ \frac{\sqrt{a+b \sinh ^2(e+f x)} \left (\sqrt{b} \sinh (e+f x) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} (a+b \cosh (2 (e+f x))+3 b)-\sqrt{a} (a-4 b) \sinh ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right )\right )}{8 b^{3/2} f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
(Sqrt[a + b*Sinh[e + f*x]^2]*(-(Sqrt[a]*(a - 4*b)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]) + Sqrt[b]*(a + 3*b
+ b*Cosh[2*(e + f*x)])*Sinh[e + f*x]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]))/(8*b^(3/2)*f*Sqrt[1 + (b*Sinh[e + f*x]
^2)/a])
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Maple [C] time = 0.09, size = 52, normalized size = 0.4 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({(b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{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)^3*(a+b*sinh(f*x+e)^2)^(1/2),x)
[Out]
`int/indef0`((b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^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 \sqrt{b \sinh \left (f x + e\right )^{2} + a} \cosh \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*sinh(f*x + e)^2 + a)*cosh(f*x + e)^3, x)
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Fricas [B] time = 2.45862, size = 8343, normalized size = 71.31 \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)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
[-1/64*(2*((a^2 - 4*a*b)*cosh(f*x + e)^4 + 4*(a^2 - 4*a*b)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 - 4*a*b)*cos
h(f*x + e)^2*sinh(f*x + e)^2 + 4*(a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 - 4*a*b)*sinh(f*x + e)^4)*
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)*cosh(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))*sin
h(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)*sinh(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)*s
inh(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)*cosh(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))*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)) + 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))/(cosh(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)) + 2*((a^2 - 4*a*b)*cosh(f*x + e)^4 + 4*(a^2 - 4*a*b)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 - 4*a*b)*cosh(
f*x + e)^2*sinh(f*x + e)^2 + 4*(a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 - 4*a*b)*sinh(f*x + e)^4)*sq
rt(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) + sinh(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)) - sqrt(2)*(b^2*cosh(f*x + e)^6 + 6*b^2*cosh(f*x + e)*
sinh(f*x + e)^5 + b^2*sinh(f*x + e)^6 + (2*a*b + 5*b^2)*cosh(f*x + e)^4 + (15*b^2*cosh(f*x + e)^2 + 2*a*b + 5*
b^2)*sinh(f*x + e)^4 + 4*(5*b^2*cosh(f*x + e)^3 + (2*a*b + 5*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - (2*a*b + 5*
b^2)*cosh(f*x + e)^2 + (15*b^2*cosh(f*x + e)^4 + 6*(2*a*b + 5*b^2)*cosh(f*x + e)^2 - 2*a*b - 5*b^2)*sinh(f*x +
e)^2 - b^2 + 2*(3*b^2*cosh(f*x + e)^5 + 2*(2*a*b + 5*b^2)*cosh(f*x + e)^3 - (2*a*b + 5*b^2)*cosh(f*x + e))*si
nh(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^2*f*cosh(f*x + e)^4 + 4*b^2*f*cosh(f*x + e)^3*sinh(f*x + e) + 6*b^2*f*cosh(f*x
+ e)^2*sinh(f*x + e)^2 + 4*b^2*f*cosh(f*x + e)*sinh(f*x + e)^3 + b^2*f*sinh(f*x + e)^4), 1/64*(4*((a^2 - 4*a*b
)*cosh(f*x + e)^4 + 4*(a^2 - 4*a*b)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 - 4*a*b)*cosh(f*x + e)^2*sinh(f*x +
e)^2 + 4*(a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 - 4*a*b)*sinh(f*x + e)^4)*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) + sinh(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))) + 4*((a^2 - 4*a*b)*cosh(f
*x + e)^4 + 4*(a^2 - 4*a*b)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 - 4*a*b)*cosh(f*x + e)^2*sinh(f*x + e)^2 +
4*(a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 - 4*a*b)*sinh(f*x + e)^4)*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)) + sqrt(2)*(b^2*
cosh(f*x + e)^6 + 6*b^2*cosh(f*x + e)*sinh(f*x + e)^5 + b^2*sinh(f*x + e)^6 + (2*a*b + 5*b^2)*cosh(f*x + e)^4
+ (15*b^2*cosh(f*x + e)^2 + 2*a*b + 5*b^2)*sinh(f*x + e)^4 + 4*(5*b^2*cosh(f*x + e)^3 + (2*a*b + 5*b^2)*cosh(f
*x + e))*sinh(f*x + e)^3 - (2*a*b + 5*b^2)*cosh(f*x + e)^2 + (15*b^2*cosh(f*x + e)^4 + 6*(2*a*b + 5*b^2)*cosh(
f*x + e)^2 - 2*a*b - 5*b^2)*sinh(f*x + e)^2 - b^2 + 2*(3*b^2*cosh(f*x + e)^5 + 2*(2*a*b + 5*b^2)*cosh(f*x + e)
^3 - (2*a*b + 5*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cos
h(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(b^2*f*cosh(f*x + e)^4 + 4*b^2*f*cosh(f*x +
e)^3*sinh(f*x + e) + 6*b^2*f*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*b^2*f*cosh(f*x + e)*sinh(f*x + e)^3 + b^2*f*s
inh(f*x + e)^4)]
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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(cosh(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (f x + e\right )^{2} + a} \cosh \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*sinh(f*x + e)^2 + a)*cosh(f*x + e)^3, x)