3.331 \(\int \frac{\cosh ^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)
Optimal. Leaf size=100 \[ -\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}-\frac{(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{x}{b^2} \]
[Out]
x/b^2 - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) - ((a - b)*Tanh
[c + d*x])/(2*a*b*d*(a - (a - b)*Tanh[c + d*x]^2))
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Rubi [A] time = 0.1384, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{3191, 414, 522, 206, 208} \[ -\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}-\frac{(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{x}{b^2} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
x/b^2 - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) - ((a - b)*Tanh
[c + d*x])/(2*a*b*d*(a - (a - b)*Tanh[c + d*x]^2))
Rule 3191
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 414
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]
Rule 522
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]
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])
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 \frac{\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-a-b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a b d}\\ &=-\frac{(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{b^2 d}-\frac{((a-b) (2 a+b)) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a b^2 d}\\ &=\frac{x}{b^2}-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}-\frac{(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.590935, size = 108, normalized size = 1.08 \[ \frac{-\frac{\left (2 a^2-a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} \sqrt{a-b}}+\frac{b (b-a) \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}+2 (c+d x)}{2 b^2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
(2*(c + d*x) - ((2*a^2 - a*b - b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a - b]) + (b*(
-a + b)*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^2*d)
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Maple [B] time = 0.062, size = 1116, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x)
[Out]
1/d/b^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*
c)^2*b+a)*tanh(1/2*d*x+1/2*c)^3+1/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2
*b+a)/a*tanh(1/2*d*x+1/2*c)^3-1/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2
*b+a)*tanh(1/2*d*x+1/2*c)+1/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/
a*tanh(1/2*d*x+1/2*c)-1/d/b^2*a/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-
b))^(1/2)+a-2*b)*a)^(1/2))+1/d/b*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*
x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/d/b^2*a/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/
2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/d/b*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/
2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/2/d/b/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(
1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/2/d/(-b*(a-b))^(1/2)/((2*(-b*(a-b))
^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/2/d/b/((2*(-b*(a-
b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2/d/(-b*(a-b))^
(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+
1/2/d/a/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2
))-1/2/d/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^
(1/2)+a-2*b)*a)^(1/2))*b-1/2/d/a/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-
b))^(1/2)-a+2*b)*a)^(1/2))-1/2/d/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x
+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))*b-1/d/b^2*ln(tanh(1/2*d*x+1/2*c)-1)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.75171, size = 3749, normalized size = 37.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(4*a*b*d*x*cosh(d*x + c)^4 + 16*a*b*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 4*a*b*d*x*sinh(d*x + c)^4 + 4*a*b
*d*x + 4*(2*(2*a^2 - a*b)*d*x + 2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 4*(6*a*b*d*x*cosh(d*x + c)^2 + 2*(2*a^2
- a*b)*d*x + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + ((2*a*b + b^2)*cosh(d*x + c)^4 + 4*(2*a*b + b^2)*cosh(d*x
+ c)*sinh(d*x + c)^3 + (2*a*b + b^2)*sinh(d*x + c)^4 + 2*(4*a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b + b^2)*c
osh(d*x + c)^2 + 4*a^2 - b^2)*sinh(d*x + c)^2 + 2*a*b + b^2 + 4*((2*a*b + b^2)*cosh(d*x + c)^3 + (4*a^2 - b^2)
*cosh(d*x + c))*sinh(d*x + c))*sqrt((a - b)/a)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3
+ b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c
)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(
d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + 2*a^2 - a*b)*sqrt((a - b)/a))/(b*cosh(d
*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(
d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) +
4*a*b - 4*b^2 + 8*(2*a*b*d*x*cosh(d*x + c)^3 + (2*(2*a^2 - a*b)*d*x + 2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh
(d*x + c))/(a*b^3*d*cosh(d*x + c)^4 + 4*a*b^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^3*d*sinh(d*x + c)^4 + a*b^
3*d + 2*(2*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 2*(3*a*b^3*d*cosh(d*x + c)^2 + (2*a^2*b^2 - a*b^3)*d)*sinh(d*x
+ c)^2 + 4*(a*b^3*d*cosh(d*x + c)^3 + (2*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*a*b*d*x*cos
h(d*x + c)^4 + 8*a*b*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*a*b*d*x*sinh(d*x + c)^4 + 2*a*b*d*x + 2*(2*(2*a^2 -
a*b)*d*x + 2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(6*a*b*d*x*cosh(d*x + c)^2 + 2*(2*a^2 - a*b)*d*x + 2*a^2
- 3*a*b + b^2)*sinh(d*x + c)^2 + ((2*a*b + b^2)*cosh(d*x + c)^4 + 4*(2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^
3 + (2*a*b + b^2)*sinh(d*x + c)^4 + 2*(4*a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b + b^2)*cosh(d*x + c)^2 + 4*a
^2 - b^2)*sinh(d*x + c)^2 + 2*a*b + b^2 + 4*((2*a*b + b^2)*cosh(d*x + c)^3 + (4*a^2 - b^2)*cosh(d*x + c))*sinh
(d*x + c))*sqrt(-(a - b)/a)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)
^2 + 2*a - b)*sqrt(-(a - b)/a)/(a - b)) + 2*a*b - 2*b^2 + 4*(2*a*b*d*x*cosh(d*x + c)^3 + (2*(2*a^2 - a*b)*d*x
+ 2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^4 + 4*a*b^3*d*cosh(d*x + c)*sinh(d
*x + c)^3 + a*b^3*d*sinh(d*x + c)^4 + a*b^3*d + 2*(2*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 2*(3*a*b^3*d*cosh(d*
x + c)^2 + (2*a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + 4*(a*b^3*d*cosh(d*x + c)^3 + (2*a^2*b^2 - a*b^3)*d*cosh(d*
x + c))*sinh(d*x + c))]
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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(d*x+c)**4/(a+b*sinh(d*x+c)**2)**2,x)
[Out]
Timed out
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Giac [B] time = 1.19301, size = 243, normalized size = 2.43 \begin{align*} \frac{d x + c}{b^{2} d} - \frac{{\left (2 \, a^{2} - a b - b^{2}\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{2 \, \sqrt{-a^{2} + a b} a b^{2} d} + \frac{2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} + b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b - b^{2}}{{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} a b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
(d*x + c)/(b^2*d) - 1/2*(2*a^2 - a*b - b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-
a^2 + a*b)*a*b^2*d) + (2*a^2*e^(2*d*x + 2*c) - 3*a*b*e^(2*d*x + 2*c) + b^2*e^(2*d*x + 2*c) + a*b - b^2)/((b*e^
(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)*a*b^2*d)