3.318 \(\int \frac{\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=77 \[ -\frac{(a-2 b) \sinh (c+d x)}{b^2 d}+\frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}+\frac{\sinh ^3(c+d x)}{3 b d} \]

[Out]

((a - b)^2*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Sinh[c + d*x])/(b^2*d) +
Sinh[c + d*x]^3/(3*b*d)

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Rubi [A]  time = 0.0933289, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3190, 390, 205} \[ -\frac{(a-2 b) \sinh (c+d x)}{b^2 d}+\frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}+\frac{\sinh ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

((a - b)^2*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Sinh[c + d*x])/(b^2*d) +
Sinh[c + d*x]^3/(3*b*d)

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a-2 b}{b^2}+\frac{x^2}{b}+\frac{a^2-2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{(a-2 b) \sinh (c+d x)}{b^2 d}+\frac{\sinh ^3(c+d x)}{3 b d}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{b^2 d}\\ &=\frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \sinh (c+d x)}{b^2 d}+\frac{\sinh ^3(c+d x)}{3 b d}\\ \end{align*}

Mathematica [A]  time = 0.244737, size = 79, normalized size = 1.03 \[ \frac{3 \sqrt{b} (7 b-4 a) \sinh (c+d x)-\frac{12 (a-b)^2 \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )}{\sqrt{a}}+b^{3/2} \sinh (3 (c+d x))}{12 b^{5/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

((-12*(a - b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 3*Sqrt[b]*(-4*a + 7*b)*Sinh[c + d*x] + b^(3
/2)*Sinh[3*(c + d*x)])/(12*b^(5/2)*d)

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Maple [B]  time = 0.059, size = 1148, normalized size = 14.9 \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)^5/(a+b*sinh(d*x+c)^2),x)

[Out]

-1/3/d/b/(tanh(1/2*d*x+1/2*c)+1)^3+1/2/d/b/(tanh(1/2*d*x+1/2*c)+1)^2+1/d/b^2/(tanh(1/2*d*x+1/2*c)+1)*a-2/d/b/(
tanh(1/2*d*x+1/2*c)+1)-1/d*a^3/b^2/(-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))+3/d*a^2/b/(-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))-3/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))-1/d*a^2/b^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))+2/d*a/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/d/((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/(-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*a^3/b^2/(-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))+3/d*a^2/b/(-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))-3/d*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*ar
ctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/d*a^2/b^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))-2/d*a/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/d/((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/(-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))-1/3/d/b/(tanh(1/2*d*x+
1/2*c)-1)^3-1/2/d/b/(tanh(1/2*d*x+1/2*c)-1)^2+1/d/b^2/(tanh(1/2*d*x+1/2*c)-1)*a-2/d/b/(tanh(1/2*d*x+1/2*c)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \,{\left (4 \, a e^{\left (4 \, c\right )} - 7 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 3 \,{\left (4 \, a e^{\left (2 \, c\right )} - 7 \, b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - b e^{\left (6 \, d x + 6 \, c\right )} + b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, b^{2} d} + \frac{1}{32} \, \int \frac{64 \,{\left ({\left (a^{2} e^{\left (3 \, c\right )} - 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a^{2} e^{c} - 2 \, a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{3} e^{\left (4 \, d x + 4 \, c\right )} + b^{3} + 2 \,{\left (2 \, a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/24*(3*(4*a*e^(4*c) - 7*b*e^(4*c))*e^(4*d*x) - 3*(4*a*e^(2*c) - 7*b*e^(2*c))*e^(2*d*x) - b*e^(6*d*x + 6*c) +
b)*e^(-3*d*x - 3*c)/(b^2*d) + 1/32*integrate(64*((a^2*e^(3*c) - 2*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (a^2
*e^c - 2*a*b*e^c + b^2*e^c)*e^(d*x))/(b^3*e^(4*d*x + 4*c) + b^3 + 2*(2*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x))
, x)

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Fricas [B]  time = 1.71961, size = 3804, normalized size = 49.4 \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)^5/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/24*(a*b^2*cosh(d*x + c)^6 + 6*a*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^2*sinh(d*x + c)^6 - 3*(4*a^2*b - 7*
a*b^2)*cosh(d*x + c)^4 + 3*(5*a*b^2*cosh(d*x + c)^2 - 4*a^2*b + 7*a*b^2)*sinh(d*x + c)^4 + 4*(5*a*b^2*cosh(d*x
+ c)^3 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a*b^2 + 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2
+ 3*(5*a*b^2*cosh(d*x + c)^4 + 4*a^2*b - 7*a*b^2 - 6*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 1
2*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b
+ b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*sqrt(-a*b)*log((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) - 4*(cosh(d*x + c)
^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c)
)*sqrt(-a*b) + b)/(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)*cos
h(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)) + 6*(a*b^2*cosh(d*x + c)^5 - 2*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^3 + (4*a^2*b - 7*a*b
^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*b^3
*d*cosh(d*x + c)*sinh(d*x + c)^2 + a*b^3*d*sinh(d*x + c)^3), 1/24*(a*b^2*cosh(d*x + c)^6 + 6*a*b^2*cosh(d*x +
c)*sinh(d*x + c)^5 + a*b^2*sinh(d*x + c)^6 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^4 + 3*(5*a*b^2*cosh(d*x + c)^
2 - 4*a^2*b + 7*a*b^2)*sinh(d*x + c)^4 + 4*(5*a*b^2*cosh(d*x + c)^3 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c))*sin
h(d*x + c)^3 - a*b^2 + 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*a*b^2*cosh(d*x + c)^4 + 4*a^2*b - 7*a*b^2
- 6*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 -
2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 - 2*a*
b + b^2)*sinh(d*x + c)^3)*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) + 24*((a^2 - 2*a*b
+ b^2)*cosh(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x
+ c)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*sqrt(a*b)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cos
h(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*sin
h(d*x + c))*sqrt(a*b)/(a*b)) + 6*(a*b^2*cosh(d*x + c)^5 - 2*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^3 + (4*a^2*b - 7
*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*a
*b^3*d*cosh(d*x + c)*sinh(d*x + c)^2 + a*b^3*d*sinh(d*x + c)^3)]

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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)**5/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError