3.4.44 \(\int \frac {\cosh (c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [344]

Optimal. Leaf size=96 \[ \frac {3 \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

1/4*sinh(d*x+c)/a/d/(a+b*sinh(d*x+c)^2)^2+3/8*sinh(d*x+c)/a^2/d/(a+b*sinh(d*x+c)^2)+3/8*arctan(sinh(d*x+c)*b^(
1/2)/a^(1/2))/a^(5/2)/d/b^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3269, 205, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*d) + Sinh[c + d*x]/(4*a*d*(a + b*Sinh[c + d*x]^
2)^2) + (3*Sinh[c + d*x])/(8*a^2*d*(a + b*Sinh[c + d*x]^2))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 3269

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]

Rubi steps

\begin {align*} \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a d}\\ &=\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 d}\\ &=\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 79, normalized size = 0.82 \begin {gather*} \frac {\frac {3 \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} \sinh (c+d x) \left (5 a+3 b \sinh ^2(c+d x)\right )}{\left (a+b \sinh ^2(c+d x)\right )^2}}{8 a^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/Sqrt[b] + (Sqrt[a]*Sinh[c + d*x]*(5*a + 3*b*Sinh[c + d*x]^2))/(a
+ b*Sinh[c + d*x]^2)^2)/(8*a^(5/2)*d)

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Maple [A]
time = 0.56, size = 86, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\frac {\sinh \left (d x +c \right )}{4 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {\frac {3 \sinh \left (d x +c \right )}{8 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {3 \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}}{a}}{d}\)	\(86\)
default	\(\frac {\frac {\sinh \left (d x +c \right )}{4 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {\frac {3 \sinh \left (d x +c \right )}{8 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {3 \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}}{a}}{d}\)	\(86\)
risch	\(\frac {\left (3 b \,{\mathrm e}^{6 d x +6 c}+20 a \,{\mathrm e}^{4 d x +4 c}-9 b \,{\mathrm e}^{4 d x +4 c}-20 a \,{\mathrm e}^{2 d x +2 c}+9 b \,{\mathrm e}^{2 d x +2 c}-3 b \right ) {\mathrm e}^{d x +c}}{4 \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2} a^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,a^{2}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4*sinh(d*x+c)/a/(a+b*sinh(d*x+c)^2)^2+3/4/a*(1/2*sinh(d*x+c)/a/(a+b*sinh(d*x+c)^2)+1/2/a/(a*b)^(1/2)*ar
ctan(b*sinh(d*x+c)/(a*b)^(1/2))))

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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(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((20*a*e^(5*c) - 9*b*e^(5*c))*e^(5*d*x) - (20*a*e^(3*c) - 9*b*e^(3*c))*e^(3*d*x) + 3*b*e^(7*d*x + 7*c) - 3
*b*e^(d*x + c))/(a^2*b^2*d*e^(8*d*x + 8*c) + a^2*b^2*d + 4*(2*a^3*b*d*e^(6*c) - a^2*b^2*d*e^(6*c))*e^(6*d*x) +
2*(8*a^4*d*e^(4*c) - 8*a^3*b*d*e^(4*c) + 3*a^2*b^2*d*e^(4*c))*e^(4*d*x) + 4*(2*a^3*b*d*e^(2*c) - a^2*b^2*d*e^
(2*c))*e^(2*d*x)) + 1/2*integrate(3/2*(e^(3*d*x + 3*c) + e^(d*x + c))/(a^2*b*e^(4*d*x + 4*c) + a^2*b + 2*(2*a^
3*e^(2*c) - a^2*b*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2111 vs. \(2 (82) = 164\).
time = 0.63, size = 3934, normalized size = 40.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(12*a*b^2*cosh(d*x + c)^7 + 84*a*b^2*cosh(d*x + c)*sinh(d*x + c)^6 + 12*a*b^2*sinh(d*x + c)^7 + 4*(20*a^
2*b - 9*a*b^2)*cosh(d*x + c)^5 + 4*(63*a*b^2*cosh(d*x + c)^2 + 20*a^2*b - 9*a*b^2)*sinh(d*x + c)^5 + 20*(21*a*
b^2*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 12*a*b^2*cosh(d*x + c) - 4*(20*a^2
*b - 9*a*b^2)*cosh(d*x + c)^3 + 4*(105*a*b^2*cosh(d*x + c)^4 - 20*a^2*b + 9*a*b^2 + 10*(20*a^2*b - 9*a*b^2)*co
sh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(63*a*b^2*cosh(d*x + c)^5 + 10*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 - 3*(20
*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^
7 + b^2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x +
c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)*
cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(d
*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a*b - b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c
))*sinh(d*x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x +
c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7
+ 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*si
nh(d*x + c))*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*cos
h(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)*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*(21*a*b^2*cosh(d*x + c)^6 + 5*(20*
a^2*b - 9*a*b^2)*cosh(d*x + c)^4 - 3*a*b^2 - 3*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^3*b^3*d
*cosh(d*x + c)^8 + 8*a^3*b^3*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*b^3*d*sinh(d*x + c)^8 + 4*(2*a^4*b^2 - a^3*
b^3)*d*cosh(d*x + c)^6 + a^3*b^3*d + 4*(7*a^3*b^3*d*cosh(d*x + c)^2 + (2*a^4*b^2 - a^3*b^3)*d)*sinh(d*x + c)^6
+ 2*(8*a^5*b - 8*a^4*b^2 + 3*a^3*b^3)*d*cosh(d*x + c)^4 + 8*(7*a^3*b^3*d*cosh(d*x + c)^3 + 3*(2*a^4*b^2 - a^3
*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^3*b^3*d*cosh(d*x + c)^4 + 30*(2*a^4*b^2 - a^3*b^3)*d*cosh(d*x
+ c)^2 + (8*a^5*b - 8*a^4*b^2 + 3*a^3*b^3)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^2 + 8
*(7*a^3*b^3*d*cosh(d*x + c)^5 + 10*(2*a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^3 + (8*a^5*b - 8*a^4*b^2 + 3*a^3*b^3)
*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^3*b^3*d*cosh(d*x + c)^6 + 15*(2*a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^
4 + 3*(8*a^5*b - 8*a^4*b^2 + 3*a^3*b^3)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - a^3*b^3)*d)*sinh(d*x + c)^2 + 8*(a^3*
b^3*d*cosh(d*x + c)^7 + 3*(2*a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^5 + (8*a^5*b - 8*a^4*b^2 + 3*a^3*b^3)*d*cosh(d
*x + c)^3 + (2*a^4*b^2 - a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(6*a*b^2*cosh(d*x + c)^7 + 42*a*b^2*cos
h(d*x + c)*sinh(d*x + c)^6 + 6*a*b^2*sinh(d*x + c)^7 + 2*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^5 + 2*(63*a*b^2*co
sh(d*x + c)^2 + 20*a^2*b - 9*a*b^2)*sinh(d*x + c)^5 + 10*(21*a*b^2*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^2)*cosh
(d*x + c))*sinh(d*x + c)^4 - 6*a*b^2*cosh(d*x + c) - 2*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 + 2*(105*a*b^2*cos
h(d*x + c)^4 - 20*a^2*b + 9*a*b^2 + 10*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(63*a*b^2*cos
h(d*x + c)^5 + 10*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 - 3*(20*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2
+ 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d
*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30
*(2*a*b - b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a*b
- b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x +
c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x + c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2
+ 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*
a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cos
h(d*x + c) + sinh(d*x + c))/a) + 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x +
c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*b^2*c
osh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (85) = 170\).
time = 30.10, size = 835, normalized size = 8.70 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x \cosh {\left (c \right )}}{\sinh ^{6}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{3} d} & \text {for}\: b = 0 \\- \frac {1}{5 b^{3} d \sinh ^{5}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sinh ^{2}{\left (c \right )}\right )^{3}} & \text {for}\: d = 0 \\\frac {3 a^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {3 a^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {10 a b \sqrt {- \frac {a}{b}} \sinh {\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {6 a b \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {6 a b \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {6 b^{2} \sqrt {- \frac {a}{b}} \sinh ^{3}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {3 b^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{4}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {3 b^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{4}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Piecewise((zoo*x*cosh(c)/sinh(c)**6, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (sinh(c + d*x)/(a**3*d), Eq(b, 0)), (-1/
(5*b**3*d*sinh(c + d*x)**5), Eq(a, 0)), (x*cosh(c)/(a + b*sinh(c)**2)**3, Eq(d, 0)), (3*a**2*log(-sqrt(-a/b) +
sinh(c + d*x))/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/
b)*sinh(c + d*x)**4) - 3*a**2*log(sqrt(-a/b) + sinh(c + d*x))/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a
/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 10*a*b*sqrt(-a/b)*sinh(c + d*x)/(16*a**4*
b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 6*
a*b*log(-sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**2/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh
(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) - 6*a*b*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*
x)**2/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c
+ d*x)**4) + 6*b**2*sqrt(-a/b)*sinh(c + d*x)**3/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d
*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 3*b**2*log(-sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**
4/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*
x)**4) - 3*b**2*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**4/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt
(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 0.94, size = 87, normalized size = 0.91 \begin {gather*} \frac {\frac {5\,\mathrm {sinh}\left (c+d\,x\right )}{8\,a}+\frac {3\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{8\,a^2}}{d\,a^2+2\,d\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+d\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {b}\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(a + b*sinh(c + d*x)^2)^3,x)

[Out]

((5*sinh(c + d*x))/(8*a) + (3*b*sinh(c + d*x)^3)/(8*a^2))/(a^2*d + b^2*d*sinh(c + d*x)^4 + 2*a*b*d*sinh(c + d*
x)^2) + (3*atan((b^(1/2)*sinh(c + d*x))/a^(1/2)))/(8*a^(5/2)*b^(1/2)*d)

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