3.340 \(\int \frac{\cosh ^5(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{3 \left (\frac{1}{a^2}-\frac{1}{b^2}\right ) \sinh (c+d x)}{8 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2} d}-\frac{(a-b) \sinh (c+d x) \cosh ^2(c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.124537, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3190, 413, 385, 205} \[ \frac{3 \left (\frac{1}{a^2}-\frac{1}{b^2}\right ) \sinh (c+d x)}{8 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2} d}-\frac{(a-b) \sinh (c+d x) \cosh ^2(c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.352346, size = 149, normalized size = 1.12 \[ \frac{-\frac{2 \sqrt{a} \sqrt{b} \left (5 a^2-2 a b-3 b^2\right ) \sinh (c+d x)}{2 a+b \cosh (2 (c+d x))-b}-\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )+\frac{8 a^{3/2} \sqrt{b} (a-b)^2 \sinh (c+d x)}{(2 a+b \cosh (2 (c+d x))-b)^2}}{8 a^{5/2} b^{5/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

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Maple [B]  time = 0.062, size = 1884, normalized size = 14.2 \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)^5/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-3/8/d/b^2*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))+3/8/d*b/a^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))-3/8/d/b^2*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))+3/8/d*b/a^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*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)^2/a^2*tanh(1/2*d*x+1/2*c)^3-3/8/d/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))+3/8/d/
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))+1/
2/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)^2*tanh(1/2*d*x+1/2*c)^7+
7/2/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)^2*tanh(1/2*d*x+1/2*c)^
5-7/2/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)^2*tanh(1/2*d*x+1/2*c
)^3-1/2/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)^2*tanh(1/2*d*x+1/2
*c)+5/4/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)^2/a*tanh(1/2*d*x+1/2
*c)+3/8/d/a^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))-3/8/d/a^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))-7/4/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)^2/a*tanh(1/
2*d*x+1/2*c)^3-5/4/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)^2/a*tanh(
1/2*d*x+1/2*c)^7+7/4/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)^2/a*tan
h(1/2*d*x+1/2*c)^5-3/4/d/b^2/(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)^2
*a*tanh(1/2*d*x+1/2*c)-9/4/d/b^2/(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)^2*a*tanh(1/2*d*x+1/2*c)^5-3/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)^2/a^2*tanh(1/2*d*x+1/2*c)^5+9/4/d/b^2/(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)^2*a*tanh(1/2*d*x+1/2*c)^3+3/4/d/b^2/(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)^2*a*tanh(1/2*d*x+1/2*c)^7-1/8/d/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arc
tanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/8/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))+1/4/d/b/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))-1/4/d/b/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))+1/8/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
))+1/8/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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (5 \, a^{2} b e^{\left (7 \, c\right )} - 2 \, a b^{2} e^{\left (7 \, c\right )} - 3 \, b^{3} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} +{\left (12 \, a^{3} e^{\left (5 \, c\right )} - 7 \, a^{2} b e^{\left (5 \, c\right )} - 14 \, a b^{2} e^{\left (5 \, c\right )} + 9 \, b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (12 \, a^{3} e^{\left (3 \, c\right )} - 7 \, a^{2} b e^{\left (3 \, c\right )} - 14 \, a b^{2} e^{\left (3 \, c\right )} + 9 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (5 \, a^{2} b e^{c} - 2 \, a b^{2} e^{c} - 3 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{2} b^{4} d e^{\left (8 \, d x + 8 \, c\right )} + a^{2} b^{4} d + 4 \,{\left (2 \, a^{3} b^{3} d e^{\left (6 \, c\right )} - a^{2} b^{4} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (8 \, a^{4} b^{2} d e^{\left (4 \, c\right )} - 8 \, a^{3} b^{3} d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (2 \, a^{3} b^{3} d e^{\left (2 \, c\right )} - a^{2} b^{4} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + \frac{1}{32} \, \int \frac{8 \,{\left ({\left (3 \, a^{2} e^{\left (3 \, c\right )} + 2 \, a b e^{\left (3 \, c\right )} + 3 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (3 \, a^{2} e^{c} + 2 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + a^{2} b^{3} + 2 \,{\left (2 \, a^{3} b^{2} e^{\left (2 \, c\right )} - a^{2} 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)^3,x, algorithm="maxima")

[Out]

-1/4*((5*a^2*b*e^(7*c) - 2*a*b^2*e^(7*c) - 3*b^3*e^(7*c))*e^(7*d*x) + (12*a^3*e^(5*c) - 7*a^2*b*e^(5*c) - 14*a
*b^2*e^(5*c) + 9*b^3*e^(5*c))*e^(5*d*x) - (12*a^3*e^(3*c) - 7*a^2*b*e^(3*c) - 14*a*b^2*e^(3*c) + 9*b^3*e^(3*c)
)*e^(3*d*x) - (5*a^2*b*e^c - 2*a*b^2*e^c - 3*b^3*e^c)*e^(d*x))/(a^2*b^4*d*e^(8*d*x + 8*c) + a^2*b^4*d + 4*(2*a
^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*b^2*d*e^(4*c) - 8*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e
^(4*c))*e^(4*d*x) + 4*(2*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) + 1/32*integrate(8*((3*a^2*e^(3*c)
+ 2*a*b*e^(3*c) + 3*b^2*e^(3*c))*e^(3*d*x) + (3*a^2*e^c + 2*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^2*b^3*e^(4*d*x +
4*c) + a^2*b^3 + 2*(2*a^3*b^2*e^(2*c) - a^2*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.45113, size = 13445, normalized size = 101.09 \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)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^7 + 28*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x +
c)*sinh(d*x + c)^6 + 4*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^7 + 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^
3 + 9*a*b^4)*cosh(d*x + c)^5 + 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 + 21*(5*a^3*b^2 - 2*a^2*b^3 - 3*
a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (12*a^4*b
- 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*
a*b^4)*cosh(d*x + c)^3 - 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 - 35*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)
*cosh(d*x + c)^4 - 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(
5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x +
c)^3 - 3*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2*b^2 + 2*a*b^3
+ 3*b^4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b^2 + 2*a*b
^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(6*a^3*b + a^2*b^2 +
4*a*b^3 - 3*b^4 + 7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b^2 + 2*a*b^
3 + 3*b^4)*cosh(d*x + c)^3 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^
4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x +
c)^4 + 24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4 + 30*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 3*a^2*b^2 + 2*a*b^3 + 3*b^4 + 8*(7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*
(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^2*b^2 + 2*a*
b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 6*a^3*b + a^2*b^2 +
4*a*b^3 - 3*b^4 + 3*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((
3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (24*a
^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x
+ c))*sinh(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*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)*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*(5*a^3*b^2 - 2*a^2*b^3 -
3*a*b^4)*cosh(d*x + c) + 4*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 - 5*a^3*b^2 + 2*a^2*b^3 + 3*a*
b^4 + 5*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 - 3*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 +
9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^3*b^5*d*cosh(d*x + c)^8 + 8*a^3*b^5*d*cosh(d*x + c)*sinh(d*x + c)
^7 + a^3*b^5*d*sinh(d*x + c)^8 + a^3*b^5*d + 4*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*a^3*b^5*d*cosh(d
*x + c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4
+ 8*(7*a^3*b^5*d*cosh(d*x + c)^3 + 3*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^3*b^5*d
*cosh(d*x + c)^4 + 30*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*
x + c)^4 + 4*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*a^3*b^5*d*cosh(d*x + c)^5 + 10*(2*a^4*b^4 - a^3*b^
5)*d*cosh(d*x + c)^3 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^3*b^5*d*c
osh(d*x + c)^6 + 15*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x
+ c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + 8*(a^3*b^5*d*cosh(d*x + c)^7 + 3*(2*a^4*b^4 - a^3*b^5)*d*
cosh(d*x + c)^5 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c
))*sinh(d*x + c)), -1/8*(2*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^7 + 14*(5*a^3*b^2 - 2*a^2*b^3 - 3*a
*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^7 + 2*(12*a^4*b - 7*a^
3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^5 + 2*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 + 21*(5*a^3*b^2
- 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x +
c)^3 + (12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(12*a^4*b - 7*a^3*b^2
- 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^3 - 2*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 - 35*(5*a^3*b^2 - 2*a
^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 - 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 2*(21*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*
a*b^4)*cosh(d*x + c)^3 - 3*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 - ((3*
a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (
3*a^2*b^2 + 2*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(6*
a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4 + 7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3
*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 2*a*b^3 + 3
*b^4)*cosh(d*x + c)^4 + 24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4 + 30*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3
*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^2*b^2 + 2*a*b^3 + 3*b^4 + 8*(7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh
(d*x + c)^5 + 10*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a
*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*
(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 6*a
^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4 + 3*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 + 8*((3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d
*x + c)^5 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (6*a^3*b + a^2*b^2 + 4*a*b^3
- 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) - ((3
*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 +
(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(6
*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4 + 7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(
3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 2*a*b^3 +
3*b^4)*cosh(d*x + c)^4 + 24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4 + 30*(6*a^3*b + a^2*b^2 + 4*a*b^3 -
3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^2*b^2 + 2*a*b^3 + 3*b^4 + 8*(7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cos
h(d*x + c)^5 + 10*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*
a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7
*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 6*
a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4 + 3*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh
(d*x + c)^2 + 8*((3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(
d*x + c)^5 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (6*a^3*b + a^2*b^2 + 4*a*b^3
- 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(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)*sinh(d*x + c))*sqrt(a*b)
/(a*b)) - 2*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c) + 2*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x
+ c)^6 - 5*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 + 5*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 - 3
*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^3*b^5*d*cosh(d*x + c)^8 + 8*
a^3*b^5*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*b^5*d*sinh(d*x + c)^8 + a^3*b^5*d + 4*(2*a^4*b^4 - a^3*b^5)*d*co
sh(d*x + c)^6 + 4*(7*a^3*b^5*d*cosh(d*x + c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^3 - 8*a
^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*a^3*b^5*d*cosh(d*x + c)^3 + 3*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(35*a^3*b^5*d*cosh(d*x + c)^4 + 30*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (8*a^5*b^
3 - 8*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*a^3*b^5*d*cos
h(d*x + c)^5 + 10*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c
))*sinh(d*x + c)^3 + 4*(7*a^3*b^5*d*cosh(d*x + c)^6 + 15*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^
3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + 8*(a^3*b^5*d*cosh(d*
x + c)^7 + 3*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 +
(2*a^4*b^4 - a^3*b^5)*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)**5/(a+b*sinh(d*x+c)**2)**3,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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError