3.83 \(\int \frac{\cosh ^3(c+d x)}{(a+b \text{sech}^2(c+d x))^2} \, dx\)

Optimal. Leaf size=125 \[ -\frac{b^3 \sinh (c+d x)}{2 a^3 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{b^2 (6 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{7/2} d (a+b)^{3/2}}+\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d} \]

[Out]

(b^2*(6*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(7/2)*(a + b)^(3/2)*d) + ((a - 2*b)*Sinh[c
+ d*x])/(a^3*d) + Sinh[c + d*x]^3/(3*a^2*d) - (b^3*Sinh[c + d*x])/(2*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)
)

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Rubi [A]  time = 0.153556, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4147, 390, 385, 205} \[ -\frac{b^3 \sinh (c+d x)}{2 a^3 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{b^2 (6 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{7/2} d (a+b)^{3/2}}+\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*(6*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(7/2)*(a + b)^(3/2)*d) + ((a - 2*b)*Sinh[c
+ d*x])/(a^3*d) + Sinh[c + d*x]^3/(3*a^2*d) - (b^3*Sinh[c + d*x])/(2*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)
)

Rule 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]

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 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 ^3(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a-2 b}{a^3}+\frac{x^2}{a^2}+\frac{b^2 (3 a+2 b)+3 a b^2 x^2}{a^3 \left (a+b+a x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d}+\frac{\operatorname{Subst}\left (\int \frac{b^2 (3 a+2 b)+3 a b^2 x^2}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{a^3 d}\\ &=\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d}-\frac{b^3 \sinh (c+d x)}{2 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac{\left (b^2 (6 a+5 b)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{2 a^3 (a+b) d}\\ &=\frac{b^2 (6 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{7/2} (a+b)^{3/2} d}+\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d}-\frac{b^3 \sinh (c+d x)}{2 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.862626, size = 113, normalized size = 0.9 \[ \frac{a^{3/2} \sinh (3 (c+d x))+3 \sqrt{a} \sinh (c+d x) \left (-\frac{4 b^3}{(a+b) (a \cosh (2 (c+d x))+a+2 b)}+3 a-8 b\right )-\frac{6 b^2 (6 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a+b} \text{csch}(c+d x)}{\sqrt{a}}\right )}{(a+b)^{3/2}}}{12 a^{7/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/(a + b)^(3/2) + 3*Sqrt[a]*(3*a - 8*b - (4*b^
3)/((a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])))*Sinh[c + d*x] + a^(3/2)*Sinh[3*(c + d*x)])/(12*a^(7/2)*d)

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Maple [B]  time = 0.101, size = 517, normalized size = 4.1 \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)^3/(a+b*sech(d*x+c)^2)^2,x)

[Out]

-1/3/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^3+1/2/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^2-1/d/a^2/(tanh(1/2*d*x+1/2*c)+1)+2/d/a
^3/(tanh(1/2*d*x+1/2*c)+1)*b+1/d*b^3/a^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c
)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)^3-1/d*b^3/a^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1
/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)+3/d*b^2/a^(
5/2)/(a+b)^(3/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+3/d*b^2/a^(5/2)/(a+b)^(3/2)
*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))+5/2/d*b^3/a^(7/2)/(a+b)^(3/2)*arctan(1/2*(2
*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+5/2/d*b^3/a^(7/2)/(a+b)^(3/2)*arctan(1/2*(2*tanh(1/2*d*x+
1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))-1/3/d/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/d/a^2/(tanh(1/2*d*x+1/2*c)-1)^2
-1/d/a^2/(tanh(1/2*d*x+1/2*c)-1)+2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} + a^{2} b -{\left (a^{3} e^{\left (10 \, c\right )} + a^{2} b e^{\left (10 \, c\right )}\right )} e^{\left (10 \, d x\right )} -{\left (11 \, a^{3} e^{\left (8 \, c\right )} - 9 \, a^{2} b e^{\left (8 \, c\right )} - 20 \, a b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 2 \,{\left (5 \, a^{3} e^{\left (6 \, c\right )} + 11 \, a^{2} b e^{\left (6 \, c\right )} - 42 \, a b^{2} e^{\left (6 \, c\right )} - 60 \, b^{3} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (5 \, a^{3} e^{\left (4 \, c\right )} + 11 \, a^{2} b e^{\left (4 \, c\right )} - 42 \, a b^{2} e^{\left (4 \, c\right )} - 60 \, b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} +{\left (11 \, a^{3} e^{\left (2 \, c\right )} - 9 \, a^{2} b e^{\left (2 \, c\right )} - 20 \, a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{24 \,{\left ({\left (a^{5} d e^{\left (7 \, c\right )} + a^{4} b d e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + 2 \,{\left (a^{5} d e^{\left (5 \, c\right )} + 3 \, a^{4} b d e^{\left (5 \, c\right )} + 2 \, a^{3} b^{2} d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (a^{5} d e^{\left (3 \, c\right )} + a^{4} b d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )}\right )}} + \frac{1}{8} \, \int \frac{8 \,{\left ({\left (6 \, a b^{2} e^{\left (3 \, c\right )} + 5 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (6 \, a b^{2} e^{c} + 5 \, b^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{5} + a^{4} b +{\left (a^{5} e^{\left (4 \, c\right )} + a^{4} b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{5} e^{\left (2 \, c\right )} + 3 \, a^{4} b e^{\left (2 \, c\right )} + 2 \, a^{3} b^{2} 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)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

-1/24*(a^3 + a^2*b - (a^3*e^(10*c) + a^2*b*e^(10*c))*e^(10*d*x) - (11*a^3*e^(8*c) - 9*a^2*b*e^(8*c) - 20*a*b^2
*e^(8*c))*e^(8*d*x) - 2*(5*a^3*e^(6*c) + 11*a^2*b*e^(6*c) - 42*a*b^2*e^(6*c) - 60*b^3*e^(6*c))*e^(6*d*x) + 2*(
5*a^3*e^(4*c) + 11*a^2*b*e^(4*c) - 42*a*b^2*e^(4*c) - 60*b^3*e^(4*c))*e^(4*d*x) + (11*a^3*e^(2*c) - 9*a^2*b*e^
(2*c) - 20*a*b^2*e^(2*c))*e^(2*d*x))/((a^5*d*e^(7*c) + a^4*b*d*e^(7*c))*e^(7*d*x) + 2*(a^5*d*e^(5*c) + 3*a^4*b
*d*e^(5*c) + 2*a^3*b^2*d*e^(5*c))*e^(5*d*x) + (a^5*d*e^(3*c) + a^4*b*d*e^(3*c))*e^(3*d*x)) + 1/8*integrate(8*(
(6*a*b^2*e^(3*c) + 5*b^3*e^(3*c))*e^(3*d*x) + (6*a*b^2*e^c + 5*b^3*e^c)*e^(d*x))/(a^5 + a^4*b + (a^5*e^(4*c) +
a^4*b*e^(4*c))*e^(4*d*x) + 2*(a^5*e^(2*c) + 3*a^4*b*e^(2*c) + 2*a^3*b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.90541, size = 13824, normalized size = 110.59 \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)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/24*((a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^10 + 10*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)*sinh(d*x + c)^9
+ (a^5 + 2*a^4*b + a^3*b^2)*sinh(d*x + c)^10 + (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^8 +
(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3 + 45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 +
8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^3 + (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c))
*sinh(d*x + c)^7 + 2*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^6 + 2*(5*a^5 + 16*
a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4 + 105*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^4 + 14*(11*a^5 + 2*a
^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x +
c)^5 + 14*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^3 + 3*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 10
2*a^2*b^3 - 60*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - a^5 - 2*a^4*b - a^3*b^2 - 2*(5*a^5 + 16*a^4*b - 31*a^3*
b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^4 + 2*(105*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^6 - 5*a^5 - 16*
a^4*b + 31*a^3*b^2 + 102*a^2*b^3 + 60*a*b^4 + 35*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^4
+ 15*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(a^5 +
2*a^4*b + a^3*b^2)*cosh(d*x + c)^7 + 7*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^5 + 5*(5*a^5
+ 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^3 - (5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*
b^3 - 60*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^2
+ (45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^8 + 28*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c
)^6 - 11*a^5 - 2*a^4*b + 29*a^3*b^2 + 20*a^2*b^3 + 30*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)
*cosh(d*x + c)^4 - 12*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 - 6*((6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^7 + 7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (6*a^2*
b^2 + 5*a*b^3)*sinh(d*x + c)^7 + 2*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^5 + (12*a^2*b^2 + 34*a*b^3 +
20*b^4 + 21*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^
3 + 2*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + (6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 +
(35*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^4 + 6*a^2*b^2 + 5*a*b^3 + 20*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x
+ c)^2)*sinh(d*x + c)^3 + (21*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^5 + 20*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh
(d*x + c)^3 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^
6 + 10*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^4 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
))*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a +
2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*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*c
osh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c
)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d
*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 2*(5*(a^5 + 2*a^4*b + a^3*b^
2)*cosh(d*x + c)^9 + 4*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^7 + 6*(5*a^5 + 16*a^4*b - 31
*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^5 - 4*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4
)*cosh(d*x + c)^3 - (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^6*b
+ a^5*b^2)*d*cosh(d*x + c)^7 + 7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^6 + (a^7 + 2*a^6*b +
a^5*b^2)*d*sinh(d*x + c)^7 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^5 + (21*(a^7 + 2*a^6*b
+ a^5*b^2)*d*cosh(d*x + c)^2 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^5 + (a^7 + 2*a^6*b
+ a^5*b^2)*d*cosh(d*x + c)^3 + 5*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2
+ 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (35*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 + 20*(a^7 + 4
*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b + a^5*b^2)*d)*sinh(d*x + c)^3 + (21*(a^7 +
2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^5 + 20*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^3 + 3*(a^7 +
2*a^6*b + a^5*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 + 10*(a^
7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2)*sinh(d
*x + c)), 1/24*((a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^10 + 10*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)*sinh(d
*x + c)^9 + (a^5 + 2*a^4*b + a^3*b^2)*sinh(d*x + c)^10 + (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x
+ c)^8 + (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3 + 45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x
+ c)^8 + 8*(15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^3 + (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(
d*x + c))*sinh(d*x + c)^7 + 2*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^6 + 2*(5*
a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4 + 105*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^4 + 14*(11*
a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^5 + 2*a^4*b + a^3*b^2)*co
sh(d*x + c)^5 + 14*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^3 + 3*(5*a^5 + 16*a^4*b - 31*a^3
*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - a^5 - 2*a^4*b - a^3*b^2 - 2*(5*a^5 + 16*a^4*b
- 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^4 + 2*(105*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^6 - 5*
a^5 - 16*a^4*b + 31*a^3*b^2 + 102*a^2*b^3 + 60*a*b^4 + 35*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*
x + c)^4 + 15*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(1
5*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^7 + 7*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^5 +
5*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^3 - (5*a^5 + 16*a^4*b - 31*a^3*b^2 -
102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*
x + c)^2 + (45*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^8 + 28*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cos
h(d*x + c)^6 - 11*a^5 - 2*a^4*b + 29*a^3*b^2 + 20*a^2*b^3 + 30*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 -
60*a*b^4)*cosh(d*x + c)^4 - 12*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 + 12*((6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^7 + 7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6
+ (6*a^2*b^2 + 5*a*b^3)*sinh(d*x + c)^7 + 2*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^5 + (12*a^2*b^2 + 3
4*a*b^3 + 20*b^4 + 21*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(6*a^2*b^2 + 5*a*b^3)*cosh
(d*x + c)^3 + 2*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + (6*a^2*b^2 + 5*a*b^3)*cosh(d*
x + c)^3 + (35*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^4 + 6*a^2*b^2 + 5*a*b^3 + 20*(6*a^2*b^2 + 17*a*b^3 + 10*b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (21*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^5 + 20*(6*a^2*b^2 + 17*a*b^3 + 10
*b^4)*cosh(d*x + c)^3 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(6*a^2*b^2 + 5*a*b^3)*cosh
(d*x + c)^6 + 10*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^4 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*si
nh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x +
c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + 12*((6*
a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^7 + 7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (6*a^2*b^2 + 5*a*
b^3)*sinh(d*x + c)^7 + 2*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^5 + (12*a^2*b^2 + 34*a*b^3 + 20*b^4 + 2
1*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 + 2*(6*a
^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + (6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 + (35*(6*a^
2*b^2 + 5*a*b^3)*cosh(d*x + c)^4 + 6*a^2*b^2 + 5*a*b^3 + 20*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^2)*s
inh(d*x + c)^3 + (21*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^5 + 20*(6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^
3 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^6 + 10*(6*
a^2*b^2 + 17*a*b^3 + 10*b^4)*cosh(d*x + c)^4 + 3*(6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(a^
2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b)) + 2*(5*(a^5 + 2*a^4*b + a^3*b^2)*
cosh(d*x + c)^9 + 4*(11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c)^7 + 6*(5*a^5 + 16*a^4*b - 31*a^
3*b^2 - 102*a^2*b^3 - 60*a*b^4)*cosh(d*x + c)^5 - 4*(5*a^5 + 16*a^4*b - 31*a^3*b^2 - 102*a^2*b^3 - 60*a*b^4)*c
osh(d*x + c)^3 - (11*a^5 + 2*a^4*b - 29*a^3*b^2 - 20*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^6*b +
a^5*b^2)*d*cosh(d*x + c)^7 + 7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^6 + (a^7 + 2*a^6*b + a^
5*b^2)*d*sinh(d*x + c)^7 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^5 + (21*(a^7 + 2*a^6*b +
a^5*b^2)*d*cosh(d*x + c)^2 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^5 + (a^7 + 2*a^6*b + a
^5*b^2)*d*cosh(d*x + c)^3 + 5*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 + 2*(a^7 + 4*a^6*b + 5*a^5*b^2 +
2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (35*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 + 20*(a^7 + 4*a^
6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b + a^5*b^2)*d)*sinh(d*x + c)^3 + (21*(a^7 + 2*a
^6*b + a^5*b^2)*d*cosh(d*x + c)^5 + 20*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^3 + 3*(a^7 + 2*
a^6*b + a^5*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 + 10*(a^7 +
4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2)*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)**3/(a+b*sech(d*x+c)**2)**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)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError