3.70 \(\int \frac{1}{(a+b \cosh (c+d x))^4} \, dx\)
Optimal. Leaf size=184 \[ \frac{a \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cosh (c+d x))}-\frac{5 a b \sinh (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cosh (c+d x))^2}-\frac{b \sinh (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cosh (c+d x))^3} \]
[Out]
(a*(2*a^2 + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*
Sinh[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cosh[c + d*x])^3) - (5*a*b*Sinh[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos
h[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sinh[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cosh[c + d*x]))
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Rubi [A] time = 0.253111, antiderivative size = 184, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used =
{2664, 2754, 12, 2659, 205} \[ \frac{a \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cosh (c+d x))}-\frac{5 a b \sinh (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cosh (c+d x))^2}-\frac{b \sinh (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cosh (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Cosh[c + d*x])^(-4),x]
[Out]
(a*(2*a^2 + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*
Sinh[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cosh[c + d*x])^3) - (5*a*b*Sinh[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos
h[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sinh[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cosh[c + d*x]))
Rule 2664
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]
Rule 2754
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 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{1}{(a+b \cosh (c+d x))^4} \, dx &=-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{\int \frac{-3 a+2 b \cosh (c+d x)}{(a+b \cosh (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}+\frac{\int \frac{2 \left (3 a^2+2 b^2\right )-5 a b \cosh (c+d x)}{(a+b \cosh (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}-\frac{\int -\frac{3 a \left (2 a^2+3 b^2\right )}{a+b \cosh (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}+\frac{\left (a \left (2 a^2+3 b^2\right )\right ) \int \frac{1}{a+b \cosh (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}-\frac{\left (i a \left (2 a^2+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=\frac{a \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac{b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac{5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac{b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.998127, size = 160, normalized size = 0.87 \[ \frac{\frac{6 a \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}-\frac{b \sinh (c+d x) \left (6 a b \left (9 a^2+b^2\right ) \cosh (c+d x)+\left (11 a^2 b^2+4 b^4\right ) \cosh (2 (c+d x))+a^2 b^2+36 a^4+8 b^4\right )}{2 (a-b)^3 (a+b)^3 (a+b \cosh (c+d x))^3}}{6 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Cosh[c + d*x])^(-4),x]
[Out]
((6*a*(2*a^2 + 3*b^2)*ArcTan[((a - b)*Tanh[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - (b*(36*a^4 +
a^2*b^2 + 8*b^4 + 6*a*b*(9*a^2 + b^2)*Cosh[c + d*x] + (11*a^2*b^2 + 4*b^4)*Cosh[2*(c + d*x)])*Sinh[c + d*x])/(
2*(a - b)^3*(a + b)^3*(a + b*Cosh[c + d*x])^3))/(6*d)
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Maple [A] time = 0.024, size = 284, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{1}{ \left ( a \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}- \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b-a-b \right ) ^{3}} \left ( -1/2\,{\frac{ \left ( 6\,{a}^{2}+3\,ab+2\,{b}^{2} \right ) b \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{ \left ( a-b \right ) \left ({a}^{3}+3\,{a}^{2}b+3\,a{b}^{2}+{b}^{3} \right ) }}+2/3\,{\frac{ \left ( 9\,{a}^{2}+{b}^{2} \right ) b \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{ \left ({a}^{2}+2\,ab+{b}^{2} \right ) \left ({a}^{2}-2\,ab+{b}^{2} \right ) }}-1/2\,{\frac{ \left ( 6\,{a}^{2}-3\,ab+2\,{b}^{2} \right ) b\tanh \left ( 1/2\,dx+c/2 \right ) }{ \left ( a+b \right ) \left ({a}^{3}-3\,{a}^{2}b+3\,a{b}^{2}-{b}^{3} \right ) }} \right ) }+{\frac{a \left ( 2\,{a}^{2}+3\,{b}^{2} \right ) }{{a}^{6}-3\,{a}^{4}{b}^{2}+3\,{a}^{2}{b}^{4}-{b}^{6}}{\it Artanh} \left ({(a-b)\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*cosh(d*x+c))^4,x)
[Out]
1/d*(-2*(-1/2*(6*a^2+3*a*b+2*b^2)*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5+2/3*(9*a^2+b^2)*b/(a
^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/2*(6*a^2-3*a*b+2*b^2)*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*
tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^2-tanh(1/2*d*x+1/2*c)^2*b-a-b)^3+a*(2*a^2+3*b^2)/(a^6-3*a^4*b^2+3*
a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))
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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(1/(a+b*cosh(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.95478, size = 12951, normalized size = 70.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cosh(d*x+c))^4,x, algorithm="fricas")
[Out]
[1/6*(22*a^4*b^3 - 14*a^2*b^5 - 8*b^7 + 6*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^5 + 6*(2*a^5*b^2 + a^3
*b^4 - 3*a*b^6)*sinh(d*x + c)^5 + 30*(2*a^6*b + a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^4 + 30*(2*a^6*b + a^4*b^3 -
3*a^2*b^5 + (2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(22*a^7 + 19*a^5*b^2 - 29*a^3*
b^4 - 12*a*b^6)*cosh(d*x + c)^3 + 4*(22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6 + 15*(2*a^5*b^2 + a^3*b^4 - 3
*a*b^6)*cosh(d*x + c)^2 + 30*(2*a^6*b + a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 12*(17*a^6*b - 1
1*a^4*b^3 - 4*a^2*b^5 - 2*b^7)*cosh(d*x + c)^2 + 12*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7 + 5*(2*a^5*b^2
+ a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^3 + 15*(2*a^6*b + a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^2 + (22*a^7 + 19*a^5*b
^2 - 29*a^3*b^4 - 12*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*((2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^6 + (2*a^3
*b^3 + 3*a*b^5)*sinh(d*x + c)^6 + 2*a^3*b^3 + 3*a*b^5 + 6*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^5 + 6*(2*a^4*b
^2 + 3*a^2*b^4 + (2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cos
h(d*x + c)^4 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 5*(2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^2 + 10*(2*a^4*b^2 + 3
*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c)^3 + 4*(4*a^6 + 12*
a^4*b^2 + 9*a^2*b^4 + 5*(2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^3 + 15*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^2 + 3
*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x
+ c)^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 5*(2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^4 + 20*(2*a^4*b^2 + 3*a^2*
b^4)*cosh(d*x + c)^3 + 6*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^2 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4)
*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c) + 6*(2*a^4*b^2 + 3*a^2*b^4 + (2*a^3*
b^3 + 3*a*b^5)*cosh(d*x + c)^5 + 5*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^4 + 2*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5
)*cosh(d*x + c)^3 + 2*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c)^2 + (8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh
(d*x + c))*sinh(d*x + c))*sqrt(a^2 - b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c)
+ 2*a^2 - b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 - b^2)*(b*cosh(d*x + c) + b*sinh(d*x +
c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) +
b)) + 30*(4*a^5*b^2 - 3*a^3*b^4 - a*b^6)*cosh(d*x + c) + 6*(20*a^5*b^2 - 15*a^3*b^4 - 5*a*b^6 + 5*(2*a^5*b^2
+ a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^4 + 20*(2*a^6*b + a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^3 + 2*(22*a^7 + 19*a^5
*b^2 - 29*a^3*b^4 - 12*a*b^6)*cosh(d*x + c)^2 + 4*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7)*cosh(d*x + c))*s
inh(d*x + c))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^6 + (a^8*b^3 - 4*a^6*b^5 +
6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*sinh(d*x + c)^6 + 6*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*
cosh(d*x + c)^5 + 3*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^4 + 6*((a^8*b^3 -
4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c) + (a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*
b^10)*d)*sinh(d*x + c)^5 + 4*(2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*d*cosh(d*x + c)^3 + 3*(
5*(a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^2 + 10*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6
- 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c) + (4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d)*sinh(d*x
+ c)^4 + 3*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^2 + 4*(5*(a^8*b^3 - 4*a^6*
b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^3 + 15*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^
10)*d*cosh(d*x + c)^2 + 3*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c) + (2*a^11 -
5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*d)*sinh(d*x + c)^3 + 6*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a
^3*b^8 + a*b^10)*d*cosh(d*x + c) + 3*(5*(a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^4
+ 20*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^3 + 6*(4*a^10*b - 15*a^8*b^3 + 20
*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^2 + 4*(2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*
d*cosh(d*x + c) + (4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d)*sinh(d*x + c)^2 + (a^8*b^3 - 4*a
^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d + 6*((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x
+ c)^5 + 5*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^4 + 2*(4*a^10*b - 15*a^8*b^3
+ 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^3 + 2*(2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b
^10)*d*cosh(d*x + c)^2 + (4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c) + (a^9*b^2 -
4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d)*sinh(d*x + c)), 1/3*(11*a^4*b^3 - 7*a^2*b^5 - 4*b^7 + 3*(2*a^5
*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^5 + 3*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*sinh(d*x + c)^5 + 15*(2*a^6*b +
a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^4 + 15*(2*a^6*b + a^4*b^3 - 3*a^2*b^5 + (2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cos
h(d*x + c))*sinh(d*x + c)^4 + 2*(22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6)*cosh(d*x + c)^3 + 2*(22*a^7 + 19
*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6 + 15*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^2 + 30*(2*a^6*b + a^4*b^3
- 3*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7)*cosh(d*x + c)^2 +
6*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7 + 5*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^3 + 15*(2*a^6*b
+ a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^2 + (22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6)*cosh(d*x + c))*sinh(d*
x + c)^2 - 3*((2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^6 + (2*a^3*b^3 + 3*a*b^5)*sinh(d*x + c)^6 + 2*a^3*b^3 + 3*a*
b^5 + 6*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^5 + 6*(2*a^4*b^2 + 3*a^2*b^4 + (2*a^3*b^3 + 3*a*b^5)*cosh(d*x +
c))*sinh(d*x + c)^5 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^4 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 +
5*(2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^2 + 10*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(4*a^
6 + 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c)^3 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4 + 5*(2*a^3*b^3 + 3*a*b^5)*cosh
(d*x + c)^3 + 15*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c))*s
inh(d*x + c)^3 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 5*(2
*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^4 + 20*(2*a^4*b^2 + 3*a^2*b^4)*cosh(d*x + c)^3 + 6*(8*a^5*b + 14*a^3*b^3 + 3
*a*b^5)*cosh(d*x + c)^2 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(2*a^4*b^2 + 3
*a^2*b^4)*cosh(d*x + c) + 6*(2*a^4*b^2 + 3*a^2*b^4 + (2*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^5 + 5*(2*a^4*b^2 + 3*
a^2*b^4)*cosh(d*x + c)^4 + 2*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^3 + 2*(4*a^6 + 12*a^4*b^2 + 9*a^2*
b^4)*cosh(d*x + c)^2 + (8*a^5*b + 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + b^2)*arctan(
-sqrt(-a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a)/(a^2 - b^2)) + 15*(4*a^5*b^2 - 3*a^3*b^4 - a*b^6)*co
sh(d*x + c) + 3*(20*a^5*b^2 - 15*a^3*b^4 - 5*a*b^6 + 5*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^4 + 20*(2
*a^6*b + a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^3 + 2*(22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6)*cosh(d*x + c)^
2 + 4*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7)*cosh(d*x + c))*sinh(d*x + c))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*
b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^6 + (a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*sinh(d*x +
c)^6 + 6*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^5 + 3*(4*a^10*b - 15*a^8*b^3 +
20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^4 + 6*((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d
*cosh(d*x + c) + (a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d)*sinh(d*x + c)^5 + 4*(2*a^11 - 5*a^9
*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*d*cosh(d*x + c)^3 + 3*(5*(a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b
^9 + b^11)*d*cosh(d*x + c)^2 + 10*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c) + (4*
a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d)*sinh(d*x + c)^4 + 3*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b
^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c)^2 + 4*(5*(a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d
*x + c)^3 + 15*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^2 + 3*(4*a^10*b - 15*a^8
*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c) + (2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b
^10)*d)*sinh(d*x + c)^3 + 6*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c) + 3*(5*(a^8
*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^4 + 20*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a
^3*b^8 + a*b^10)*d*cosh(d*x + c)^3 + 6*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c
)^2 + 4*(2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*d*cosh(d*x + c) + (4*a^10*b - 15*a^8*b^3 + 2
0*a^6*b^5 - 10*a^4*b^7 + b^11)*d)*sinh(d*x + c)^2 + (a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d + 6
*((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^5 + 5*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6
- 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^4 + 2*(4*a^10*b - 15*a^8*b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*
x + c)^3 + 2*(2*a^11 - 5*a^9*b^2 + 10*a^5*b^6 - 10*a^3*b^8 + 3*a*b^10)*d*cosh(d*x + c)^2 + (4*a^10*b - 15*a^8*
b^3 + 20*a^6*b^5 - 10*a^4*b^7 + b^11)*d*cosh(d*x + c) + (a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)
*d)*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(1/(a+b*cosh(d*x+c))**4,x)
[Out]
Timed out
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Giac [A] time = 1.2372, size = 451, normalized size = 2.45 \begin{align*} \frac{{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \arctan \left (\frac{b e^{\left (d x + c\right )} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{{\left (a^{6} d - 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d - b^{6} d\right )} \sqrt{-a^{2} + b^{2}}} + \frac{6 \, a^{3} b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 9 \, a b^{4} e^{\left (5 \, d x + 5 \, c\right )} + 30 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 45 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 44 \, a^{5} e^{\left (3 \, d x + 3 \, c\right )} + 82 \, a^{3} b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b^{4} e^{\left (3 \, d x + 3 \, c\right )} + 102 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 36 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 60 \, a^{3} b^{2} e^{\left (d x + c\right )} + 15 \, a b^{4} e^{\left (d x + c\right )} + 11 \, a^{2} b^{3} + 4 \, b^{5}}{3 \,{\left (a^{6} d - 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d - b^{6} d\right )}{\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} + b\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*cosh(d*x+c))^4,x, algorithm="giac")
[Out]
(2*a^3 + 3*a*b^2)*arctan((b*e^(d*x + c) + a)/sqrt(-a^2 + b^2))/((a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)*sq
rt(-a^2 + b^2)) + 1/3*(6*a^3*b^2*e^(5*d*x + 5*c) + 9*a*b^4*e^(5*d*x + 5*c) + 30*a^4*b*e^(4*d*x + 4*c) + 45*a^2
*b^3*e^(4*d*x + 4*c) + 44*a^5*e^(3*d*x + 3*c) + 82*a^3*b^2*e^(3*d*x + 3*c) + 24*a*b^4*e^(3*d*x + 3*c) + 102*a^
4*b*e^(2*d*x + 2*c) + 36*a^2*b^3*e^(2*d*x + 2*c) + 12*b^5*e^(2*d*x + 2*c) + 60*a^3*b^2*e^(d*x + c) + 15*a*b^4*
e^(d*x + c) + 11*a^2*b^3 + 4*b^5)/((a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)*(b*e^(2*d*x + 2*c) + 2*a*e^(d*x
+ c) + b)^3)