3.256 \(\int \frac{A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx\)
Optimal. Leaf size=250 \[ -\frac{\left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \tanh ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{7/2}}-\frac{\left (11 a^2 A c-2 a^3 C+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^3 (a+c \sinh (d+e x))}-\frac{\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac{B}{3 c e (a+c \sinh (d+e x))^3} \]
[Out]
-(((2*a^3*A - 3*a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTanh[(c - a*Tanh[(d + e*x)/2])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^
(7/2)*e)) - B/(3*c*e*(a + c*Sinh[d + e*x])^3) - ((A*c - a*C)*Cosh[d + e*x])/(3*(a^2 + c^2)*e*(a + c*Sinh[d + e
*x])^3) - ((5*a*A*c - 2*a^2*C + 3*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^2*e*(a + c*Sinh[d + e*x])^2) - ((11*a^2
*A*c - 4*A*c^3 - 2*a^3*C + 13*a*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^3*e*(a + c*Sinh[d + e*x]))
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Rubi [A] time = 0.439518, antiderivative size = 250, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used
= {4376, 2754, 12, 2660, 618, 204, 2668, 32} \[ -\frac{\left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \tanh ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{7/2}}-\frac{\left (11 a^2 A c-2 a^3 C+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^3 (a+c \sinh (d+e x))}-\frac{\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac{B}{3 c e (a+c \sinh (d+e x))^3} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^4,x]
[Out]
-(((2*a^3*A - 3*a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTanh[(c - a*Tanh[(d + e*x)/2])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^
(7/2)*e)) - B/(3*c*e*(a + c*Sinh[d + e*x])^3) - ((A*c - a*C)*Cosh[d + e*x])/(3*(a^2 + c^2)*e*(a + c*Sinh[d + e
*x])^3) - ((5*a*A*c - 2*a^2*C + 3*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^2*e*(a + c*Sinh[d + e*x])^2) - ((11*a^2
*A*c - 4*A*c^3 - 2*a^3*C + 13*a*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^3*e*(a + c*Sinh[d + e*x]))
Rule 4376
Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] :> With[{e = FreeFactors[Sin[c*(a +
b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[
Sin[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Cos] || EqQ[F, cos])
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 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 2668
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rubi steps
\begin{align*} \int \frac{A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx &=B \int \frac{\cosh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx+\int \frac{A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx\\ &=-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\int \frac{-3 (a A+c C)+2 (A c-a C) \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx}{3 \left (a^2+c^2\right )}+\frac{B \operatorname{Subst}\left (\int \frac{1}{(a+x)^4} \, dx,x,c \sinh (d+e x)\right )}{c e}\\ &=-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}+\frac{\int \frac{2 \left (3 a^2 A-2 A c^2+5 a c C\right )-\left (5 a A c-2 a^2 C+3 c^2 C\right ) \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx}{6 \left (a^2+c^2\right )^2}\\ &=-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac{\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))}-\frac{\int -\frac{3 \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )}{a+c \sinh (d+e x)} \, dx}{6 \left (a^2+c^2\right )^3}\\ &=-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac{\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))}+\frac{\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \int \frac{1}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^3}\\ &=-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac{\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))}-\frac{\left (i \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-2 i c x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^3 e}\\ &=-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac{\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))}+\frac{\left (2 i \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2+c^2\right )-x^2} \, dx,x,-2 i c+2 a \tan \left (\frac{1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^3 e}\\ &=-\frac{\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \tanh ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2+c^2}}\right )}{\left (a^2+c^2\right )^{7/2} e}-\frac{B}{3 c e (a+c \sinh (d+e x))^3}-\frac{(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac{\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac{\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))}\\ \end{align*}
Mathematica [A] time = 1.53869, size = 235, normalized size = 0.94 \[ \frac{-\frac{2 \left (a^2+c^2\right )^2 \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^3}+\frac{6 \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \tan ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{-a^2-c^2}}\right )}{\sqrt{-a^2-c^2}}+\frac{\left (a^2+c^2\right ) \left (2 a^2 C-5 a A c-3 c^2 C\right ) \cosh (d+e x)}{(a+c \sinh (d+e x))^2}+\frac{\left (-11 a^2 A c+2 a^3 C-13 a c^2 C+4 A c^3\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{6 e \left (a^2+c^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^4,x]
[Out]
((6*(2*a^3*A - 3*a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTan[(c - a*Tanh[(d + e*x)/2])/Sqrt[-a^2 - c^2]])/Sqrt[-a^2 -
c^2] - (2*(a^2 + c^2)^2*(B*(a^2 + c^2) + c*(A*c - a*C)*Cosh[d + e*x]))/(c*(a + c*Sinh[d + e*x])^3) + ((a^2 + c
^2)*(-5*a*A*c + 2*a^2*C - 3*c^2*C)*Cosh[d + e*x])/(a + c*Sinh[d + e*x])^2 + ((-11*a^2*A*c + 4*A*c^3 + 2*a^3*C
- 13*a*c^2*C)*Cosh[d + e*x])/(a + c*Sinh[d + e*x]))/(6*(a^2 + c^2)^3*e)
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Maple [B] time = 0.134, size = 844, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^4,x)
[Out]
1/e*(-2*(-1/2*(9*A*a^4*c^2+6*A*a^2*c^4+2*A*c^6-2*B*a^6-6*B*a^4*c^2-6*B*a^2*c^4-2*B*c^6-4*C*a^5*c+C*a^3*c^3)/a/
(a^6+3*a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)^5-1/2*(6*A*a^6*c-27*A*a^4*c^3-12*A*a^2*c^5-4*A*c^7+4*B*a^6*c
+12*B*a^4*c^3+12*B*a^2*c^5+4*B*c^7-2*C*a^7+14*C*a^5*c^2-11*C*a^3*c^4-2*C*a*c^6)/(a^6+3*a^4*c^2+3*a^2*c^4+c^6)/
a^2*tanh(1/2*e*x+1/2*d)^4+1/3/a^3*(54*A*a^6*c^2-21*A*a^4*c^4-4*A*a^2*c^6-4*A*c^8-6*B*a^8-14*B*a^6*c^2-6*B*a^4*
c^4+6*B*a^2*c^6+4*B*c^8-18*C*a^7*c+42*C*a^5*c^3-17*C*a^3*c^5-2*C*a*c^7)/(a^6+3*a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2
*e*x+1/2*d)^3+1/a^2*(6*A*a^6*c-20*A*a^4*c^3-3*A*a^2*c^5-2*A*c^7+2*B*a^6*c+6*B*a^4*c^3+6*B*a^2*c^5+2*B*c^7-2*C*
a^7+10*C*a^5*c^2-14*C*a^3*c^4-C*a*c^6)/(a^6+3*a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)^2-1/2/a*(27*A*a^4*c^2
+4*A*a^2*c^4+2*A*c^6-2*B*a^6-6*B*a^4*c^2-6*B*a^2*c^4-2*B*c^6-8*C*a^5*c+19*C*a^3*c^3+2*C*a*c^5)/(a^6+3*a^4*c^2+
3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)-1/6*(18*A*a^4*c+5*A*a^2*c^3+2*A*c^5-6*C*a^5+10*C*a^3*c^2+C*a*c^4)/(a^6+3*a^
4*c^2+3*a^2*c^4+c^6))/(a*tanh(1/2*e*x+1/2*d)^2-2*c*tanh(1/2*e*x+1/2*d)-a)^3+(2*A*a^3-3*A*a*c^2+4*C*a^2*c-C*c^3
)/(a^6+3*a^4*c^2+3*a^2*c^4+c^6)/(a^2+c^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*e*x+1/2*d)-2*c)/(a^2+c^2)^(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((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.64937, size = 9709, normalized size = 38.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^4,x, algorithm="fricas")
[Out]
1/6*(4*C*a^5*c^3 - 22*A*a^4*c^4 - 22*C*a^3*c^5 - 14*A*a^2*c^6 - 26*C*a*c^7 + 8*A*c^8 + 6*(2*A*a^5*c^3 + 4*C*a^
4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d)^5 + 6*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^
5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*sinh(e*x + d)^5 + 30*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5
- 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d)^4 + 30*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*
c^6 - C*a*c^7 + (2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d))*sinh(
e*x + d)^4 - 4*(4*(B + C)*a^8 - 22*A*a^7*c + 4*(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*
A*a^3*c^5 + (16*B + 39*C)*a^2*c^6 - 12*A*a*c^7 + 4*B*c^8)*cosh(e*x + d)^3 - 4*(4*(B + C)*a^8 - 22*A*a^7*c + 4*
(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*A*a^3*c^5 + (16*B + 39*C)*a^2*c^6 - 12*A*a*c^7
+ 4*B*c^8 - 15*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d)^2 - 30*
(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d))*sinh(e*x + d)^3 +
12*(4*C*a^7*c - 17*A*a^6*c^2 - 13*C*a^5*c^3 - 11*A*a^4*c^4 - 13*C*a^3*c^5 + 4*A*a^2*c^6 + 4*C*a*c^7 - 2*A*c^8
)*cosh(e*x + d)^2 + 12*(4*C*a^7*c - 17*A*a^6*c^2 - 13*C*a^5*c^3 - 11*A*a^4*c^4 - 13*C*a^3*c^5 + 4*A*a^2*c^6 +
4*C*a*c^7 - 2*A*c^8 + 5*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d
)^3 + 15*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d)^2 - (4*(B
+ C)*a^8 - 22*A*a^7*c + 4*(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*A*a^3*c^5 + (16*B +
39*C)*a^2*c^6 - 12*A*a*c^7 + 4*B*c^8)*cosh(e*x + d))*sinh(e*x + d)^2 + 3*(2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^
6 - C*c^7 - (2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x + d)^6 - (2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A
*a*c^6 - C*c^7)*sinh(e*x + d)^6 - 6*(2*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6)*cosh(e*x + d)^5 - 6*(2
*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6 + (2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x +
d))*sinh(e*x + d)^5 - 3*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7)*cosh(e*x
+ d)^4 - 3*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7 + 5*(2*A*a^3*c^4 + 4*
C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x + d)^2 + 10*(2*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6)*cosh(e
*x + d))*sinh(e*x + d)^4 - 4*(4*A*a^6*c + 8*C*a^5*c^2 - 12*A*a^4*c^3 - 14*C*a^3*c^4 + 9*A*a^2*c^5 + 3*C*a*c^6)
*cosh(e*x + d)^3 - 4*(4*A*a^6*c + 8*C*a^5*c^2 - 12*A*a^4*c^3 - 14*C*a^3*c^4 + 9*A*a^2*c^5 + 3*C*a*c^6 + 5*(2*A
*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x + d)^3 + 15*(2*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*
a*c^6)*cosh(e*x + d)^2 + 3*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7)*cosh(
e*x + d))*sinh(e*x + d)^3 + 3*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7)*co
sh(e*x + d)^2 + 3*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7 - 5*(2*A*a^3*c^
4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x + d)^4 - 20*(2*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6)*
cosh(e*x + d)^3 - 6*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c^7)*cosh(e*x + d
)^2 - 4*(4*A*a^6*c + 8*C*a^5*c^2 - 12*A*a^4*c^3 - 14*C*a^3*c^4 + 9*A*a^2*c^5 + 3*C*a*c^6)*cosh(e*x + d))*sinh(
e*x + d)^2 - 6*(2*A*a^4*c^3 + 4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6)*cosh(e*x + d) - 6*(2*A*a^4*c^3 + 4*C*a^3*c^
4 - 3*A*a^2*c^5 - C*a*c^6 + (2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*cosh(e*x + d)^5 + 5*(2*A*a^4*c^3 +
4*C*a^3*c^4 - 3*A*a^2*c^5 - C*a*c^6)*cosh(e*x + d)^4 + 2*(8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2
*c^5 + 3*A*a*c^6 + C*c^7)*cosh(e*x + d)^3 + 2*(4*A*a^6*c + 8*C*a^5*c^2 - 12*A*a^4*c^3 - 14*C*a^3*c^4 + 9*A*a^2
*c^5 + 3*C*a*c^6)*cosh(e*x + d)^2 - (8*A*a^5*c^2 + 16*C*a^4*c^3 - 14*A*a^3*c^4 - 8*C*a^2*c^5 + 3*A*a*c^6 + C*c
^7)*cosh(e*x + d))*sinh(e*x + d))*sqrt(a^2 + c^2)*log((c^2*cosh(e*x + d)^2 + c^2*sinh(e*x + d)^2 + 2*a*c*cosh(
e*x + d) + 2*a^2 + c^2 + 2*(c^2*cosh(e*x + d) + a*c)*sinh(e*x + d) + 2*sqrt(a^2 + c^2)*(c*cosh(e*x + d) + c*si
nh(e*x + d) + a))/(c*cosh(e*x + d)^2 + c*sinh(e*x + d)^2 + 2*a*cosh(e*x + d) + 2*(c*cosh(e*x + d) + a)*sinh(e*
x + d) - c)) - 6*(4*C*a^6*c^2 - 20*A*a^5*c^3 - 18*C*a^4*c^4 - 15*A*a^3*c^5 - 23*C*a^2*c^6 + 5*A*a*c^7 - C*c^8)
*cosh(e*x + d) - 6*(4*C*a^6*c^2 - 20*A*a^5*c^3 - 18*C*a^4*c^4 - 15*A*a^3*c^5 - 23*C*a^2*c^6 + 5*A*a*c^7 - C*c^
8 - 5*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d)^4 - 20*(2*A*a^6*
c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d)^3 + 2*(4*(B + C)*a^8 - 22*A
*a^7*c + 4*(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*A*a^3*c^5 + (16*B + 39*C)*a^2*c^6 -
12*A*a*c^7 + 4*B*c^8)*cosh(e*x + d)^2 - 4*(4*C*a^7*c - 17*A*a^6*c^2 - 13*C*a^5*c^3 - 11*A*a^4*c^4 - 13*C*a^3*c
^5 + 4*A*a^2*c^6 + 4*C*a*c^7 - 2*A*c^8)*cosh(e*x + d))*sinh(e*x + d))/((a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^
2*c^10 + c^12)*e*cosh(e*x + d)^6 + (a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10 + c^12)*e*sinh(e*x + d)^6 + 6
*(a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d)^5 + 3*(4*a^10*c^2 + 15*a^8*c^4 + 20*a^
6*c^6 + 10*a^4*c^8 - c^12)*e*cosh(e*x + d)^4 + 6*((a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10 + c^12)*e*cosh
(e*x + d) + (a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e)*sinh(e*x + d)^5 + 4*(2*a^11*c + 5*a^9*c^
3 - 10*a^5*c^7 - 10*a^3*c^9 - 3*a*c^11)*e*cosh(e*x + d)^3 + 3*(5*(a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10
+ c^12)*e*cosh(e*x + d)^2 + 10*(a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d) + (4*a^
10*c^2 + 15*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8 - c^12)*e)*sinh(e*x + d)^4 - 3*(4*a^10*c^2 + 15*a^8*c^4 + 20*a^6
*c^6 + 10*a^4*c^8 - c^12)*e*cosh(e*x + d)^2 + 4*(5*(a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10 + c^12)*e*cos
h(e*x + d)^3 + 15*(a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d)^2 + 3*(4*a^10*c^2 + 1
5*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8 - c^12)*e*cosh(e*x + d) + (2*a^11*c + 5*a^9*c^3 - 10*a^5*c^7 - 10*a^3*c^9
- 3*a*c^11)*e)*sinh(e*x + d)^3 + 6*(a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d) + 3*
(5*(a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10 + c^12)*e*cosh(e*x + d)^4 + 20*(a^9*c^3 + 4*a^7*c^5 + 6*a^5*c
^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d)^3 + 6*(4*a^10*c^2 + 15*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8 - c^12)*e*co
sh(e*x + d)^2 + 4*(2*a^11*c + 5*a^9*c^3 - 10*a^5*c^7 - 10*a^3*c^9 - 3*a*c^11)*e*cosh(e*x + d) - (4*a^10*c^2 +
15*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8 - c^12)*e)*sinh(e*x + d)^2 - (a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^1
0 + c^12)*e + 6*((a^8*c^4 + 4*a^6*c^6 + 6*a^4*c^8 + 4*a^2*c^10 + c^12)*e*cosh(e*x + d)^5 + 5*(a^9*c^3 + 4*a^7*
c^5 + 6*a^5*c^7 + 4*a^3*c^9 + a*c^11)*e*cosh(e*x + d)^4 + 2*(4*a^10*c^2 + 15*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8
- c^12)*e*cosh(e*x + d)^3 + 2*(2*a^11*c + 5*a^9*c^3 - 10*a^5*c^7 - 10*a^3*c^9 - 3*a*c^11)*e*cosh(e*x + d)^2 -
(4*a^10*c^2 + 15*a^8*c^4 + 20*a^6*c^6 + 10*a^4*c^8 - c^12)*e*cosh(e*x + d) + (a^9*c^3 + 4*a^7*c^5 + 6*a^5*c^7
+ 4*a^3*c^9 + a*c^11)*e)*sinh(e*x + d))
________________________________________________________________________________________
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((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))**4,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.20456, size = 988, normalized size = 3.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^4,x, algorithm="giac")
[Out]
1/2*(2*A*a^3 + 4*C*a^2*c - 3*A*a*c^2 - C*c^3)*log(abs(2*c*e^(x*e + d) + 2*a - 2*sqrt(a^2 + c^2))/abs(2*c*e^(x*
e + d) + 2*a + 2*sqrt(a^2 + c^2)))/((a^6*e + 3*a^4*c^2*e + 3*a^2*c^4*e + c^6*e)*sqrt(a^2 + c^2)) + 1/3*(6*A*a^
3*c^3*e^(5*x*e + 5*d) + 12*C*a^2*c^4*e^(5*x*e + 5*d) - 9*A*a*c^5*e^(5*x*e + 5*d) - 3*C*c^6*e^(5*x*e + 5*d) + 3
0*A*a^4*c^2*e^(4*x*e + 4*d) + 60*C*a^3*c^3*e^(4*x*e + 4*d) - 45*A*a^2*c^4*e^(4*x*e + 4*d) - 15*C*a*c^5*e^(4*x*
e + 4*d) - 8*B*a^6*e^(3*x*e + 3*d) - 8*C*a^6*e^(3*x*e + 3*d) + 44*A*a^5*c*e^(3*x*e + 3*d) - 24*B*a^4*c^2*e^(3*
x*e + 3*d) + 64*C*a^4*c^2*e^(3*x*e + 3*d) - 82*A*a^3*c^3*e^(3*x*e + 3*d) - 24*B*a^2*c^4*e^(3*x*e + 3*d) - 78*C
*a^2*c^4*e^(3*x*e + 3*d) + 24*A*a*c^5*e^(3*x*e + 3*d) - 8*B*c^6*e^(3*x*e + 3*d) + 24*C*a^5*c*e^(2*x*e + 2*d) -
102*A*a^4*c^2*e^(2*x*e + 2*d) - 102*C*a^3*c^3*e^(2*x*e + 2*d) + 36*A*a^2*c^4*e^(2*x*e + 2*d) + 24*C*a*c^5*e^(
2*x*e + 2*d) - 12*A*c^6*e^(2*x*e + 2*d) - 12*C*a^4*c^2*e^(x*e + d) + 60*A*a^3*c^3*e^(x*e + d) + 66*C*a^2*c^4*e
^(x*e + d) - 15*A*a*c^5*e^(x*e + d) + 3*C*c^6*e^(x*e + d) + 2*C*a^3*c^3 - 11*A*a^2*c^4 - 13*C*a*c^5 + 4*A*c^6)
/((a^6*c*e + 3*a^4*c^3*e + 3*a^2*c^5*e + c^7*e)*(c*e^(2*x*e + 2*d) + 2*a*e^(x*e + d) - c)^3)