3.3.56 \(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx\) [256]

Optimal. Leaf size=250 \[ -\frac {\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{7/2} e}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}-\frac {(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac {\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac {\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))} \]

[Out]

-(2*A*a^3-3*A*a*c^2+4*C*a^2*c-C*c^3)*arctanh((c-a*tanh(1/2*e*x+1/2*d))/(a^2+c^2)^(1/2))/(a^2+c^2)^(7/2)/e-1/3*
B/c/e/(a+c*sinh(e*x+d))^3-1/3*(A*c-C*a)*cosh(e*x+d)/(a^2+c^2)/e/(a+c*sinh(e*x+d))^3-1/6*(5*A*a*c-2*C*a^2+3*C*c
^2)*cosh(e*x+d)/(a^2+c^2)^2/e/(a+c*sinh(e*x+d))^2-1/6*(11*A*a^2*c-4*A*c^3-2*C*a^3+13*C*a*c^2)*cosh(e*x+d)/(a^2
+c^2)^3/e/(a+c*sinh(e*x+d))

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Rubi [A]
time = 0.33, antiderivative size = 250, normalized size of antiderivative = 1.00, 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 = {4461, 2833, 12, 2739, 632, 210, 2747, 32} \begin {gather*} -\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 {\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 (-2 a^3 C+11 a^2 A 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 {B}{3 c e (a+c \sinh (d+e x))^3} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 2739

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 2747

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 2833

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 4461

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])

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 \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.13, size = 235, normalized size = 0.94 \begin {gather*} \frac {\frac {6 \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \text {ArcTan}\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 {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 {\left (a^2+c^2\right ) \left (-5 a A c+2 a^2 C-3 c^2 C\right ) \cosh (d+e x)}{(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)}{a+c \sinh (d+e x)}}{6 \left (a^2+c^2\right )^3 e} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(237)=474\).
time = 6.22, size = 844, normalized size = 3.38 Too large to display

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,method=_RETURNVERBOSE)

[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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1290 vs. \(2 (242) = 484\).
time = 0.53, size = 1290, normalized size = 5.16 \begin {gather*} \frac {1}{6} \, {\left (\frac {3 \, {\left (2 \, a^{2} - 3 \, c^{2}\right )} a e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )} \sqrt {a^{2} + c^{2}}} - \frac {2 \, {\left (11 \, a^{2} c^{3} - 4 \, c^{5} + 15 \, {\left (4 \, a^{3} c^{2} - a c^{4}\right )} e^{\left (-x e - d\right )} + 6 \, {\left (17 \, a^{4} c - 6 \, a^{2} c^{3} + 2 \, c^{5}\right )} e^{\left (-2 \, x e - 2 \, d\right )} + 2 \, {\left (22 \, a^{5} - 41 \, a^{3} c^{2} + 12 \, a c^{4}\right )} e^{\left (-3 \, x e - 3 \, d\right )} - 15 \, {\left (2 \, a^{4} c - 3 \, a^{2} c^{3}\right )} e^{\left (-4 \, x e - 4 \, d\right )} + 3 \, {\left (2 \, a^{3} c^{2} - 3 \, a c^{4}\right )} e^{\left (-5 \, x e - 5 \, d\right )}\right )} e^{\left (-1\right )}}{a^{6} c^{3} + 3 \, a^{4} c^{5} + 3 \, a^{2} c^{7} + c^{9} + 6 \, {\left (a^{7} c^{2} + 3 \, a^{5} c^{4} + 3 \, a^{3} c^{6} + a c^{8}\right )} e^{\left (-x e - d\right )} + 3 \, {\left (4 \, a^{8} c + 11 \, a^{6} c^{3} + 9 \, a^{4} c^{5} + a^{2} c^{7} - c^{9}\right )} e^{\left (-2 \, x e - 2 \, d\right )} + 4 \, {\left (2 \, a^{9} + 3 \, a^{7} c^{2} - 3 \, a^{5} c^{4} - 7 \, a^{3} c^{6} - 3 \, a c^{8}\right )} e^{\left (-3 \, x e - 3 \, d\right )} - 3 \, {\left (4 \, a^{8} c + 11 \, a^{6} c^{3} + 9 \, a^{4} c^{5} + a^{2} c^{7} - c^{9}\right )} e^{\left (-4 \, x e - 4 \, d\right )} + 6 \, {\left (a^{7} c^{2} + 3 \, a^{5} c^{4} + 3 \, a^{3} c^{6} + a c^{8}\right )} e^{\left (-5 \, x e - 5 \, d\right )} - {\left (a^{6} c^{3} + 3 \, a^{4} c^{5} + 3 \, a^{2} c^{7} + c^{9}\right )} e^{\left (-6 \, x e - 6 \, d\right )}}\right )} A + \frac {1}{6} \, {\left (\frac {3 \, {\left (4 \, a^{2} c - c^{3}\right )} e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )} \sqrt {a^{2} + c^{2}}} + \frac {2 \, {\left (2 \, a^{3} c^{3} - 13 \, a c^{5} + 3 \, {\left (4 \, a^{4} c^{2} - 22 \, a^{2} c^{4} - c^{6}\right )} e^{\left (-x e - d\right )} + 6 \, {\left (4 \, a^{5} c - 17 \, a^{3} c^{3} + 4 \, a c^{5}\right )} e^{\left (-2 \, x e - 2 \, d\right )} + 2 \, {\left (4 \, a^{6} - 32 \, a^{4} c^{2} + 39 \, a^{2} c^{4}\right )} e^{\left (-3 \, x e - 3 \, d\right )} + 15 \, {\left (4 \, a^{3} c^{3} - a c^{5}\right )} e^{\left (-4 \, x e - 4 \, d\right )} - 3 \, {\left (4 \, a^{2} c^{4} - c^{6}\right )} e^{\left (-5 \, x e - 5 \, d\right )}\right )} e^{\left (-1\right )}}{a^{6} c^{4} + 3 \, a^{4} c^{6} + 3 \, a^{2} c^{8} + c^{10} + 6 \, {\left (a^{7} c^{3} + 3 \, a^{5} c^{5} + 3 \, a^{3} c^{7} + a c^{9}\right )} e^{\left (-x e - d\right )} + 3 \, {\left (4 \, a^{8} c^{2} + 11 \, a^{6} c^{4} + 9 \, a^{4} c^{6} + a^{2} c^{8} - c^{10}\right )} e^{\left (-2 \, x e - 2 \, d\right )} + 4 \, {\left (2 \, a^{9} c + 3 \, a^{7} c^{3} - 3 \, a^{5} c^{5} - 7 \, a^{3} c^{7} - 3 \, a c^{9}\right )} e^{\left (-3 \, x e - 3 \, d\right )} - 3 \, {\left (4 \, a^{8} c^{2} + 11 \, a^{6} c^{4} + 9 \, a^{4} c^{6} + a^{2} c^{8} - c^{10}\right )} e^{\left (-4 \, x e - 4 \, d\right )} + 6 \, {\left (a^{7} c^{3} + 3 \, a^{5} c^{5} + 3 \, a^{3} c^{7} + a c^{9}\right )} e^{\left (-5 \, x e - 5 \, d\right )} - {\left (a^{6} c^{4} + 3 \, a^{4} c^{6} + 3 \, a^{2} c^{8} + c^{10}\right )} e^{\left (-6 \, x e - 6 \, d\right )}}\right )} C - \frac {8 \, B e^{\left (-3 \, x e - 3 \, d - 1\right )}}{3 \, {\left (6 \, a c^{3} e^{\left (-x e - d\right )} + 6 \, a c^{3} e^{\left (-5 \, x e - 5 \, d\right )} - c^{4} e^{\left (-6 \, x e - 6 \, d\right )} + c^{4} + 3 \, {\left (4 \, a^{2} c^{2} - c^{4}\right )} e^{\left (-2 \, x e - 2 \, d\right )} + 4 \, {\left (2 \, a^{3} c - 3 \, a c^{3}\right )} e^{\left (-3 \, x e - 3 \, d\right )} - 3 \, {\left (4 \, a^{2} c^{2} - c^{4}\right )} e^{\left (-4 \, x e - 4 \, d\right )}\right )}} \end {gather*}

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]

1/6*(3*(2*a^2 - 3*c^2)*a*e^(-1)*log((c*e^(-x*e - d) - a - sqrt(a^2 + c^2))/(c*e^(-x*e - d) - a + sqrt(a^2 + c^
2)))/((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*sqrt(a^2 + c^2)) - 2*(11*a^2*c^3 - 4*c^5 + 15*(4*a^3*c^2 - a*c^4)*e^
(-x*e - d) + 6*(17*a^4*c - 6*a^2*c^3 + 2*c^5)*e^(-2*x*e - 2*d) + 2*(22*a^5 - 41*a^3*c^2 + 12*a*c^4)*e^(-3*x*e
- 3*d) - 15*(2*a^4*c - 3*a^2*c^3)*e^(-4*x*e - 4*d) + 3*(2*a^3*c^2 - 3*a*c^4)*e^(-5*x*e - 5*d))*e^(-1)/(a^6*c^3
+ 3*a^4*c^5 + 3*a^2*c^7 + c^9 + 6*(a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*e^(-x*e - d) + 3*(4*a^8*c + 11*a^
6*c^3 + 9*a^4*c^5 + a^2*c^7 - c^9)*e^(-2*x*e - 2*d) + 4*(2*a^9 + 3*a^7*c^2 - 3*a^5*c^4 - 7*a^3*c^6 - 3*a*c^8)*
e^(-3*x*e - 3*d) - 3*(4*a^8*c + 11*a^6*c^3 + 9*a^4*c^5 + a^2*c^7 - c^9)*e^(-4*x*e - 4*d) + 6*(a^7*c^2 + 3*a^5*
c^4 + 3*a^3*c^6 + a*c^8)*e^(-5*x*e - 5*d) - (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*e^(-6*x*e - 6*d)))*A + 1/6
*(3*(4*a^2*c - c^3)*e^(-1)*log((c*e^(-x*e - d) - a - sqrt(a^2 + c^2))/(c*e^(-x*e - d) - a + sqrt(a^2 + c^2)))/
((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*sqrt(a^2 + c^2)) + 2*(2*a^3*c^3 - 13*a*c^5 + 3*(4*a^4*c^2 - 22*a^2*c^4 -
c^6)*e^(-x*e - d) + 6*(4*a^5*c - 17*a^3*c^3 + 4*a*c^5)*e^(-2*x*e - 2*d) + 2*(4*a^6 - 32*a^4*c^2 + 39*a^2*c^4)*
e^(-3*x*e - 3*d) + 15*(4*a^3*c^3 - a*c^5)*e^(-4*x*e - 4*d) - 3*(4*a^2*c^4 - c^6)*e^(-5*x*e - 5*d))*e^(-1)/(a^6
*c^4 + 3*a^4*c^6 + 3*a^2*c^8 + c^10 + 6*(a^7*c^3 + 3*a^5*c^5 + 3*a^3*c^7 + a*c^9)*e^(-x*e - d) + 3*(4*a^8*c^2
+ 11*a^6*c^4 + 9*a^4*c^6 + a^2*c^8 - c^10)*e^(-2*x*e - 2*d) + 4*(2*a^9*c + 3*a^7*c^3 - 3*a^5*c^5 - 7*a^3*c^7 -
3*a*c^9)*e^(-3*x*e - 3*d) - 3*(4*a^8*c^2 + 11*a^6*c^4 + 9*a^4*c^6 + a^2*c^8 - c^10)*e^(-4*x*e - 4*d) + 6*(a^7
*c^3 + 3*a^5*c^5 + 3*a^3*c^7 + a*c^9)*e^(-5*x*e - 5*d) - (a^6*c^4 + 3*a^4*c^6 + 3*a^2*c^8 + c^10)*e^(-6*x*e -
6*d)))*C - 8/3*B*e^(-3*x*e - 3*d - 1)/(6*a*c^3*e^(-x*e - d) + 6*a*c^3*e^(-5*x*e - 5*d) - c^4*e^(-6*x*e - 6*d)
+ c^4 + 3*(4*a^2*c^2 - c^4)*e^(-2*x*e - 2*d) + 4*(2*a^3*c - 3*a*c^3)*e^(-3*x*e - 3*d) - 3*(4*a^2*c^2 - c^4)*e^
(-4*x*e - 4*d))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5966 vs. \(2 (242) = 484\).
time = 0.51, size = 5966, normalized size = 23.86 \begin {gather*} \text {Too large to display} \end {gather*}

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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh
(1) + 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(x*cosh(1) +
x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x
*sinh(1) + d)^6 - (2*A*a^3*c^4 + 4*C*a^2*c^5 - 3*A*a*c^6 - C*c^7)*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*si
nh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1
) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d)^3 + 3*(8*A*a^5*c^2 + 1
6*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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cos
h(1) + x*sinh(1) + 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
(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + 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(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d)
)*sqrt(a^2 + c^2)*log((c^2*cosh(x*cosh(1) + x*sinh(1) + d)^2 + c^2*sinh(x*cosh(1) + x*sinh(1) + d)^2 + 2*a*c*c
osh(x*cosh(1) + x*sinh(1) + d) + 2*a^2 + c^2 + 2*(c^2*cosh(x*cosh(1) + x*sinh(1) + d) + a*c)*sinh(x*cosh(1) +
x*sinh(1) + d) + 2*sqrt(a^2 + c^2)*(c*cosh(x*co...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (239) = 478\).
time = 0.49, size = 685, normalized size = 2.74 \begin {gather*} \frac {\frac {3 \, {\left (2 \, A a^{3} + 4 \, C a^{2} c - 3 \, A a c^{2} - C c^{3}\right )} \log \left (\frac {{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )} \sqrt {a^{2} + c^{2}}} + \frac {2 \, {\left (6 \, A a^{3} c^{3} e^{\left (5 \, e x + 5 \, d\right )} + 12 \, C a^{2} c^{4} e^{\left (5 \, e x + 5 \, d\right )} - 9 \, A a c^{5} e^{\left (5 \, e x + 5 \, d\right )} - 3 \, C c^{6} e^{\left (5 \, e x + 5 \, d\right )} + 30 \, A a^{4} c^{2} e^{\left (4 \, e x + 4 \, d\right )} + 60 \, C a^{3} c^{3} e^{\left (4 \, e x + 4 \, d\right )} - 45 \, A a^{2} c^{4} e^{\left (4 \, e x + 4 \, d\right )} - 15 \, C a c^{5} e^{\left (4 \, e x + 4 \, d\right )} - 8 \, B a^{6} e^{\left (3 \, e x + 3 \, d\right )} - 8 \, C a^{6} e^{\left (3 \, e x + 3 \, d\right )} + 44 \, A a^{5} c e^{\left (3 \, e x + 3 \, d\right )} - 24 \, B a^{4} c^{2} e^{\left (3 \, e x + 3 \, d\right )} + 64 \, C a^{4} c^{2} e^{\left (3 \, e x + 3 \, d\right )} - 82 \, A a^{3} c^{3} e^{\left (3 \, e x + 3 \, d\right )} - 24 \, B a^{2} c^{4} e^{\left (3 \, e x + 3 \, d\right )} - 78 \, C a^{2} c^{4} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, A a c^{5} e^{\left (3 \, e x + 3 \, d\right )} - 8 \, B c^{6} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, C a^{5} c e^{\left (2 \, e x + 2 \, d\right )} - 102 \, A a^{4} c^{2} e^{\left (2 \, e x + 2 \, d\right )} - 102 \, C a^{3} c^{3} e^{\left (2 \, e x + 2 \, d\right )} + 36 \, A a^{2} c^{4} e^{\left (2 \, e x + 2 \, d\right )} + 24 \, C a c^{5} e^{\left (2 \, e x + 2 \, d\right )} - 12 \, A c^{6} e^{\left (2 \, e x + 2 \, d\right )} - 12 \, C a^{4} c^{2} e^{\left (e x + d\right )} + 60 \, A a^{3} c^{3} e^{\left (e x + d\right )} + 66 \, C a^{2} c^{4} e^{\left (e x + d\right )} - 15 \, A a c^{5} e^{\left (e x + d\right )} + 3 \, C c^{6} e^{\left (e x + d\right )} + 2 \, C a^{3} c^{3} - 11 \, A a^{2} c^{4} - 13 \, C a c^{5} + 4 \, A c^{6}\right )}}{{\left (a^{6} c + 3 \, a^{4} c^{3} + 3 \, a^{2} c^{5} + c^{7}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}^{3}}}{6 \, e} \end {gather*}

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]

1/6*(3*(2*A*a^3 + 4*C*a^2*c - 3*A*a*c^2 - C*c^3)*log(abs(2*c*e^(e*x + d) + 2*a - 2*sqrt(a^2 + c^2))/abs(2*c*e^
(e*x + d) + 2*a + 2*sqrt(a^2 + c^2)))/((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*sqrt(a^2 + c^2)) + 2*(6*A*a^3*c^3*e
^(5*e*x + 5*d) + 12*C*a^2*c^4*e^(5*e*x + 5*d) - 9*A*a*c^5*e^(5*e*x + 5*d) - 3*C*c^6*e^(5*e*x + 5*d) + 30*A*a^4
*c^2*e^(4*e*x + 4*d) + 60*C*a^3*c^3*e^(4*e*x + 4*d) - 45*A*a^2*c^4*e^(4*e*x + 4*d) - 15*C*a*c^5*e^(4*e*x + 4*d
) - 8*B*a^6*e^(3*e*x + 3*d) - 8*C*a^6*e^(3*e*x + 3*d) + 44*A*a^5*c*e^(3*e*x + 3*d) - 24*B*a^4*c^2*e^(3*e*x + 3
*d) + 64*C*a^4*c^2*e^(3*e*x + 3*d) - 82*A*a^3*c^3*e^(3*e*x + 3*d) - 24*B*a^2*c^4*e^(3*e*x + 3*d) - 78*C*a^2*c^
4*e^(3*e*x + 3*d) + 24*A*a*c^5*e^(3*e*x + 3*d) - 8*B*c^6*e^(3*e*x + 3*d) + 24*C*a^5*c*e^(2*e*x + 2*d) - 102*A*
a^4*c^2*e^(2*e*x + 2*d) - 102*C*a^3*c^3*e^(2*e*x + 2*d) + 36*A*a^2*c^4*e^(2*e*x + 2*d) + 24*C*a*c^5*e^(2*e*x +
2*d) - 12*A*c^6*e^(2*e*x + 2*d) - 12*C*a^4*c^2*e^(e*x + d) + 60*A*a^3*c^3*e^(e*x + d) + 66*C*a^2*c^4*e^(e*x +
d) - 15*A*a*c^5*e^(e*x + d) + 3*C*c^6*e^(e*x + 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 + 3*a^4*c^3 + 3*a^2*c^5 + c^7)*(c*e^(2*e*x + 2*d) + 2*a*e^(e*x + d) - c)^3))/e

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+c\,\mathrm {sinh}\left (d+e\,x\right )\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^4,x)

[Out]

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^4, x)

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