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

Optimal. Leaf size=184 \[ -\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{2 a^3 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{\left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 b^3 d}+\frac{x \left (4 a^2 b^2+8 a^4-b^4\right )}{8 b^5}-\frac{a \sinh ^2(c+d x) \cosh (c+d x)}{3 b^2 d}+\frac{\sinh ^3(c+d x) \cosh (c+d x)}{4 b d} \]

[Out]

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

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Rubi [A]  time = 0.79298, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2889, 3050, 3049, 3023, 2735, 2660, 618, 204} \[ -\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{2 a^3 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^5 d}+\frac{\left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 b^3 d}+\frac{x \left (4 a^2 b^2+8 a^4-b^4\right )}{8 b^5}-\frac{a \sinh ^2(c+d x) \cosh (c+d x)}{3 b^2 d}+\frac{\sinh ^3(c+d x) \cosh (c+d x)}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2889

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3050

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n
*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0
] && NeQ[c, 0])))

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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])

Rubi steps

\begin{align*} \int \frac{\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac{\sinh ^3(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac{\int \frac{\sinh ^2(c+d x) \left (-3 a+b \sinh (c+d x)-4 a \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{4 b}\\ &=-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac{\int \frac{\sinh (c+d x) \left (8 a^2-a b \sinh (c+d x)+3 \left (4 a^2+b^2\right ) \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{12 b^2}\\ &=\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac{\int \frac{-3 a \left (4 a^2+b^2\right )+b \left (4 a^2-3 b^2\right ) \sinh (c+d x)-8 a \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^3}\\ &=-\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac{i \int \frac{3 i a b \left (4 a^2+b^2\right )-3 i \left (8 a^4+4 a^2 b^2-b^4\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^4}\\ &=\frac{\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{1}{a+b \sinh (c+d x)} \, dx}{b^5}\\ &=\frac{\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac{\left (2 i a^3 \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{b^5 d}\\ &=\frac{\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac{\left (4 i a^3 \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{b^5 d}\\ &=\frac{\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac{2 a^3 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^5 d}-\frac{a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac{a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac{\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}\\ \end{align*}

Mathematica [A]  time = 1.98925, size = 153, normalized size = 0.83 \[ \frac{-24 a b \left (4 a^2+b^2\right ) \cosh (c+d x)+3 \left (4 \left (4 a^2 b^2+8 a^4-b^4\right ) (c+d x)+8 a^2 b^2 \sinh (2 (c+d x))+64 a^3 \sqrt{-a^2-b^2} \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a^2-b^2}}\right )+b^4 \sinh (4 (c+d x))\right )-8 a b^3 \cosh (3 (c+d x))}{96 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*b*(4*a^2 + b^2)*Cosh[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)] + 3*(4*(8*a^4 + 4*a^2*b^2 - b^4)*(c + d*x) +
64*a^3*Sqrt[-a^2 - b^2]*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]] + 8*a^2*b^2*Sinh[2*(c + d*x)] + b^4
*Sinh[4*(c + d*x)]))/(96*b^5*d)

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Maple [B]  time = 0.049, size = 624, 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(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

-1/8/d/b*ln(tanh(1/2*d*x+1/2*c)+1)+1/8/d/b*ln(tanh(1/2*d*x+1/2*c)-1)-3/8/d/b/(tanh(1/2*d*x+1/2*c)+1)^2+3/8/d/b
/(tanh(1/2*d*x+1/2*c)-1)^2-1/d*a^4/b^5*ln(tanh(1/2*d*x+1/2*c)-1)-1/2/d/b^3/(tanh(1/2*d*x+1/2*c)+1)^2*a^2-1/d/b
^4/(tanh(1/2*d*x+1/2*c)+1)*a^3+1/d*a^4/b^5*ln(tanh(1/2*d*x+1/2*c)+1)-1/3/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^3*a+1/3
/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^3*a+1/2/d/b^3/(tanh(1/2*d*x+1/2*c)-1)^2*a^2+1/d/b^4/(tanh(1/2*d*x+1/2*c)-1)*a^3
+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^2*a+1/2/d/b^3/(tanh(1/2*d*x+1/2*c)+1)*a^2+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)
^2*a+1/2/d/b^3/(tanh(1/2*d*x+1/2*c)-1)*a^2-1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)*a+1/2/d/b^3*ln(tanh(1/2*d*x+1/2*c
)+1)*a^2+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)*a-1/2/d/b^3*ln(tanh(1/2*d*x+1/2*c)-1)*a^2-2/d*a^3*(a^2+b^2)^(1/2)/b
^5*arctanh(1/2*(2*a*tanh(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))+1/4/d/b/(tanh(1/2*d*x+1/2*c)-1)^4-1/4/d/b/(tanh(
1/2*d*x+1/2*c)+1)^4+1/2/d/b/(tanh(1/2*d*x+1/2*c)+1)^3+1/2/d/b/(tanh(1/2*d*x+1/2*c)-1)^3+1/8/d/b/(tanh(1/2*d*x+
1/2*c)+1)+1/8/d/b/(tanh(1/2*d*x+1/2*c)-1)

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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(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.29543, size = 2803, normalized size = 15.23 \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)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/192*(3*b^4*cosh(d*x + c)^8 + 3*b^4*sinh(d*x + c)^8 - 8*a*b^3*cosh(d*x + c)^7 + 24*a^2*b^2*cosh(d*x + c)^6 +
8*(3*b^4*cosh(d*x + c) - a*b^3)*sinh(d*x + c)^7 + 24*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c)^4 + 4*(21*b^4
*cosh(d*x + c)^2 - 14*a*b^3*cosh(d*x + c) + 6*a^2*b^2)*sinh(d*x + c)^6 - 24*a^2*b^2*cosh(d*x + c)^2 - 24*(4*a^
3*b + a*b^3)*cosh(d*x + c)^5 + 24*(7*b^4*cosh(d*x + c)^3 - 7*a*b^3*cosh(d*x + c)^2 + 6*a^2*b^2*cosh(d*x + c) -
4*a^3*b - a*b^3)*sinh(d*x + c)^5 - 8*a*b^3*cosh(d*x + c) + 2*(105*b^4*cosh(d*x + c)^4 - 140*a*b^3*cosh(d*x +
c)^3 + 180*a^2*b^2*cosh(d*x + c)^2 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x - 60*(4*a^3*b + a*b^3)*cosh(d*x + c))*si
nh(d*x + c)^4 - 3*b^4 - 24*(4*a^3*b + a*b^3)*cosh(d*x + c)^3 + 8*(21*b^4*cosh(d*x + c)^5 - 35*a*b^3*cosh(d*x +
c)^4 + 60*a^2*b^2*cosh(d*x + c)^3 - 12*a^3*b - 3*a*b^3 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c) - 30*
(4*a^3*b + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 12*(7*b^4*cosh(d*x + c)^6 - 14*a*b^3*cosh(d*x + c)^5 + 30
*a^2*b^2*cosh(d*x + c)^4 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c)^2 - 2*a^2*b^2 - 20*(4*a^3*b + a*b^3)
*cosh(d*x + c)^3 - 6*(4*a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 192*(a^3*cosh(d*x + c)^4 + 4*a^3*cosh(
d*x + c)^3*sinh(d*x + c) + 6*a^3*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*s
inh(d*x + c)^4)*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)) + 8*(
3*b^4*cosh(d*x + c)^7 - 7*a*b^3*cosh(d*x + c)^6 + 18*a^2*b^2*cosh(d*x + c)^5 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*
x*cosh(d*x + c)^3 - 6*a^2*b^2*cosh(d*x + c) - 15*(4*a^3*b + a*b^3)*cosh(d*x + c)^4 - a*b^3 - 9*(4*a^3*b + a*b^
3)*cosh(d*x + c)^2)*sinh(d*x + c))/(b^5*d*cosh(d*x + c)^4 + 4*b^5*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*d*co
sh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*d*sinh(d*x + c)^4)

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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)**2*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.20274, size = 379, normalized size = 2.06 \begin{align*} \frac{{\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )}{\left (d x + c\right )}}{8 \, b^{5} d} - \frac{{\left (24 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (d x + c\right )} + 3 \, b^{4} + 24 \,{\left (4 \, a^{3} b + a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{192 \, b^{5} d} - \frac{{\left (a^{5} + a^{3} b^{2}\right )} \log \left (\frac{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b^{5} d} + \frac{3 \, b^{3} d^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} d^{3} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a^{2} b d^{3} e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a^{3} d^{3} e^{\left (d x + c\right )} - 24 \, a b^{2} d^{3} e^{\left (d x + c\right )}}{192 \, b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/8*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5*d) - 1/192*(24*a^2*b^2*e^(2*d*x + 2*c) + 8*a*b^3*e^(d*x + c) + 3*
b^4 + 24*(4*a^3*b + a*b^3)*e^(3*d*x + 3*c))*e^(-4*d*x - 4*c)/(b^5*d) - (a^5 + a^3*b^2)*log(abs(2*b*e^(d*x + c)
+ 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^5*d) + 1/192*(3
*b^3*d^3*e^(4*d*x + 4*c) - 8*a*b^2*d^3*e^(3*d*x + 3*c) + 24*a^2*b*d^3*e^(2*d*x + 2*c) - 96*a^3*d^3*e^(d*x + c)
- 24*a*b^2*d^3*e^(d*x + c))/(b^4*d^4)