3.4.19 \(\int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [319]

Optimal. Leaf size=81 \[ -\frac {(2 a-3 b) x}{2 b^2}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d} \]

[Out]

-1/2*(2*a-3*b)*x/b^2+1/2*cosh(d*x+c)*sinh(d*x+c)/b/d+(a-b)^(3/2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/b^2/
d/a^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3270, 425, 536, 212, 214} \begin {gather*} \frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}-\frac {x (2 a-3 b)}{2 b^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]

[Out]

-1/2*((2*a - 3*b)*x)/b^2 + ((a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^2*d) + (Cos
h[c + d*x]*Sinh[c + d*x])/(2*b*d)

Rule 212

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

Rule 214

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

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\text {Subst}\left (\int \frac {-a+2 b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 b d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {(2 a-3 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 b^2 d}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{b^2 d}\\ &=-\frac {(2 a-3 b) x}{2 b^2}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 80, normalized size = 0.99 \begin {gather*} \frac {4 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} (-2 (2 a-3 b) (c+d x)+b \sinh (2 (c+d x)))}{4 \sqrt {a} b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(69)=138\).
time = 1.64, size = 316, normalized size = 3.90

method	result	size
risch	\(-\frac {a x}{b^{2}}+\frac {3 x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 a d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 a d b}\)	\(253\)
derivativedivides	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}+\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\)	\(316\)
default	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}+\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\)	\(316\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2/b/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/b/(tanh(1/2*d*x+1/2*c)-1)+1/2*(2*a-3*b)/b^2*ln(tanh(1/2*d*x+1/2*c)-1)
+2/b^2*(a^2-2*a*b+b^2)*a*(1/2*(-(-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*ar
ctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2*(-(-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/
((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)))-1/2/
b/(tanh(1/2*d*x+1/2*c)+1)^2+1/2/b/(tanh(1/2*d*x+1/2*c)+1)+1/2/b^2*(-2*a+3*b)*ln(tanh(1/2*d*x+1/2*c)+1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (69) = 138\).
time = 0.42, size = 875, normalized size = 10.80 \begin {gather*} \left [-\frac {4 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a - b}{a}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} - a b\right )} \sqrt {\frac {a - b}{a}}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) + 4 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}}, -\frac {4 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a - b}{a}} \arctan \left (-\frac {{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sqrt {-\frac {a - b}{a}}}{2 \, {\left (a - b\right )}}\right ) + 4 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/8*(4*(2*a - 3*b)*d*x*cosh(d*x + c)^2 - b*cosh(d*x + c)^4 - 4*b*cosh(d*x + c)*sinh(d*x + c)^3 - b*sinh(d*x
+ c)^4 + 2*(2*(2*a - 3*b)*d*x - 3*b*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^2 + 2*(a - b)*
cosh(d*x + c)*sinh(d*x + c) + (a - b)*sinh(d*x + c)^2)*sqrt((a - b)/a)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d
*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2
*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh
(d*x + c) + 4*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + 2*a^2 - a*b)*sq
rt((a - b)/a))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d
*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c)
)*sinh(d*x + c) + b)) + 4*(2*(2*a - 3*b)*d*x*cosh(d*x + c) - b*cosh(d*x + c)^3)*sinh(d*x + c) + b)/(b^2*d*cosh
(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2), -1/8*(4*(2*a - 3*b)*d*x*cosh(d*x +
c)^2 - b*cosh(d*x + c)^4 - 4*b*cosh(d*x + c)*sinh(d*x + c)^3 - b*sinh(d*x + c)^4 + 2*(2*(2*a - 3*b)*d*x - 3*b
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a - b)*cosh(d*x + c)^2 + 2*(a - b)*cosh(d*x + c)*sinh(d*x + c) + (a -
b)*sinh(d*x + c)^2)*sqrt(-(a - b)/a)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh
(d*x + c)^2 + 2*a - b)*sqrt(-(a - b)/a)/(a - b)) + 4*(2*(2*a - 3*b)*d*x*cosh(d*x + c) - b*cosh(d*x + c)^3)*sin
h(d*x + c) + b)/(b^2*d*cosh(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2)]

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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(cosh(d*x+c)**4/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]
time = 1.69, size = 138, normalized size = 1.70 \begin {gather*} -\frac {\frac {4 \, {\left (d x + c\right )} {\left (2 \, a - 3 \, b\right )}}{b^{2}} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{b} - \frac {{\left (4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2}} - \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/8*(4*(d*x + c)*(2*a - 3*b)/b^2 - e^(2*d*x + 2*c)/b - (4*a*e^(2*d*x + 2*c) - 6*b*e^(2*d*x + 2*c) - b)*e^(-2*
d*x - 2*c)/b^2 - 8*(a^2 - 2*a*b + b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 +
a*b)*b^2))/d

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Mupad [B]
time = 1.61, size = 300, normalized size = 3.70 \begin {gather*} \frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}-\frac {x\,\left (2\,a-3\,b\right )}{2\,b^2}-\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}-\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d}+\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}+\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2),x)

[Out]

exp(2*c + 2*d*x)/(8*b*d) - exp(- 2*c - 2*d*x)/(8*b*d) - (x*(2*a - 3*b))/(2*b^2) - (log((4*(a - b)^3*(2*a*b - b
^2 + 8*a^2*exp(2*c + 2*d*x) + b^2*exp(2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a*b^6) - (8*(a - b)^(7/2)*(b +
4*a*exp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(1/2)*b^6))*(a - b)^(3/2))/(2*a^(1/2)*b^2*d) + (log((4*(a - b
)^3*(2*a*b - b^2 + 8*a^2*exp(2*c + 2*d*x) + b^2*exp(2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a*b^6) + (8*(a -
b)^(7/2)*(b + 4*a*exp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(1/2)*b^6))*(a - b)^(3/2))/(2*a^(1/2)*b^2*d)

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