3.4.40 \(\int \frac {\cosh ^5(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [340]

Optimal. Leaf size=133 \[ \frac {\left (3 a^2+2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2} d}-\frac {(a-b) \cosh ^2(c+d x) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \sinh (c+d x)}{8 d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

1/8*(3*a^2+2*a*b+3*b^2)*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(5/2)/b^(5/2)/d-1/4*(a-b)*cosh(d*x+c)^2*sinh(d*x
+c)/a/b/d/(a+b*sinh(d*x+c)^2)^2+3/8*(1/a^2-1/b^2)*sinh(d*x+c)/d/(a+b*sinh(d*x+c)^2)

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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 424, 393, 211} \begin {gather*} \frac {3 \left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \sinh (c+d x)}{8 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\left (3 a^2+2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2} d}-\frac {(a-b) \sinh (c+d x) \cosh ^2(c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

Rule 211

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

Rule 393

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

Rule 424

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

Rule 3269

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

Rubi steps

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

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Mathematica [A]
time = 0.26, size = 149, normalized size = 1.12 \begin {gather*} \frac {-\left (\left (3 a^2+2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )\right )+\frac {8 a^{3/2} (a-b)^2 \sqrt {b} \sinh (c+d x)}{(2 a-b+b \cosh (2 (c+d x)))^2}-\frac {2 \sqrt {a} \sqrt {b} \left (5 a^2-2 a b-3 b^2\right ) \sinh (c+d x)}{2 a-b+b \cosh (2 (c+d x))}}{8 a^{5/2} b^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(119)=238\).
time = 1.90, size = 400, normalized size = 3.01

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (3 a^{2}+2 a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \,b^{2}}-\frac {\left (9 a^{3}-14 a^{2} b -7 a \,b^{2}+12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -7 a \,b^{2}+12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} b^{2}}-\frac {\left (3 a^{2}+2 a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \,b^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (3 a^{2}+2 a b +3 b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a \,b^{2}}}{d}\)	\(400\)
default	\(\frac {\frac {\frac {\left (3 a^{2}+2 a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \,b^{2}}-\frac {\left (9 a^{3}-14 a^{2} b -7 a \,b^{2}+12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -7 a \,b^{2}+12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} b^{2}}-\frac {\left (3 a^{2}+2 a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \,b^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (3 a^{2}+2 a b +3 b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a \,b^{2}}}{d}\)	\(400\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (5 a^{2} b \,{\mathrm e}^{6 d x +6 c}-2 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 b^{3} {\mathrm e}^{6 d x +6 c}+12 a^{3} {\mathrm e}^{4 d x +4 c}-7 a^{2} b \,{\mathrm e}^{4 d x +4 c}-14 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 b^{3} {\mathrm e}^{4 d x +4 c}-12 a^{3} {\mathrm e}^{2 d x +2 c}+7 a^{2} b \,{\mathrm e}^{2 d x +2 c}+14 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 b^{3} {\mathrm e}^{2 d x +2 c}-5 a^{2} b +2 a \,b^{2}+3 b^{3}\right )}{4 b^{2} a^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{8 \sqrt {-a b}\, d b a}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{8 \sqrt {-a b}\, d b a}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{16 \sqrt {-a b}\, d \,a^{2}}\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*(1/8*(3*a^2+2*a*b-5*b^2)/a/b^2*tanh(1/2*d*x+1/2*c)^7-1/8*(9*a^3-14*a^2*b-7*a*b^2+12*b^3)/a^2/b^2*tanh(1
/2*d*x+1/2*c)^5+1/8*(9*a^3-14*a^2*b-7*a*b^2+12*b^3)/a^2/b^2*tanh(1/2*d*x+1/2*c)^3-1/8*(3*a^2+2*a*b-5*b^2)/a/b^
2*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+1/4/a
*(3*a^2+2*a*b+3*b^2)/b^2*(1/2*(-a+(-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*
arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2*(a+(-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))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/4*((5*a^2*b*e^(7*c) - 2*a*b^2*e^(7*c) - 3*b^3*e^(7*c))*e^(7*d*x) + (12*a^3*e^(5*c) - 7*a^2*b*e^(5*c) - 14*a
*b^2*e^(5*c) + 9*b^3*e^(5*c))*e^(5*d*x) - (12*a^3*e^(3*c) - 7*a^2*b*e^(3*c) - 14*a*b^2*e^(3*c) + 9*b^3*e^(3*c)
)*e^(3*d*x) - (5*a^2*b*e^c - 2*a*b^2*e^c - 3*b^3*e^c)*e^(d*x))/(a^2*b^4*d*e^(8*d*x + 8*c) + a^2*b^4*d + 4*(2*a
^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*b^2*d*e^(4*c) - 8*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e
^(4*c))*e^(4*d*x) + 4*(2*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) + 1/32*integrate(8*((3*a^2*e^(3*c)
+ 2*a*b*e^(3*c) + 3*b^2*e^(3*c))*e^(3*d*x) + (3*a^2*e^c + 2*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^2*b^3*e^(4*d*x +
4*c) + a^2*b^3 + 2*(2*a^3*b^2*e^(2*c) - a^2*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3266 vs. \(2 (119) = 238\).
time = 0.43, size = 5844, normalized size = 43.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^7 + 28*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x +
c)*sinh(d*x + c)^6 + 4*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^7 + 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^
3 + 9*a*b^4)*cosh(d*x + c)^5 + 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 + 21*(5*a^3*b^2 - 2*a^2*b^3 - 3*
a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (12*a^4*b
- 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*
a*b^4)*cosh(d*x + c)^3 - 4*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 - 35*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)
*cosh(d*x + c)^4 - 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(
5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 10*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x +
c)^3 - 3*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2*b^2 + 2*a*b^3
+ 3*b^4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b^2 + 2*a*b
^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(6*a^3*b + a^2*b^2 +
4*a*b^3 - 3*b^4 + 7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b^2 + 2*a*b^
3 + 3*b^4)*cosh(d*x + c)^3 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^
4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x +
c)^4 + 24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4 + 30*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 3*a^2*b^2 + 2*a*b^3 + 3*b^4 + 8*(7*(3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*
(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^2*b^2 + 2*a*
b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 6*a^3*b + a^2*b^2 +
4*a*b^3 - 3*b^4 + 3*(24*a^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((
3*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (24*a
^4 - 8*a^3*b + 17*a^2*b^2 - 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (6*a^3*b + a^2*b^2 + 4*a*b^3 - 3*b^4)*cosh(d*x
+ c))*sinh(d*x + c))*sqrt(-a*b)*log((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) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3
+ (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/(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*(5*a^3*b^2 - 2*a^2*b^3 -
3*a*b^4)*cosh(d*x + c) + 4*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 - 5*a^3*b^2 + 2*a^2*b^3 + 3*a*
b^4 + 5*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 - 3*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 +
9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^3*b^5*d*cosh(d*x + c)^8 + 8*a^3*b^5*d*cosh(d*x + c)*sinh(d*x + c)
^7 + a^3*b^5*d*sinh(d*x + c)^8 + a^3*b^5*d + 4*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*a^3*b^5*d*cosh(d
*x + c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4
+ 8*(7*a^3*b^5*d*cosh(d*x + c)^3 + 3*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^3*b^5*d
*cosh(d*x + c)^4 + 30*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*
x + c)^4 + 4*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*a^3*b^5*d*cosh(d*x + c)^5 + 10*(2*a^4*b^4 - a^3*b^
5)*d*cosh(d*x + c)^3 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^3*b^5*d*c
osh(d*x + c)^6 + 15*(2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x
+ c)^2 + (2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + 8*(a^3*b^5*d*cosh(d*x + c)^7 + 3*(2*a^4*b^4 - a^3*b^5)*d*
cosh(d*x + c)^5 + (8*a^5*b^3 - 8*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c
))*sinh(d*x + c)), -1/8*(2*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^7 + 14*(5*a^3*b^2 - 2*a^2*b^3 - 3*a
*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^7 + 2*(12*a^4*b - 7*a^
3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^5 + 2*(12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4 + 21*(5*a^3*b^2
- 2*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(5*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4)*cosh(d*x +
c)^3 + (12*a^4*b - 7*a^3*b^2 - 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(12*a^4*b - 7*a^3*b^2
- 14*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^3 - 2*(12...

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^5}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^3,x)

[Out]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^3, x)

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