3.3.42 \(\int \frac {\sinh ^7(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [242]

Optimal. Leaf size=210 \[ \frac {\left (3 \sqrt {a}-4 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{7/4} d}-\frac {\left (3 \sqrt {a}+4 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{7/4} d}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

-1/4*a*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)+1/8*arctan(b^(1/4)*cosh
(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(3*a^(1/2)-4*b^(1/2))/b^(7/4)/d/(a^(1/2)-b^(1/2))^(3/2)-1/8*arctanh(b^(1/4)*c
osh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(3*a^(1/2)+4*b^(1/2))/b^(7/4)/d/(a^(1/2)+b^(1/2))^(3/2)

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Rubi [A]
time = 0.27, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1219, 1180, 211, 214} \begin {gather*} \frac {\left (3 \sqrt {a}-4 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 b^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\left (3 \sqrt {a}+4 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 b^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 b d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

((3*Sqrt[a] - 4*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)
*b^(7/4)*d) - ((3*Sqrt[a] + 4*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] +
Sqrt[b])^(3/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]
^2 - b*Cosh[c + d*x]^4))

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 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 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2
*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {4 a (a-2 b)-2 a (3 a-4 b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\left (3 a-\sqrt {a} \sqrt {b}-4 b\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 (a-b) b d}-\frac {\left (3 a+\sqrt {a} \sqrt {b}-4 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 (a-b) b d}\\ &=\frac {\left (3 \sqrt {a}-4 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{7/4} d}-\frac {\left (3 \sqrt {a}+4 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{7/4} d}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.49, size = 737, normalized size = 3.51 \begin {gather*} -\frac {-\frac {16 a (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}+\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {3 a c-4 b c+3 a d x-4 b d x+6 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-5 a c \text {$\#$1}^2+12 b c \text {$\#$1}^2-5 a d x \text {$\#$1}^2+12 b d x \text {$\#$1}^2-10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+5 a c \text {$\#$1}^4-12 b c \text {$\#$1}^4+5 a d x \text {$\#$1}^4-12 b d x \text {$\#$1}^4+10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-3 a c \text {$\#$1}^6+4 b c \text {$\#$1}^6-3 a d x \text {$\#$1}^6+4 b d x \text {$\#$1}^6-6 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6+8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{32 (a-b) b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/32*((-16*a*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]
) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (3*a*c - 4*b*c + 3*a*d*x - 4*b*d*x + 6
*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 8*b*Log[-Cosh[(
c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 5*a*c*#1^2 + 12*b*c*#1^2 - 5*
a*d*x*#1^2 + 12*b*d*x*#1^2 - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c
+ d*x)/2]*#1]*#1^2 + 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2
]*#1]*#1^2 + 5*a*c*#1^4 - 12*b*c*#1^4 + 5*a*d*x*#1^4 - 12*b*d*x*#1^4 + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c +
d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2]
+ Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 3*a*c*#1^6 + 4*b*c*#1^6 - 3*a*d*x*#1^6 + 4*b*d*x*#1^6 -
6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 + 8*b*Log[
-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^
3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/((a - b)*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(164)=328\).
time = 5.59, size = 434, normalized size = 2.07

method	result	size
derivativedivides	\(\frac {128 a^{2} \left (\frac {\frac {-\frac {\left (a b -\sqrt {a b}\, a +2 \sqrt {a b}\, b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} \left (a -b \right )}-\frac {\sqrt {a b}+b}{2 a \left (a -b \right )}}{\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \sqrt {a b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+1}+\frac {\left (3 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \left (a -b \right ) \sqrt {\sqrt {a b}\, a -a b}}}{256 a \,b^{2}}-\frac {\frac {\frac {\left (\sqrt {a b}\, a -2 \sqrt {a b}\, b +a b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} \left (a -b \right )}+\frac {b -\sqrt {a b}}{2 a \left (a -b \right )}}{\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4 \sqrt {a b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}-\frac {\left (3 \sqrt {a b}\, a -4 \sqrt {a b}\, b +a b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \left (a -b \right ) \sqrt {-\sqrt {a b}\, a -a b}}}{256 a \,b^{2}}\right )}{d}\)	\(434\)
default	\(\frac {128 a^{2} \left (\frac {\frac {-\frac {\left (a b -\sqrt {a b}\, a +2 \sqrt {a b}\, b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} \left (a -b \right )}-\frac {\sqrt {a b}+b}{2 a \left (a -b \right )}}{\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \sqrt {a b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+1}+\frac {\left (3 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \left (a -b \right ) \sqrt {\sqrt {a b}\, a -a b}}}{256 a \,b^{2}}-\frac {\frac {\frac {\left (\sqrt {a b}\, a -2 \sqrt {a b}\, b +a b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} \left (a -b \right )}+\frac {b -\sqrt {a b}}{2 a \left (a -b \right )}}{\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4 \sqrt {a b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}-\frac {\left (3 \sqrt {a b}\, a -4 \sqrt {a b}\, b +a b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \left (a -b \right ) \sqrt {-\sqrt {a b}\, a -a b}}}{256 a \,b^{2}}\right )}{d}\)	\(434\)
risch	\(\frac {a \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c}-5 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+1\right )}{2 b d \left (a -b \right ) \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{3} b^{7} d^{4}-196608 a^{2} b^{8} d^{4}+196608 a \,b^{9} d^{4}-65536 d^{4} b^{10}\right ) \textit {\_Z}^{4}+\left (1536 a^{2} b^{4} d^{2}-7680 a \,b^{5} d^{2}+8192 b^{6} d^{2}\right ) \textit {\_Z}^{2}-81 a^{2}+288 a b -256 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (-\frac {24576 a^{4} b^{5} d^{3}}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}+\frac {114688 a^{3} b^{6} d^{3}}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}-\frac {196608 a^{2} b^{7} d^{3}}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}+\frac {147456 a \,b^{8} d^{3}}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}-\frac {40960 b^{9} d^{3}}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {864 a^{3} b^{2} d}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}+\frac {4928 a^{2} b^{3} d}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}-\frac {9184 a \,b^{4} d}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}+\frac {5632 b^{5} d}{81 a^{3}-405 a^{2} b +680 a \,b^{2}-384 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(555\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

128/d*a^2*(1/256/a/b^2*((-1/2*(a*b-(a*b)^(1/2)*a+2*(a*b)^(1/2)*b)/a^2/(a-b)*tanh(1/2*d*x+1/2*c)^2-1/2*((a*b)^(
1/2)+b)/a/(a-b))/(tanh(1/2*d*x+1/2*c)^4-2*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)/a*tanh(1/2*d*x+1/2*c)^2+1)+1/4*(
3*(a*b)^(1/2)*a-4*(a*b)^(1/2)*b-a*b)/a/(a-b)/((a*b)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2+4
*(a*b)^(1/2)-2*a)/((a*b)^(1/2)*a-a*b)^(1/2)))-1/256/a/b^2*((1/2*((a*b)^(1/2)*a-2*(a*b)^(1/2)*b+a*b)/a^2/(a-b)*
tanh(1/2*d*x+1/2*c)^2+1/2*(b-(a*b)^(1/2))/a/(a-b))/(tanh(1/2*d*x+1/2*c)^4-4*(a*b)^(1/2)/a*tanh(1/2*d*x+1/2*c)^
2-2*tanh(1/2*d*x+1/2*c)^2+1)-1/4*(3*(a*b)^(1/2)*a-4*(a*b)^(1/2)*b+a*b)/a/(a-b)/(-(a*b)^(1/2)*a-a*b)^(1/2)*arct
an(1/4*(-2*a*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)+2*a)/(-(a*b)^(1/2)*a-a*b)^(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(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/2*(a*e^(7*d*x + 7*c) - 5*a*e^(5*d*x + 5*c) - 5*a*e^(3*d*x + 3*c) + a*e^(d*x + c))/(a*b^2*d - b^3*d + (a*b^2
*d*e^(8*c) - b^3*d*e^(8*c))*e^(8*d*x) - 4*(a*b^2*d*e^(6*c) - b^3*d*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b*d*e^(4*c) -
11*a*b^2*d*e^(4*c) + 3*b^3*d*e^(4*c))*e^(4*d*x) - 4*(a*b^2*d*e^(2*c) - b^3*d*e^(2*c))*e^(2*d*x)) + 1/128*inte
grate(64*((3*a*e^(7*c) - 4*b*e^(7*c))*e^(7*d*x) - (5*a*e^(5*c) - 12*b*e^(5*c))*e^(5*d*x) + (5*a*e^(3*c) - 12*b
*e^(3*c))*e^(3*d*x) - (3*a*e^c - 4*b*e^c)*e^(d*x))/(a*b^2 - b^3 + (a*b^2*e^(8*c) - b^3*e^(8*c))*e^(8*d*x) - 4*
(a*b^2*e^(6*c) - b^3*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b*e^(4*c) - 11*a*b^2*e^(4*c) + 3*b^3*e^(4*c))*e^(4*d*x) - 4
*(a*b^2*e^(2*c) - 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. 6266 vs. \(2 (161) = 322\).
time = 0.56, size = 6266, normalized size = 29.84 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(8*a*cosh(d*x + c)^7 + 56*a*cosh(d*x + c)*sinh(d*x + c)^6 + 8*a*sinh(d*x + c)^7 - 40*a*cosh(d*x + c)^5 +
8*(21*a*cosh(d*x + c)^2 - 5*a)*sinh(d*x + c)^5 + 40*(7*a*cosh(d*x + c)^3 - 5*a*cosh(d*x + c))*sinh(d*x + c)^4
- 40*a*cosh(d*x + c)^3 + 40*(7*a*cosh(d*x + c)^4 - 10*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^3 + 8*(21*a*cosh(d
*x + c)^5 - 50*a*cosh(d*x + c)^3 - 15*a*cosh(d*x + c))*sinh(d*x + c)^2 - ((a*b^2 - b^3)*d*cosh(d*x + c)^8 + 8*
(a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - b^3)*d*sinh(d*x + c)^8 - 4*(a*b^2 - b^3)*d*cosh(d*x +
c)^6 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b - 11*a*b^2 + 3*b^
3)*d*cosh(d*x + c)^4 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - 3*(a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(35*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 30*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d
)*sinh(d*x + c)^4 - 4*(a*b^2 - b^3)*d*cosh(d*x + c)^2 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^5 - 10*(a*b^2 - b^3
)*d*cosh(d*x + c)^3 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^2 - b^3)*d*cos
h(d*x + c)^6 - 15*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 3*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^2 - (a*b^2
- b^3)*d)*sinh(d*x + c)^2 + (a*b^2 - b^3)*d + 8*((a*b^2 - b^3)*d*cosh(d*x + c)^7 - 3*(a*b^2 - b^3)*d*cosh(d*x
+ c)^5 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^3 - (a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-
((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)
/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16
*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-81*a^3 + 405*a^2*b - 680*a*b^2 + 384*b^3 - (81*a^3 - 4
05*a^2*b + 680*a*b^2 - 384*b^3)*cosh(d*x + c)^2 - 2*(81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cosh(d*x + c)*s
inh(d*x + c) - (81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*sinh(d*x + c)^2 + 2*(2*(9*a^3*b^2 - 47*a^2*b^3 + 82*
a*b^4 - 48*b^5)*d*cosh(d*x + c) + 2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d*sinh(d*x + c) - ((3*a^4*b^5
- 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*cosh(d*x + c) + (3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a
*b^8 + 5*b^9)*d^3*sinh(d*x + c))*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7
- 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)))*sqrt(-((a^3*b^3 - 3*a^2*b^4 +
3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 +
15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*
b^4 + 3*a*b^5 - b^6)*d^2))) + ((a*b^2 - b^3)*d*cosh(d*x + c)^8 + 8*(a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)
^7 + (a*b^2 - b^3)*d*sinh(d*x + c)^8 - 4*(a*b^2 - b^3)*d*cosh(d*x + c)^6 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^
2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a*b^2 - b^3)*d
*cosh(d*x + c)^3 - 3*(a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b^2 - b^3)*d*cosh(d*x + c)^4 -
30*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d)*sinh(d*x + c)^4 - 4*(a*b^2 - b^3)*d*cosh(
d*x + c)^2 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^5 - 10*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - (8*a^2*b - 11*a*b^2 +
3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^6 - 15*(a*b^2 - b^3)*d*cosh(d*x
+ c)^4 - 3*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a*b^2 - b^3)*d
+ 8*((a*b^2 - b^3)*d*cosh(d*x + c)^7 - 3*(a*b^2 - b^3)*d*cosh(d*x + c)^5 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cos
h(d*x + c)^3 - (a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*
sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^
3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6
)*d^2))*log(-81*a^3 + 405*a^2*b - 680*a*b^2 + 384*b^3 - (81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cosh(d*x +
c)^2 - 2*(81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cosh(d*x + c)*sinh(d*x + c) - (81*a^3 - 405*a^2*b + 680*a*
b^2 - 384*b^3)*sinh(d*x + c)^2 - 2*(2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d*cosh(d*x + c) + 2*(9*a^3*
b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d*sinh(d*x + c) - ((3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b
^9)*d^3*cosh(d*x + c) + (3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*sinh(d*x + c))*sqrt((81*a
^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 1
5*a^2*b^11 - 6*a*b^12 + b^13)*d^4)))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b
+ 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6
*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) - ((a*b^2 - b^3
)*d*cosh(d*x + c)^8 + 8*(a*b^2 - b^3)*d*cosh(d*...

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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(sinh(d*x+c)**7/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (161) = 322\).
time = 0.73, size = 1009, normalized size = 4.80 \begin {gather*} -\frac {\frac {{\left ({\left (12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} - \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b - 20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left (a b - b^{2}\right )}^{2} {\left | b \right |} - 2 \, {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{2} - 7 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{3} - 7 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{4} + 10 \, \sqrt {-b^{2} + \sqrt {a b} b} b^{5}\right )} {\left | -a b + b^{2} \right |} {\left | b \right |} - {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} - 3 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4} - 6 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{5} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{2} - b^{3} + \sqrt {{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} {\left (a b^{2} - b^{3}\right )} + {\left (a b^{2} - b^{3}\right )}^{2}}}{a b^{2} - b^{3}}}}\right )}{{\left (4 \, a^{4} b^{5} - 7 \, a^{3} b^{6} - 3 \, a^{2} b^{7} + 11 \, a b^{8} - 5 \, b^{9}\right )} {\left | -a b + b^{2} \right |}} - \frac {{\left ({\left (12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} - \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b - 20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left (a b - b^{2}\right )}^{2} {\left | b \right |} + 2 \, {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{2} - 7 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{3} - 7 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{4} + 10 \, \sqrt {-b^{2} - \sqrt {a b} b} b^{5}\right )} {\left | -a b + b^{2} \right |} {\left | b \right |} - {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} - 3 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4} - 6 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{5} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{2} - b^{3} - \sqrt {{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} {\left (a b^{2} - b^{3}\right )} + {\left (a b^{2} - b^{3}\right )}^{2}}}{a b^{2} - b^{3}}}}\right )}{{\left (4 \, a^{4} b^{5} - 7 \, a^{3} b^{6} - 3 \, a^{2} b^{7} + 11 \, a b^{8} - 5 \, b^{9}\right )} {\left | -a b + b^{2} \right |}} + \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 8 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )} {\left (a b - b^{2}\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/8*(((12*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 - sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b - 20*sqrt(a*b)*sqrt
(-b^2 + sqrt(a*b)*b)*b^2)*(a*b - b^2)^2*abs(b) - 2*(4*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^2 - 7*sqrt(-b^2 + sqrt(a*
b)*b)*a^2*b^3 - 7*sqrt(-b^2 + sqrt(a*b)*b)*a*b^4 + 10*sqrt(-b^2 + sqrt(a*b)*b)*b^5)*abs(-a*b + b^2)*abs(b) - (
4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^3 - 3*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^4 - 6*sqrt(a*b)*sqrt
(-b^2 + sqrt(a*b)*b)*a*b^5 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^6)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d
*x - c))/sqrt(-(a*b^2 - b^3 + sqrt((a^2*b - 2*a*b^2 + b^3)*(a*b^2 - b^3) + (a*b^2 - b^3)^2))/(a*b^2 - b^3)))/(
(4*a^4*b^5 - 7*a^3*b^6 - 3*a^2*b^7 + 11*a*b^8 - 5*b^9)*abs(-a*b + b^2)) - ((12*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)
*b)*a^2 - sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b - 20*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^2)*(a*b - b^2)^2*ab
s(b) + 2*(4*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^2 - 7*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^3 - 7*sqrt(-b^2 - sqrt(a*b)*b)
*a*b^4 + 10*sqrt(-b^2 - sqrt(a*b)*b)*b^5)*abs(-a*b + b^2)*abs(b) - (4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b
^3 - 3*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^4 - 6*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^5 + 5*sqrt(a*b)*s
qrt(-b^2 - sqrt(a*b)*b)*b^6)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a*b^2 - b^3 - sqrt((a^2*b
- 2*a*b^2 + b^3)*(a*b^2 - b^3) + (a*b^2 - b^3)^2))/(a*b^2 - b^3)))/((4*a^4*b^5 - 7*a^3*b^6 - 3*a^2*b^7 + 11*a*
b^8 - 5*b^9)*abs(-a*b + b^2)) + 4*(a*(e^(d*x + c) + e^(-d*x - c))^3 - 8*a*(e^(d*x + c) + e^(-d*x - c)))/((b*(e
^(d*x + c) + e^(-d*x - c))^4 - 8*b*(e^(d*x + c) + e^(-d*x - c))^2 - 16*a + 16*b)*(a*b - b^2)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^7}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4)^2,x)

[Out]

int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4)^2, x)

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