3.3.29 \(\int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [229]

Optimal. Leaf size=148 \[ -\frac {a \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{7/4} d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{7/4} d}+\frac {\cosh (c+d x)}{b d}-\frac {\cosh ^3(c+d x)}{3 b d} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1184, 1180, 211, 214} \begin {gather*} -\frac {a \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cosh ^3(c+d x)}{3 b d}+\frac {\cosh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.33, size = 390, normalized size = 2.64 \begin {gather*} \frac {18 \cosh (c+d x)-2 \cosh (3 (c+d x))-3 a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{24 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*Cosh[c + d*x] - 2*Cosh[3*(c + d*x)] - 3*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8
& , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] +
3*c*#1^2 + 3*d*x*#1^2 + 6*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 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin
h[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*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) & ])/(24*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(110)=220\).
time = 1.69, size = 263, normalized size = 1.78

method	result	size
risch	\(-\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}+\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}+\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a \,b^{7} d^{4}-256 b^{8} d^{4}\right ) \textit {\_Z}^{4}+32 a^{2} b^{4} d^{2} \textit {\_Z}^{2}-a^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {128 b^{5} d^{3}}{a^{2}}-\frac {128 b^{6} d^{3}}{a^{3}}\right ) \textit {\_R}^{3}+\frac {16 d \,b^{2} \textit {\_R}}{a}\right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(172\)
derivativedivides	\(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8 a^{2} \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{b}}{d}\)	\(263\)
default	\(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8 a^{2} \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{b}}{d}\)	\(263\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/3/b/(tanh(1/2*d*x+1/2*c)+1)^3+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)+1/3/b/(tan
h(1/2*d*x+1/2*c)-1)^3+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)+8*a^2/b*(-1/16*(a*b)^(1/2)
/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))-1/16*(a*b)^(1/2)/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))))

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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),x, algorithm="maxima")

[Out]

-1/24*(e^(6*d*x + 6*c) - 9*e^(4*d*x + 4*c) - 9*e^(2*d*x + 2*c) + 1)*e^(-3*d*x - 3*c)/(b*d) - 1/128*integrate(2
56*(a*e^(7*d*x + 7*c) - 3*a*e^(5*d*x + 5*c) + 3*a*e^(3*d*x + 3*c) - a*e^(d*x + c))/(b^2*e^(8*d*x + 8*c) - 4*b^
2*e^(6*d*x + 6*c) - 4*b^2*e^(2*d*x + 2*c) + b^2 - 2*(8*a*b*e^(4*c) - 3*b^2*e^(4*c))*e^(4*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (110) = 220\).
time = 0.45, size = 1617, normalized size = 10.93 \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),x, algorithm="fricas")

[Out]

-1/24*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6 + 3*(5*cosh(d*x + c)^2 - 3)*sinh(d*
x + c)^4 - 9*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 - 9*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*cosh(d*x + c)^4
- 18*cosh(d*x + c)^2 - 3)*sinh(d*x + c)^2 - 6*(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b
*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8
+ b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sin
h(d*x + c)^2 + a^3 + 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x + c) - ((a*b^5 - b^6)*d^3*cosh(d*x + c) +
(a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5
/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + 6*(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^
2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5
/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*si
nh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 - 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x + c) - ((a*b^5 - b^6
)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^
3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - 6*(b*d*cosh(d*x + c)^3 +
3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(((a*b^3
- b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*
a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 + 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x
+ c) + ((a*b^5 - b^6)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)
*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) + 6*(b*
d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x +
c)^3)*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*c
osh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 - 2*(a^2*b^2*d*cosh(d*x + c) +
a^2*b^2*d*sinh(d*x + c) + ((a*b^5 - b^6)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b
^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 -
b^4)*d^2))) - 9*cosh(d*x + c)^2 + 6*(cosh(d*x + c)^5 - 6*cosh(d*x + c)^3 - 3*cosh(d*x + c))*sinh(d*x + c) + 1)
/(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d
*x + c)^3)

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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),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (110) = 220\).
time = 0.59, size = 753, normalized size = 5.09 \begin {gather*} -\frac {\frac {12 \, {\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b\right )} b^{2} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{2} + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b^{4} - \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}}}\right )}{4 \, a^{2} b^{5} + a b^{6} - 5 \, b^{7}} - \frac {6 \, {\left (4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{4} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} b^{5} + {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{2} + 4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a^{2} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b\right )} b^{2} {\left | b \right |} - {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b^{2} - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{3} + 4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{2} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} b^{3}\right )} b^{2} - {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b^{3} - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{4}\right )} {\left | b \right |}\right )} \log \left (2 \, \sqrt {\frac {b^{4} + \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{4 \, a^{2} b^{6} - 7 \, a b^{7} + 3 \, b^{8}} - \frac {6 \, {\left ({\left (4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a^{2} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b\right )} b^{2} + {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b^{2} - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{3}\right )} {\left | b \right |}\right )} \log \left ({\left | -2 \, \sqrt {\frac {b^{4} + \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} \right |}\right )}{4 \, a^{2} b^{5} - 7 \, a b^{6} + 3 \, b^{7}} + \frac {b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{b^{3}}}{24 \, 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),x, algorithm="giac")

[Out]

-1/24*(12*((4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b)*b^2 + (4*sqrt
(-b^2 + sqrt(a*b)*b)*a^2*b^2 + 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^3)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c
))/sqrt(-(b^4 - sqrt(b^8 + (a*b^3 - b^4)*b^4))/b^4))/(4*a^2*b^5 + a*b^6 - 5*b^7) - 6*(4*sqrt(a*b)*sqrt(b^2 + s
qrt(a*b)*b)*a*b^4 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*b^5 + (4*sqrt(b^2 + sqrt(a*b)*b)*a^2*b - 3*sqrt(b^2 +
sqrt(a*b)*b)*a*b^2 + 4*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a^2 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a*b)*b^2*ab
s(b) - (4*sqrt(b^2 + sqrt(a*b)*b)*a^2*b^2 - 3*sqrt(b^2 + sqrt(a*b)*b)*a*b^3 + 4*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)
*b)*a*b^2 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*b^3)*b^2 - (4*sqrt(b^2 + sqrt(a*b)*b)*a^2*b^3 - 3*sqrt(b^2 + s
qrt(a*b)*b)*a*b^4)*abs(b))*log(2*sqrt((b^4 + sqrt(b^8 + (a*b^3 - b^4)*b^4))/b^4) + e^(d*x + c) + e^(-d*x - c))
/(4*a^2*b^6 - 7*a*b^7 + 3*b^8) - 6*((4*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a^2 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b
)*b)*a*b)*b^2 + (4*sqrt(b^2 + sqrt(a*b)*b)*a^2*b^2 - 3*sqrt(b^2 + sqrt(a*b)*b)*a*b^3)*abs(b))*log(abs(-2*sqrt(
(b^4 + sqrt(b^8 + (a*b^3 - b^4)*b^4))/b^4) + e^(d*x + c) + e^(-d*x - c)))/(4*a^2*b^5 - 7*a*b^6 + 3*b^7) + (b^2
*(e^(d*x + c) + e^(-d*x - c))^3 - 12*b^2*(e^(d*x + c) + e^(-d*x - c)))/b^3)/d

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Mupad [B]
time = 9.80, size = 1124, normalized size = 7.59 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^8\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^{11}\,{\left (a-b\right )}^2}-\frac {8388608\,a^7\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^9\,d\,{\mathrm {e}}^{c+d\,x}}{b^{13}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^{10}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{15}\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{16\,\left (b^8\,d^2-a\,b^7\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^8\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^{11}\,{\left (a-b\right )}^2}+\frac {8388608\,a^7\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^9\,d\,{\mathrm {e}}^{c+d\,x}}{b^{13}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^{10}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{15}\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{16\,\left (b^8\,d^2-a\,b^7\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^8\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^{11}\,{\left (a-b\right )}^2}-\frac {8388608\,a^7\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^9\,d\,{\mathrm {e}}^{c+d\,x}}{b^{13}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^{10}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{15}\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{16\,\left (b^8\,d^2-a\,b^7\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^8\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^{11}\,{\left (a-b\right )}^2}+\frac {8388608\,a^7\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^9\,d\,{\mathrm {e}}^{c+d\,x}}{b^{13}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{b^7\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^{10}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{15}\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{16\,\left (b^8\,d^2-a\,b^7\,d^2\right )}}+\frac {3\,{\mathrm {e}}^{c+d\,x}}{8\,b\,d}+\frac {3\,{\mathrm {e}}^{-c-d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d} \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),x)

[Out]

log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) - (8388608*a^7*d^3*exp(c + d*x)*(a
+ b)*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b
^7*d^2*(a - b)))^(1/2))/4 + (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^
2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(((a^5*b^7)^(1/2) + a^2*
b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) - log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a -
b)^2) + (8388608*a^7*d^3*exp(c + d*x)*(a + b)*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(
a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b
)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(
b^15*(a - b)^2))*(((a^5*b^7)^(1/2) + a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) + log((((((4194304*a^8*d^2*(ex
p(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) - (8388608*a^7*d^3*exp(c + d*x)*(a + b)*(((a^5*b^7)^(1/2) - a^
2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 + (2
097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*
a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(-((a^5*b^7)^(1/2) - a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)
))^(1/2) - log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) + (8388608*a^7*d^3*exp(
c + d*x)*(a + b)*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(((a^5*b^7)^(1/2) - a^
2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)
/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(-((a^5*b^7)^(1/
2) - a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) + (3*exp(c + d*x))/(8*b*d) + (3*exp(- c - d*x))/(8*b*d) - exp(
- 3*c - 3*d*x)/(24*b*d) - exp(3*c + 3*d*x)/(24*b*d)

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