Integrand size = 24, antiderivative size = 148 \[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {a \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{7/4} d}+\frac {a \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{7/4} d}+\frac {\cosh (c+d x)}{b d}-\frac {\cosh ^3(c+d x)}{3 b d} \] Output:
-1/2*a*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/(a^(1/2)-b^(1/2 ))^(1/2)/b^(7/4)/d+1/2*a*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/ 2))/(a^(1/2)+b^(1/2))^(1/2)/b^(7/4)/d+cosh(d*x+c)/b/d-1/3*cosh(d*x+c)^3/b/ d
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.58 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.64 \[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {18 \cosh (c+d x)-2 \cosh (3 (c+d x))-3 a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{24 b d} \] Input:
Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4),x]
Output:
(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*L og[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(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)
Time = 0.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 26, 3694, 1484, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^7}{a-b \sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^7}{a-b \sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^3}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -\frac {\int \left (\frac {\cosh ^2(c+d x)}{b}-\frac {1}{b}+\frac {a-a \cosh ^2(c+d x)}{b \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{7/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {a \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{7/4} \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\cosh ^3(c+d x)}{3 b}-\frac {\cosh (c+d x)}{b}}{d}\) |
Input:
Int[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4),x]
Output:
-(((a*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqr t[a] - Sqrt[b]]*b^(7/4)) - (a*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(7/4)) - Cosh[c + d*x]/b + Cosh [c + d*x]^3/(3*b))/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{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]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.90 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}+\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}+\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a \,b^{7} d^{4}-256 b^{8} d^{4}\right ) \textit {\_Z}^{4}+32 a^{2} b^{4} d^{2} \textit {\_Z}^{2}-a^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {128 b^{5} d^{3}}{a^{2}}-\frac {128 b^{6} d^{3}}{a^{3}}\right ) \textit {\_R}^{3}+\frac {16 d \,b^{2} \textit {\_R}}{a}\right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(172\) |
| derivativedivides | \(\frac {\frac {8 a^{2} \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{b}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(263\) |
| default | \(\frac {\frac {8 a^{2} \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{b}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(263\) |
Input:
int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)
Output:
-1/24/b/d*exp(3*d*x+3*c)+3/8/b/d*exp(d*x+c)+3/8/b/d*exp(-d*x-c)-1/24/b/d*e xp(-3*d*x-3*c)+sum(_R*ln(exp(2*d*x+2*c)+((128/a^2*b^5*d^3-128/a^3*b^6*d^3) *_R^3+16/a*d*b^2*_R)*exp(d*x+c)+1),_R=RootOf((256*a*b^7*d^4-256*b^8*d^4)*_ Z^4+32*a^2*b^4*d^2*_Z^2-a^4))
Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (110) = 220\).
Time = 0.13 (sec) , antiderivative size = 1617, normalized size of antiderivative = 10.93 \[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")
Output:
-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*co sh(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*c osh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*s inh(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)))*sqr t(-((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*sin h(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*co...
Timed out. \[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**7/(a-b*sinh(d*x+c)**4),x)
Output:
Timed out
\[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")
Output:
-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(256*(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)
\[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="giac")
Output:
sage0*x
Time = 10.66 (sec) , antiderivative size = 1124, normalized size of antiderivative = 7.59 \[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4),x)
Output:
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*(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 + ...
\[ \int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {-e^{6 d x +6 c} b +9 e^{4 d x +4 c} b -768 e^{3 d x +6 c} \left (\int \frac {e^{3 d x}}{e^{8 d x +8 c} b -4 e^{6 d x +6 c} b -16 e^{4 d x +4 c} a +6 e^{4 d x +4 c} b -4 e^{2 d x +2 c} b +b}d x \right ) a^{2} d -48 e^{3 d x +6 c} \left (\int \frac {e^{3 d x}}{e^{8 d x +8 c} b -4 e^{6 d x +6 c} b -16 e^{4 d x +4 c} a +6 e^{4 d x +4 c} b -4 e^{2 d x +2 c} b +b}d x \right ) a b d -768 e^{3 d x +4 c} \left (\int \frac {e^{d x}}{e^{8 d x +8 c} b -4 e^{6 d x +6 c} b -16 e^{4 d x +4 c} a +6 e^{4 d x +4 c} b -4 e^{2 d x +2 c} b +b}d x \right ) a^{2} d +144 e^{3 d x +4 c} \left (\int \frac {e^{d x}}{e^{8 d x +8 c} b -4 e^{6 d x +6 c} b -16 e^{4 d x +4 c} a +6 e^{4 d x +4 c} b -4 e^{2 d x +2 c} b +b}d x \right ) a b d +48 e^{3 d x +3 c} \left (\int \frac {1}{e^{11 d x +11 c} b -4 e^{9 d x +9 c} b -16 e^{7 d x +7 c} a +6 e^{7 d x +7 c} b -4 e^{5 d x +5 c} b +e^{3 d x +3 c} b}d x \right ) a b d -144 e^{3 d x +2 c} \left (\int \frac {1}{e^{9 d x +8 c} b -4 e^{7 d x +6 c} b -16 e^{5 d x +4 c} a +6 e^{5 d x +4 c} b -4 e^{3 d x +2 c} b +e^{d x} b}d x \right ) a b d +48 e^{2 d x +2 c} a +9 e^{2 d x +2 c} b +16 a -b}{24 e^{3 d x +3 c} b^{2} d} \] Input:
int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x)
Output:
( - e**(6*c + 6*d*x)*b + 9*e**(4*c + 4*d*x)*b - 768*e**(6*c + 3*d*x)*int(e **(3*d*x)/(e**(8*c + 8*d*x)*b - 4*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x) *a + 6*e**(4*c + 4*d*x)*b - 4*e**(2*c + 2*d*x)*b + b),x)*a**2*d - 48*e**(6 *c + 3*d*x)*int(e**(3*d*x)/(e**(8*c + 8*d*x)*b - 4*e**(6*c + 6*d*x)*b - 16 *e**(4*c + 4*d*x)*a + 6*e**(4*c + 4*d*x)*b - 4*e**(2*c + 2*d*x)*b + b),x)* a*b*d - 768*e**(4*c + 3*d*x)*int(e**(d*x)/(e**(8*c + 8*d*x)*b - 4*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x)*a + 6*e**(4*c + 4*d*x)*b - 4*e**(2*c + 2* d*x)*b + b),x)*a**2*d + 144*e**(4*c + 3*d*x)*int(e**(d*x)/(e**(8*c + 8*d*x )*b - 4*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x)*a + 6*e**(4*c + 4*d*x)*b - 4*e**(2*c + 2*d*x)*b + b),x)*a*b*d + 48*e**(3*c + 3*d*x)*int(1/(e**(11*c + 11*d*x)*b - 4*e**(9*c + 9*d*x)*b - 16*e**(7*c + 7*d*x)*a + 6*e**(7*c + 7*d*x)*b - 4*e**(5*c + 5*d*x)*b + e**(3*c + 3*d*x)*b),x)*a*b*d - 144*e**(2 *c + 3*d*x)*int(1/(e**(8*c + 9*d*x)*b - 4*e**(6*c + 7*d*x)*b - 16*e**(4*c + 5*d*x)*a + 6*e**(4*c + 5*d*x)*b - 4*e**(2*c + 3*d*x)*b + e**(d*x)*b),x)* a*b*d + 48*e**(2*c + 2*d*x)*a + 9*e**(2*c + 2*d*x)*b + 16*a - b)/(24*e**(3 *c + 3*d*x)*b**2*d)