Integrand size = 18, antiderivative size = 202 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {6 e^3 F^{c (a+b x)} \cosh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 F^{c (a+b x)} \log (F) \sinh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^3(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)} \] Output:
-6*e^3*F^(c*(b*x+a))*cosh(e*x+d)/(9*e^4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln( F)^4)+6*b*c*e^2*F^(c*(b*x+a))*ln(F)*sinh(e*x+d)/(9*e^4-10*b^2*c^2*e^2*ln(F )^2+b^4*c^4*ln(F)^4)+3*e*F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^2/(9*e^2-b^ 2*c^2*ln(F)^2)-b*c*F^(c*(b*x+a))*ln(F)*sinh(e*x+d)^3/(9*e^2-b^2*c^2*ln(F)^ 2)
Time = 0.47 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (3 \cosh (3 (d+e x)) \left (e^3-b^2 c^2 e \log ^2(F)\right )+3 \cosh (d+e x) \left (-9 e^3+b^2 c^2 e \log ^2(F)\right )+2 b c \log (F) \left (13 e^2-b^2 c^2 \log ^2(F)+\cosh (2 (d+e x)) \left (-e^2+b^2 c^2 \log ^2(F)\right )\right ) \sinh (d+e x)\right )}{4 \left (9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \] Input:
Integrate[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
Output:
(F^(c*(a + b*x))*(3*Cosh[3*(d + e*x)]*(e^3 - b^2*c^2*e*Log[F]^2) + 3*Cosh[ d + e*x]*(-9*e^3 + b^2*c^2*e*Log[F]^2) + 2*b*c*Log[F]*(13*e^2 - b^2*c^2*Lo g[F]^2 + Cosh[2*(d + e*x)]*(-e^2 + b^2*c^2*Log[F]^2))*Sinh[d + e*x]))/(4*( 9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
Time = 0.44 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5999, 5997}
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.
\(\displaystyle \int \sinh ^3(d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 5999 |
\(\displaystyle -\frac {6 e^2 \int F^{c (a+b x)} \sinh (d+e x)dx}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}\) |
\(\Big \downarrow \) 5997 |
\(\displaystyle -\frac {b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {6 e^2 \left (\frac {e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}\right )}{9 e^2-b^2 c^2 \log ^2(F)}\) |
Input:
Int[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
Output:
(3*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F ]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2 - b^2*c^2*Log[F ]^2) - (6*e^2*((e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)))/(9 *e^2 - b^2*c^2*Log[F]^2)
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] : > Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2 )), x] + Simp[e*F^(c*(a + b*x))*(Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)), x ] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symb ol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Sinh[d + e*x]^n/(e^2*n^2 - b^2*c ^2*Log[F]^2)), x] + (Simp[e*n*F^(c*(a + b*x))*Cosh[d + e*x]*(Sinh[d + e*x]^ (n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] - Simp[n*(n - 1)*(e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)) Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n , 1]
Time = 1.36 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(-\frac {3 F^{c \left (b x +a \right )} \left (\left (\ln \left (F \right )^{2} b^{2} c^{2} e -e^{3}\right ) \cosh \left (3 e x +3 d \right )+\frac {\left (-\ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right ) b c \,e^{2}\right ) \sinh \left (3 e x +3 d \right )}{3}+\left (b c \ln \left (F \right )-3 e \right ) \left (b c \ln \left (F \right )+3 e \right ) \left (\sinh \left (e x +d \right ) \ln \left (F \right ) b c -e \cosh \left (e x +d \right )\right )\right )}{4 \left (9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}\right )}\) | \(147\) |
risch | \(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}-3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}+3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-\ln \left (F \right ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}-\ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}+27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}+3 e^{3} {\mathrm e}^{6 e x +6 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e -27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}-27 e^{3} {\mathrm e}^{4 e x +4 d}+\ln \left (F \right ) b c \,e^{2}-27 e^{3} {\mathrm e}^{2 e x +2 d}+3 e^{3}\right ) {\mathrm e}^{-3 e x -3 d} F^{c \left (b x +a \right )}}{8 \left (b c \ln \left (F \right )-e \right ) \left (b c \ln \left (F \right )-3 e \right ) \left (e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+3 e \right )}\) | \(326\) |
orering | \(\frac {4 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{3}}{9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{3}+3 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )\right )}{9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {4 b c \ln \left (F \right ) \left (F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \sinh \left (e x +d \right )^{3}+6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+6 F^{c \left (b x +a \right )} \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+3 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{3} e^{2}\right )}{9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \sinh \left (e x +d \right )^{3}+9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+9 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{3} e^{2}+6 F^{c \left (b x +a \right )} e^{3} \cosh \left (e x +d \right )^{3}+21 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e^{3} \cosh \left (e x +d \right )}{9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}\) | \(537\) |
Input:
int(F^(c*(b*x+a))*sinh(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-3/4*F^(c*(b*x+a))*((ln(F)^2*b^2*c^2*e-e^3)*cosh(3*e*x+3*d)+1/3*(-ln(F)^3* b^3*c^3+ln(F)*b*c*e^2)*sinh(3*e*x+3*d)+(b*c*ln(F)-3*e)*(b*c*ln(F)+3*e)*(si nh(e*x+d)*ln(F)*b*c-e*cosh(e*x+d)))/(9*e^4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^4* ln(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 2228 vs. \(2 (199) = 398\).
Time = 0.17 (sec) , antiderivative size = 2228, normalized size of antiderivative = 11.03 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="fricas")
Output:
1/8*((3*e^3*cosh(e*x + d)^6 - 27*e^3*cosh(e*x + d)^4 + (b^3*c^3*log(F)^3 - 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c ^3*cosh(e*x + d)*log(F)^3 - 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2*c osh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 - 27*e^3*cosh(e *x + d)^2 + 3*(15*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^2 - b^3*c ^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^2 - 9*b*c*e^2)*log(F))*sinh(e*x + d)^4 + (b^3* c^3*cosh(e*x + d)^6 - 3*b^3*c^3*cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^ 2 - b^3*c^3)*log(F)^3 + 4*(15*e^3*cosh(e*x + d)^3 - 27*e^3*cosh(e*x + d) + (5*b^3*c^3*cosh(e*x + d)^3 - 3*b^3*c^3*cosh(e*x + d))*log(F)^3 - 3*(5*b^2 *c^2*e*cosh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*co sh(e*x + d)^3 - 27*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3 + 3*e^3 - 3*(b^2*c^2*e*cosh(e*x + d)^6 - b^2*c^2*e*cosh(e*x + d)^4 - b^2*c^2*e*cos h(e*x + d)^2 + b^2*c^2*e)*log(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 - 54*e^3*co sh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^4 - 6*b^3*c^3*cosh(e*x + d)^2 + b ^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^4 - 6*b^2*c^2*e*cos h(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 - 54*b*c*e ^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c*e^2*cosh(e* x + d)^6 - 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*cosh(e*x + d)^2 - b*c*e ^2)*log(F) + 6*(3*e^3*cosh(e*x + d)^5 - 18*e^3*cosh(e*x + d)^3 - 9*e^3*...
Leaf count of result is larger than twice the leaf count of optimal. 1525 vs. \(2 (199) = 398\).
Time = 3.61 (sec) , antiderivative size = 1525, normalized size of antiderivative = 7.55 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*sinh(e*x+d)**3,x)
Output:
Piecewise((x*sinh(d)**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d)**3, Eq( b, 0) & Eq(e, 0)), (x*sinh(d)**3, Eq(c, 0) & Eq(e, 0)), (-3*F**(a*c + b*c* x)*x*sinh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) - d)*cosh(b*c*x*log(F) - d)**2/8 - 3*F**(a*c + b*c*x)*x*cosh(b*c*x*log(F) - d)**3/8 + F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**3/(8*b*c*log(F)) - 3* F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/(4*b*c*l og(F)) + 3*F**(a*c + b*c*x)*cosh(b*c*x*log(F) - d)**3/(8*b*c*log(F)), Eq(e , -b*c*log(F))), (-F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**3/8 + 3*F* *(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**2*cosh(b*c*x*log(F)/3 - d)/8 - 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)*cosh(b*c*x*log(F)/3 - d)**2/ 8 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/3 - d)**3/8 - 9*F**(a*c + b*c*x)* sinh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)) + 3*F**(a*c + b*c*x)*sinh(b*c*x *log(F)/3 - d)**2*cosh(b*c*x*log(F)/3 - d)/(4*b*c*log(F)) - F**(a*c + b*c* x)*cosh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)), Eq(e, -b*c*log(F)/3)), (F** (a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 + d)**3/8 - 3*F**(a*c + b*c*x)*x*sinh( b*c*x*log(F)/3 + d)**2*cosh(b*c*x*log(F)/3 + d)/8 + 3*F**(a*c + b*c*x)*x*s inh(b*c*x*log(F)/3 + d)*cosh(b*c*x*log(F)/3 + d)**2/8 - F**(a*c + b*c*x)*x *cosh(b*c*x*log(F)/3 + d)**3/8 - F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + d) **3/(8*b*c*log(F)) + 3*F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + d)**2*cos...
Time = 0.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.66 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 3 \, e\right )}} - \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \, {\left (b c \log \left (F\right ) + e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \] Input:
integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="maxima")
Output:
1/8*F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) - 3/8*F^(a*c )*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + 3/8*F^(a*c)*e^(b*c*x*log(F ) - e*x)/(b*c*e^d*log(F) - e*e^d) - 1/8*F^(a*c)*e^(b*c*x*log(F) - 3*e*x)/( b*c*e^(3*d)*log(F) - 3*e*e^(3*d))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1211, normalized size of antiderivative = 6.00 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="giac")
Output:
1/4*(2*(b*c*log(abs(F)) + 3*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1 /2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(ab s(F)) + 3*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2* pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4 *(b*c*log(abs(F)) + 3*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)* x + 3*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*s gn(F) - 1/2*I*pi*a*c)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*e) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn (F) + 1/2*I*pi*a*c)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 3/4*(2*(b *c*log(abs(F)) + e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*s gn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + e)^ 2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/ 2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs (F)) + e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 3*I*(-I* e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi *a*c)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e) + I*e^( -1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a *c)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e))*e^(a*c* log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 3/4*(2*(b*c*log(abs(F)) - ...
Time = 2.58 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {F^{a\,c+b\,c\,x}\,\left (-b^3\,c^3\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^2-6\,b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )+7\,b\,c\,e^2\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,\ln \left (F\right )+6\,e^3\,{\mathrm {cosh}\left (d+e\,x\right )}^3-9\,e^3\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \] Input:
int(F^(c*(a + b*x))*sinh(d + e*x)^3,x)
Output:
-(F^(a*c + b*c*x)*(6*e^3*cosh(d + e*x)^3 - 9*e^3*cosh(d + e*x)*sinh(d + e* x)^2 - b^3*c^3*sinh(d + e*x)^3*log(F)^3 + 7*b*c*e^2*sinh(d + e*x)^3*log(F) + 3*b^2*c^2*e*cosh(d + e*x)*sinh(d + e*x)^2*log(F)^2 - 6*b*c*e^2*cosh(d + e*x)^2*sinh(d + e*x)*log(F)))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2* log(F)^2)
\[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \sinh \left (e x +d \right )^{3}d x \right ) \] Input:
int(F^(c*(b*x+a))*sinh(e*x+d)^3,x)
Output:
f**(a*c)*int(f**(b*c*x)*sinh(d + e*x)**3,x)