3.4.16 \(\int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [316]

3.4.16.1 Optimal result

Integrand size = 23, antiderivative size = 108 \[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {(a-b)^3 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2} d}+\frac {\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^5(c+d x)}{5 b d} \]

output

(a^2-3*a*b+3*b^2)*sinh(d*x+c)/b^3/d-1/3*(a-3*b)*sinh(d*x+c)^3/b^2/d+1/5*si 
nh(d*x+c)^5/b/d-(a-b)^3*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/b^(7/2)/d/a^(1 
/2)
 

3.4.16.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {30 \sqrt {b} \left (8 a^2-22 a b+19 b^2\right ) \sinh (c+d x)+5 b^{3/2} (-4 a+9 b) \sinh (3 (c+d x))+\frac {3 \left (80 (a-b)^3 \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+\sqrt {a} b^{5/2} \sinh (5 (c+d x))\right )}{\sqrt {a}}}{240 b^{7/2} d} \]

input

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

output

(30*Sqrt[b]*(8*a^2 - 22*a*b + 19*b^2)*Sinh[c + d*x] + 5*b^(3/2)*(-4*a + 9* 
b)*Sinh[3*(c + d*x)] + (3*(80*(a - b)^3*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqr 
t[b]] + Sqrt[a]*b^(5/2)*Sinh[5*(c + d*x)]))/Sqrt[a])/(240*b^(7/2)*d)
 

3.4.16.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 300, 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.

\(\displaystyle \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i c+i d x)^7}{a-b \sin (i c+i d x)^2}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^3}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 300

\(\displaystyle \frac {\int \left (\frac {\sinh ^4(c+d x)}{b}-\frac {(a-3 b) \sinh ^2(c+d x)}{b^2}+\frac {a^2-3 b a+3 b^2}{b^3}+\frac {-a^3+3 b a^2-3 b^2 a+b^3}{b^3 \left (b \sinh ^2(c+d x)+a\right )}\right )d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3}-\frac {(a-b)^3 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(a-3 b) \sinh ^3(c+d x)}{3 b^2}+\frac {\sinh ^5(c+d x)}{5 b}}{d}\)

input

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

output

(-(((a - b)^3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) 
+ ((a^2 - 3*a*b + 3*b^2)*Sinh[c + d*x])/b^3 - ((a - 3*b)*Sinh[c + d*x]^3)/ 
(3*b^2) + Sinh[c + d*x]^5/(5*b))/d
 

3.4.16.3.1 Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3669

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f   S 
ubst[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]
 

3.4.16.4 Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14

\[\frac {\frac {\frac {b^{2} \sinh \left (d x +c \right )^{5}}{5}-\frac {a b \sinh \left (d x +c \right )^{3}}{3}+b^{2} \sinh \left (d x +c \right )^{3}+\sinh \left (d x +c \right ) a^{2}-3 a b \sinh \left (d x +c \right )+3 b^{2} \sinh \left (d x +c \right )}{b^{3}}+\frac {\left (-a^{3}+3 a^{2} b -3 a \,b^{2}+b^{3}\right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}}{d}\]

input

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

output

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

3.4.16.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1548 vs. \(2 (96) = 192\).

Time = 0.30 (sec) , antiderivative size = 3066, normalized size of antiderivative = 28.39 \[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \]

input

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

output

[1/480*(3*a*b^3*cosh(d*x + c)^10 + 30*a*b^3*cosh(d*x + c)*sinh(d*x + c)^9 
+ 3*a*b^3*sinh(d*x + c)^10 - 5*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^8 + 5*( 
27*a*b^3*cosh(d*x + c)^2 - 4*a^2*b^2 + 9*a*b^3)*sinh(d*x + c)^8 + 40*(9*a* 
b^3*cosh(d*x + c)^3 - (4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 
 + 30*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^6 + 10*(63*a*b^3*cos 
h(d*x + c)^4 + 24*a^3*b - 66*a^2*b^2 + 57*a*b^3 - 14*(4*a^2*b^2 - 9*a*b^3) 
*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*a*b^3*cosh(d*x + c)^5 - 70*(4*a 
^2*b^2 - 9*a*b^3)*cosh(d*x + c)^3 + 45*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*c 
osh(d*x + c))*sinh(d*x + c)^5 - 30*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh( 
d*x + c)^4 + 10*(63*a*b^3*cosh(d*x + c)^6 - 35*(4*a^2*b^2 - 9*a*b^3)*cosh( 
d*x + c)^4 - 24*a^3*b + 66*a^2*b^2 - 57*a*b^3 + 45*(8*a^3*b - 22*a^2*b^2 + 
 19*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 3*a*b^3 + 40*(9*a*b^3*cosh(d 
*x + c)^7 - 7*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 15*(8*a^3*b - 22*a^2 
*b^2 + 19*a*b^3)*cosh(d*x + c)^3 - 3*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cos 
h(d*x + c))*sinh(d*x + c)^3 + 5*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2 + 5* 
(27*a*b^3*cosh(d*x + c)^8 - 28*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^6 + 90* 
(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^4 + 4*a^2*b^2 - 9*a*b^3 - 
36*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24 
0*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(a^3 - 3*a^2*b + 3* 
a*b^2 - b^3)*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^3 - 3*a^2*b + 3*a*b^...
 

3.4.16.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \]

input

integrate(cosh(d*x+c)**7/(a+b*sinh(d*x+c)**2),x)
 

output

Timed out
 

3.4.16.7 Maxima [F]

\[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]

input

integrate(cosh(d*x+c)^7/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
 

output

1/480*(3*b^2*e^(10*d*x + 10*c) - 3*b^2 - 5*(4*a*b*e^(8*c) - 9*b^2*e^(8*c)) 
*e^(8*d*x) + 30*(8*a^2*e^(6*c) - 22*a*b*e^(6*c) + 19*b^2*e^(6*c))*e^(6*d*x 
) - 30*(8*a^2*e^(4*c) - 22*a*b*e^(4*c) + 19*b^2*e^(4*c))*e^(4*d*x) + 5*(4* 
a*b*e^(2*c) - 9*b^2*e^(2*c))*e^(2*d*x))*e^(-5*d*x - 5*c)/(b^3*d) - 1/128*i 
ntegrate(256*((a^3*e^(3*c) - 3*a^2*b*e^(3*c) + 3*a*b^2*e^(3*c) - b^3*e^(3* 
c))*e^(3*d*x) + (a^3*e^c - 3*a^2*b*e^c + 3*a*b^2*e^c - b^3*e^c)*e^(d*x))/( 
b^4*e^(4*d*x + 4*c) + b^4 + 2*(2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), 
x)
 

3.4.16.8 Giac [F]

\[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]

input

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

output

sage0*x
 

3.4.16.9 Mupad [B] (verification not implemented)

Time = 2.29 (sec) , antiderivative size = 954, normalized size of antiderivative = 8.83 \[ \int \frac {\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,b\,d}-\frac {{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,{\left (a-b\right )}^3\,\sqrt {a\,b^7\,d^2}}{2\,a\,b^3\,d\,\sqrt {{\left (a-b\right )}^6}}\right )+2\,\mathrm {atan}\left (\frac {a\,b^8\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (12\,a^3\,b^5\,d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}-8\,a^2\,b^6\,d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}-8\,a^4\,b^4\,d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}+2\,a^5\,b^3\,d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}+2\,a\,b^7\,d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}\right )}{a^2\,b^{15}\,d^2\,{\left (a-b\right )}^3}-\frac {2\,\left (a^7\,\sqrt {a\,b^7\,d^2}-b^7\,\sqrt {a\,b^7\,d^2}+7\,a\,b^6\,\sqrt {a\,b^7\,d^2}-7\,a^6\,b\,\sqrt {a\,b^7\,d^2}-21\,a^2\,b^5\,\sqrt {a\,b^7\,d^2}+35\,a^3\,b^4\,\sqrt {a\,b^7\,d^2}-35\,a^4\,b^3\,\sqrt {a\,b^7\,d^2}+21\,a^5\,b^2\,\sqrt {a\,b^7\,d^2}\right )}{a^2\,b^{11}\,d\,\sqrt {{\left (a-b\right )}^6}\,\sqrt {a\,b^7\,d^2}}\right )\,\sqrt {a\,b^7\,d^2}}{4\,a^4-16\,a^3\,b+24\,a^2\,b^2-16\,a\,b^3+4\,b^4}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^7\,\sqrt {a\,b^7\,d^2}-b^7\,\sqrt {a\,b^7\,d^2}+7\,a\,b^6\,\sqrt {a\,b^7\,d^2}-7\,a^6\,b\,\sqrt {a\,b^7\,d^2}-21\,a^2\,b^5\,\sqrt {a\,b^7\,d^2}+35\,a^3\,b^4\,\sqrt {a\,b^7\,d^2}-35\,a^4\,b^3\,\sqrt {a\,b^7\,d^2}+21\,a^5\,b^2\,\sqrt {a\,b^7\,d^2}\right )}{a\,b^3\,d\,\sqrt {{\left (a-b\right )}^6}\,\left (4\,a^4-16\,a^3\,b+24\,a^2\,b^2-16\,a\,b^3+4\,b^4\right )}\right )\right )\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}}{2\,\sqrt {a\,b^7\,d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-22\,a\,b+19\,b^2\right )}{16\,b^3\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a^2-22\,a\,b+19\,b^2\right )}{16\,b^3\,d}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (4\,a-9\,b\right )}{96\,b^2\,d}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (4\,a-9\,b\right )}{96\,b^2\,d} \]

input

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

output

exp(5*c + 5*d*x)/(160*b*d) - exp(- 5*c - 5*d*x)/(160*b*d) - ((2*atan((exp( 
d*x)*exp(c)*(a - b)^3*(a*b^7*d^2)^(1/2))/(2*a*b^3*d*((a - b)^6)^(1/2))) + 
2*atan((a*b^8*exp(d*x)*exp(c)*((4*(12*a^3*b^5*d*(a^6 - 6*a^5*b - 6*a*b^5 + 
 b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)^(1/2) - 8*a^2*b^6*d*(a^6 - 6* 
a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)^(1/2) - 8*a^ 
4*b^4*d*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4* 
b^2)^(1/2) + 2*a^5*b^3*d*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20* 
a^3*b^3 + 15*a^4*b^2)^(1/2) + 2*a*b^7*d*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 1 
5*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)^(1/2)))/(a^2*b^15*d^2*(a - b)^3) - (2 
*(a^7*(a*b^7*d^2)^(1/2) - b^7*(a*b^7*d^2)^(1/2) + 7*a*b^6*(a*b^7*d^2)^(1/2 
) - 7*a^6*b*(a*b^7*d^2)^(1/2) - 21*a^2*b^5*(a*b^7*d^2)^(1/2) + 35*a^3*b^4* 
(a*b^7*d^2)^(1/2) - 35*a^4*b^3*(a*b^7*d^2)^(1/2) + 21*a^5*b^2*(a*b^7*d^2)^ 
(1/2)))/(a^2*b^11*d*((a - b)^6)^(1/2)*(a*b^7*d^2)^(1/2)))*(a*b^7*d^2)^(1/2 
))/(4*a^4 - 16*a^3*b - 16*a*b^3 + 4*b^4 + 24*a^2*b^2) + (2*exp(3*c)*exp(3* 
d*x)*(a^7*(a*b^7*d^2)^(1/2) - b^7*(a*b^7*d^2)^(1/2) + 7*a*b^6*(a*b^7*d^2)^ 
(1/2) - 7*a^6*b*(a*b^7*d^2)^(1/2) - 21*a^2*b^5*(a*b^7*d^2)^(1/2) + 35*a^3* 
b^4*(a*b^7*d^2)^(1/2) - 35*a^4*b^3*(a*b^7*d^2)^(1/2) + 21*a^5*b^2*(a*b^7*d 
^2)^(1/2)))/(a*b^3*d*((a - b)^6)^(1/2)*(4*a^4 - 16*a^3*b - 16*a*b^3 + 4*b^ 
4 + 24*a^2*b^2))))*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^ 
3 + 15*a^4*b^2)^(1/2))/(2*(a*b^7*d^2)^(1/2)) + (exp(c + d*x)*(8*a^2 - 2...