3.208 \(\int \sinh ^3(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=183 \[ \frac {b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac {5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}-\frac {3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac {(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{15}(c+d x)}{15 d}-\frac {7 b^3 \cosh ^{13}(c+d x)}{13 d} \]

[Out]

-(a+b)^3*cosh(d*x+c)/d+1/3*(a+b)^2*(a+7*b)*cosh(d*x+c)^3/d-3/5*b*(a+b)*(3*a+7*b)*cosh(d*x+c)^5/d+1/7*b*(3*a^2+
30*a*b+35*b^2)*cosh(d*x+c)^7/d-5/9*b^2*(3*a+7*b)*cosh(d*x+c)^9/d+3/11*b^2*(a+7*b)*cosh(d*x+c)^11/d-7/13*b^3*co
sh(d*x+c)^13/d+1/15*b^3*cosh(d*x+c)^15/d

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Rubi [A]  time = 0.18, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3215, 1153} \[ \frac {b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac {5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}-\frac {3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac {(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{15}(c+d x)}{15 d}-\frac {7 b^3 \cosh ^{13}(c+d x)}{13 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-(((a + b)^3*Cosh[c + d*x])/d) + ((a + b)^2*(a + 7*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*(a + b)*(3*a + 7*b)*Cosh[c
+ d*x]^5)/(5*d) + (b*(3*a^2 + 30*a*b + 35*b^2)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*(3*a + 7*b)*Cosh[c + d*x]^9)/(
9*d) + (3*b^2*(a + 7*b)*Cosh[c + d*x]^11)/(11*d) - (7*b^3*Cosh[c + d*x]^13)/(13*d) + (b^3*Cosh[c + d*x]^15)/(1
5*d)

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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[p, 0] && IGtQ[q, -2]

Rule 3215

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 \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b-2 b x^2+b x^4\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left ((a+b)^3-(a+b)^2 (a+7 b) x^2+3 b (a+b) (3 a+7 b) x^4-b \left (3 a^2+30 a b+35 b^2\right ) x^6+5 b^2 (3 a+7 b) x^8-3 b^2 (a+7 b) x^{10}+7 b^3 x^{12}-b^3 x^{14}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac {b \left (3 a^2+30 a b+35 b^2\right ) \cosh ^7(c+d x)}{7 d}-\frac {5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}+\frac {3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac {7 b^3 \cosh ^{13}(c+d x)}{13 d}+\frac {b^3 \cosh ^{15}(c+d x)}{15 d}\\ \end {align*}

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Mathematica [A]  time = 2.46, size = 185, normalized size = 1.01 \[ \frac {b \left (-27027 \left (1792 a^2+2640 a b+1001 b^2\right ) \cosh (5 (c+d x))+19305 \left (256 a^2+880 a b+455 b^2\right ) \cosh (7 (c+d x))-7 b (715 (528 a+455 b) \cosh (9 (c+d x))-1755 (16 a+35 b) \cosh (11 (c+d x))+7425 b \cosh (13 (c+d x))-429 b \cosh (15 (c+d x)))\right )-135135 \left (4096 a^3+8960 a^2 b+7392 a b^2+2145 b^3\right ) \cosh (c+d x)+15015 \left (4096 a^3+16128 a^2 b+15840 a b^2+5005 b^3\right ) \cosh (3 (c+d x))}{738017280 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-135135*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*Cosh[c + d*x] + 15015*(4096*a^3 + 16128*a^2*b + 15840
*a*b^2 + 5005*b^3)*Cosh[3*(c + d*x)] + b*(-27027*(1792*a^2 + 2640*a*b + 1001*b^2)*Cosh[5*(c + d*x)] + 19305*(2
56*a^2 + 880*a*b + 455*b^2)*Cosh[7*(c + d*x)] - 7*b*(715*(528*a + 455*b)*Cosh[9*(c + d*x)] - 1755*(16*a + 35*b
)*Cosh[11*(c + d*x)] + 7425*b*Cosh[13*(c + d*x)] - 429*b*Cosh[15*(c + d*x)])))/(738017280*d)

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fricas [B]  time = 1.21, size = 795, normalized size = 4.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/738017280*(3003*b^3*cosh(d*x + c)^15 + 45045*b^3*cosh(d*x + c)*sinh(d*x + c)^14 - 51975*b^3*cosh(d*x + c)^13
+ 15015*(91*b^3*cosh(d*x + c)^3 - 45*b^3*cosh(d*x + c))*sinh(d*x + c)^12 + 12285*(16*a*b^2 + 35*b^3)*cosh(d*x
+ c)^11 + 9009*(1001*b^3*cosh(d*x + c)^5 - 1650*b^3*cosh(d*x + c)^3 + 15*(16*a*b^2 + 35*b^3)*cosh(d*x + c))*s
inh(d*x + c)^10 - 5005*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^9 + 45045*(429*b^3*cosh(d*x + c)^7 - 1485*b^3*cosh(
d*x + c)^5 + 45*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^3 - (528*a*b^2 + 455*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 1
9305*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^7 + 15015*(1001*b^3*cosh(d*x + c)^9 - 5940*b^3*cosh(d*x +
c)^7 + 378*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^5 - 28*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^3 + 9*(256*a^2*b + 88
0*a*b^2 + 455*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 27027*(1792*a^2*b + 2640*a*b^2 + 1001*b^3)*cosh(d*x + c)^5
+ 45045*(91*b^3*cosh(d*x + c)^11 - 825*b^3*cosh(d*x + c)^9 + 90*(16*a*b^2 + 35*b^3)*cosh(d*x + c)^7 - 14*(528
*a*b^2 + 455*b^3)*cosh(d*x + c)^5 + 15*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^3 - 3*(1792*a^2*b + 264
0*a*b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 15015*(4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*c
osh(d*x + c)^3 + 45045*(7*b^3*cosh(d*x + c)^13 - 90*b^3*cosh(d*x + c)^11 + 15*(16*a*b^2 + 35*b^3)*cosh(d*x + c
)^9 - 4*(528*a*b^2 + 455*b^3)*cosh(d*x + c)^7 + 9*(256*a^2*b + 880*a*b^2 + 455*b^3)*cosh(d*x + c)^5 - 6*(1792*
a^2*b + 2640*a*b^2 + 1001*b^3)*cosh(d*x + c)^3 + (4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*cosh(d*x +
c))*sinh(d*x + c)^2 - 135135*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*cosh(d*x + c))/d

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giac [B]  time = 0.40, size = 446, normalized size = 2.44 \[ \frac {b^{3} e^{\left (15 \, d x + 15 \, c\right )}}{491520 \, d} - \frac {15 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )}}{425984 \, d} - \frac {15 \, b^{3} e^{\left (-13 \, d x - 13 \, c\right )}}{425984 \, d} + \frac {b^{3} e^{\left (-15 \, d x - 15 \, c\right )}}{491520 \, d} + \frac {3 \, {\left (16 \, a b^{2} + 35 \, b^{3}\right )} e^{\left (11 \, d x + 11 \, c\right )}}{360448 \, d} - \frac {{\left (528 \, a b^{2} + 455 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{294912 \, d} + \frac {3 \, {\left (256 \, a^{2} b + 880 \, a b^{2} + 455 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{229376 \, d} - \frac {3 \, {\left (1792 \, a^{2} b + 2640 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{163840 \, d} + \frac {{\left (4096 \, a^{3} + 16128 \, a^{2} b + 15840 \, a b^{2} + 5005 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{98304 \, d} - \frac {3 \, {\left (4096 \, a^{3} + 8960 \, a^{2} b + 7392 \, a b^{2} + 2145 \, b^{3}\right )} e^{\left (d x + c\right )}}{32768 \, d} - \frac {3 \, {\left (4096 \, a^{3} + 8960 \, a^{2} b + 7392 \, a b^{2} + 2145 \, b^{3}\right )} e^{\left (-d x - c\right )}}{32768 \, d} + \frac {{\left (4096 \, a^{3} + 16128 \, a^{2} b + 15840 \, a b^{2} + 5005 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{98304 \, d} - \frac {3 \, {\left (1792 \, a^{2} b + 2640 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{163840 \, d} + \frac {3 \, {\left (256 \, a^{2} b + 880 \, a b^{2} + 455 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{229376 \, d} - \frac {{\left (528 \, a b^{2} + 455 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{294912 \, d} + \frac {3 \, {\left (16 \, a b^{2} + 35 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{360448 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/491520*b^3*e^(15*d*x + 15*c)/d - 15/425984*b^3*e^(13*d*x + 13*c)/d - 15/425984*b^3*e^(-13*d*x - 13*c)/d + 1/
491520*b^3*e^(-15*d*x - 15*c)/d + 3/360448*(16*a*b^2 + 35*b^3)*e^(11*d*x + 11*c)/d - 1/294912*(528*a*b^2 + 455
*b^3)*e^(9*d*x + 9*c)/d + 3/229376*(256*a^2*b + 880*a*b^2 + 455*b^3)*e^(7*d*x + 7*c)/d - 3/163840*(1792*a^2*b
+ 2640*a*b^2 + 1001*b^3)*e^(5*d*x + 5*c)/d + 1/98304*(4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*e^(3*d*
x + 3*c)/d - 3/32768*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*e^(d*x + c)/d - 3/32768*(4096*a^3 + 8960*
a^2*b + 7392*a*b^2 + 2145*b^3)*e^(-d*x - c)/d + 1/98304*(4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*e^(-
3*d*x - 3*c)/d - 3/163840*(1792*a^2*b + 2640*a*b^2 + 1001*b^3)*e^(-5*d*x - 5*c)/d + 3/229376*(256*a^2*b + 880*
a*b^2 + 455*b^3)*e^(-7*d*x - 7*c)/d - 1/294912*(528*a*b^2 + 455*b^3)*e^(-9*d*x - 9*c)/d + 3/360448*(16*a*b^2 +
35*b^3)*e^(-11*d*x - 11*c)/d

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maple [A]  time = 0.13, size = 218, normalized size = 1.19 \[ \frac {b^{3} \left (-\frac {2048}{6435}+\frac {\left (\sinh ^{14}\left (d x +c \right )\right )}{15}-\frac {14 \left (\sinh ^{12}\left (d x +c \right )\right )}{195}+\frac {56 \left (\sinh ^{10}\left (d x +c \right )\right )}{715}-\frac {112 \left (\sinh ^{8}\left (d x +c \right )\right )}{1287}+\frac {128 \left (\sinh ^{6}\left (d x +c \right )\right )}{1287}-\frac {256 \left (\sinh ^{4}\left (d x +c \right )\right )}{2145}+\frac {1024 \left (\sinh ^{2}\left (d x +c \right )\right )}{6435}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {256}{693}+\frac {\left (\sinh ^{10}\left (d x +c \right )\right )}{11}-\frac {10 \left (\sinh ^{8}\left (d x +c \right )\right )}{99}+\frac {80 \left (\sinh ^{6}\left (d x +c \right )\right )}{693}-\frac {32 \left (\sinh ^{4}\left (d x +c \right )\right )}{231}+\frac {128 \left (\sinh ^{2}\left (d x +c \right )\right )}{693}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{35}\right ) \cosh \left (d x +c \right )+a^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(b^3*(-2048/6435+1/15*sinh(d*x+c)^14-14/195*sinh(d*x+c)^12+56/715*sinh(d*x+c)^10-112/1287*sinh(d*x+c)^8+12
8/1287*sinh(d*x+c)^6-256/2145*sinh(d*x+c)^4+1024/6435*sinh(d*x+c)^2)*cosh(d*x+c)+3*a*b^2*(-256/693+1/11*sinh(d
*x+c)^10-10/99*sinh(d*x+c)^8+80/693*sinh(d*x+c)^6-32/231*sinh(d*x+c)^4+128/693*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^
2*b*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c)+a^3*(-2/3+1/3*sinh(d*x+c)^2)*
cosh(d*x+c))

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maxima [B]  time = 0.34, size = 501, normalized size = 2.74 \[ -\frac {1}{210862080} \, b^{3} {\left (\frac {{\left (7425 \, e^{\left (-2 \, d x - 2 \, c\right )} - 61425 \, e^{\left (-4 \, d x - 4 \, c\right )} + 325325 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1254825 \, e^{\left (-8 \, d x - 8 \, c\right )} + 3864861 \, e^{\left (-10 \, d x - 10 \, c\right )} - 10735725 \, e^{\left (-12 \, d x - 12 \, c\right )} + 41409225 \, e^{\left (-14 \, d x - 14 \, c\right )} - 429\right )} e^{\left (15 \, d x + 15 \, c\right )}}{d} + \frac {41409225 \, e^{\left (-d x - c\right )} - 10735725 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3864861 \, e^{\left (-5 \, d x - 5 \, c\right )} - 1254825 \, e^{\left (-7 \, d x - 7 \, c\right )} + 325325 \, e^{\left (-9 \, d x - 9 \, c\right )} - 61425 \, e^{\left (-11 \, d x - 11 \, c\right )} + 7425 \, e^{\left (-13 \, d x - 13 \, c\right )} - 429 \, e^{\left (-15 \, d x - 15 \, c\right )}}{d}\right )} - \frac {1}{473088} \, a b^{2} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a^{2} b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/210862080*b^3*((7425*e^(-2*d*x - 2*c) - 61425*e^(-4*d*x - 4*c) + 325325*e^(-6*d*x - 6*c) - 1254825*e^(-8*d*
x - 8*c) + 3864861*e^(-10*d*x - 10*c) - 10735725*e^(-12*d*x - 12*c) + 41409225*e^(-14*d*x - 14*c) - 429)*e^(15
*d*x + 15*c)/d + (41409225*e^(-d*x - c) - 10735725*e^(-3*d*x - 3*c) + 3864861*e^(-5*d*x - 5*c) - 1254825*e^(-7
*d*x - 7*c) + 325325*e^(-9*d*x - 9*c) - 61425*e^(-11*d*x - 11*c) + 7425*e^(-13*d*x - 13*c) - 429*e^(-15*d*x -
15*c))/d) - 1/473088*a*b^2*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76230*e^(
-8*d*x - 8*c) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x -
3*c) + 22869*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 3/
4480*a^2*b*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225
*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/24*a^3*(e^(3*d*x + 3*c
)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

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mupad [B]  time = 1.33, size = 266, normalized size = 1.45 \[ -\frac {-\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^3\,\mathrm {cosh}\left (c+d\,x\right )-\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {9\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {5\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {30\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+6\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-5\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )-\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{15}}{15}+\frac {7\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {21\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {35\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-5\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7+\frac {21\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {7\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^3*(a + b*sinh(c + d*x)^4)^3,x)

[Out]

-(a^3*cosh(c + d*x) + b^3*cosh(c + d*x) - (a^3*cosh(c + d*x)^3)/3 - (7*b^3*cosh(c + d*x)^3)/3 + (21*b^3*cosh(c
+ d*x)^5)/5 - 5*b^3*cosh(c + d*x)^7 + (35*b^3*cosh(c + d*x)^9)/9 - (21*b^3*cosh(c + d*x)^11)/11 + (7*b^3*cosh
(c + d*x)^13)/13 - (b^3*cosh(c + d*x)^15)/15 - 5*a*b^2*cosh(c + d*x)^3 - 3*a^2*b*cosh(c + d*x)^3 + 6*a*b^2*cos
h(c + d*x)^5 + (9*a^2*b*cosh(c + d*x)^5)/5 - (30*a*b^2*cosh(c + d*x)^7)/7 - (3*a^2*b*cosh(c + d*x)^7)/7 + (5*a
*b^2*cosh(c + d*x)^9)/3 - (3*a*b^2*cosh(c + d*x)^11)/11 + 3*a*b^2*cosh(c + d*x) + 3*a^2*b*cosh(c + d*x))/d

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sympy [A]  time = 139.98, size = 484, normalized size = 2.64 \[ \begin {cases} \frac {a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {6 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {48 a^{2} b \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 a b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {10 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {16 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {96 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{21 d} - \frac {256 a b^{2} \cosh ^{11}{\left (c + d x \right )}}{231 d} + \frac {b^{3} \sinh ^{14}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {14 b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {56 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{d} + \frac {128 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{9 d} - \frac {256 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{33 d} + \frac {1024 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{13}{\left (c + d x \right )}}{429 d} - \frac {2048 b^{3} \cosh ^{15}{\left (c + d x \right )}}{6435 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right )^{3} \sinh ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Piecewise((a**3*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**3*cosh(c + d*x)**3/(3*d) + 3*a**2*b*sinh(c + d*x)**6*c
osh(c + d*x)/d - 6*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 24*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**5/(5
*d) - 48*a**2*b*cosh(c + d*x)**7/(35*d) + 3*a*b**2*sinh(c + d*x)**10*cosh(c + d*x)/d - 10*a*b**2*sinh(c + d*x)
**8*cosh(c + d*x)**3/d + 16*a*b**2*sinh(c + d*x)**6*cosh(c + d*x)**5/d - 96*a*b**2*sinh(c + d*x)**4*cosh(c + d
*x)**7/(7*d) + 128*a*b**2*sinh(c + d*x)**2*cosh(c + d*x)**9/(21*d) - 256*a*b**2*cosh(c + d*x)**11/(231*d) + b*
*3*sinh(c + d*x)**14*cosh(c + d*x)/d - 14*b**3*sinh(c + d*x)**12*cosh(c + d*x)**3/(3*d) + 56*b**3*sinh(c + d*x
)**10*cosh(c + d*x)**5/(5*d) - 16*b**3*sinh(c + d*x)**8*cosh(c + d*x)**7/d + 128*b**3*sinh(c + d*x)**6*cosh(c
+ d*x)**9/(9*d) - 256*b**3*sinh(c + d*x)**4*cosh(c + d*x)**11/(33*d) + 1024*b**3*sinh(c + d*x)**2*cosh(c + d*x
)**13/(429*d) - 2048*b**3*cosh(c + d*x)**15/(6435*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**3*sinh(c)**3, True))

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