3.207 \(\int \sinh ^5(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=220 \[ \frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d} \]

[Out]

(a+b)^3*cosh(d*x+c)/d-2/3*(a+b)^2*(a+4*b)*cosh(d*x+c)^3/d+1/5*(a+b)*(a^2+17*a*b+28*b^2)*cosh(d*x+c)^5/d-4/7*b*
(3*a^2+15*a*b+14*b^2)*cosh(d*x+c)^7/d+1/9*b*(3*a^2+45*a*b+70*b^2)*cosh(d*x+c)^9/d-2/11*b^2*(9*a+28*b)*cosh(d*x
+c)^11/d+1/13*b^2*(3*a+28*b)*cosh(d*x+c)^13/d-8/15*b^3*cosh(d*x+c)^15/d+1/17*b^3*cosh(d*x+c)^17/d

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Rubi [A]  time = 0.22, antiderivative size = 220, 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} \[ \frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

((a + b)^3*Cosh[c + d*x])/d - (2*(a + b)^2*(a + 4*b)*Cosh[c + d*x]^3)/(3*d) + ((a + b)*(a^2 + 17*a*b + 28*b^2)
*Cosh[c + d*x]^5)/(5*d) - (4*b*(3*a^2 + 15*a*b + 14*b^2)*Cosh[c + d*x]^7)/(7*d) + (b*(3*a^2 + 45*a*b + 70*b^2)
*Cosh[c + d*x]^9)/(9*d) - (2*b^2*(9*a + 28*b)*Cosh[c + d*x]^11)/(11*d) + (b^2*(3*a + 28*b)*Cosh[c + d*x]^13)/(
13*d) - (8*b^3*Cosh[c + d*x]^15)/(15*d) + (b^3*Cosh[c + d*x]^17)/(17*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 ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^2 \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)^3-2 (a+b)^2 (a+4 b) x^2+(a+b) \left (a^2+17 a b+28 b^2\right ) x^4-4 b \left (3 a^2+15 a b+14 b^2\right ) x^6+b \left (3 a^2+45 a b+70 b^2\right ) x^8-2 b^2 (9 a+28 b) x^{10}+b^2 (3 a+28 b) x^{12}-8 b^3 x^{14}+b^3 x^{16}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d}\\ \end {align*}

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Mathematica [A]  time = 2.28, size = 288, normalized size = 1.31 \[ \frac {627314688 a^3 \cosh (5 (c+d x))[phone] a^2 b \cosh (5 (c+d x))-756138240 a^2 b \cosh (7 (c+d x))[phone] a^2 b \cosh (9 (c+d x))+1531530 \left (20480 a^3+48384 a^2 b+41184 a b^2+12155 b^3\right ) \cosh (c+d x)-2042040 \left (2560 a^3+8064 a^2 b+7722 a b^2+2431 b^3\right ) \cosh (3 (c+d x))[phone] a b^2 \cosh (5 (c+d x))-1501774560 a b^2 \cosh (7 (c+d x))[phone] a b^2 \cosh (9 (c+d x))-43439760 a b^2 \cosh (11 (c+d x))+2827440 a b^2 \cosh (13 (c+d x))[phone] b^3 \cosh (5 (c+d x))-676936260 b^3 \cosh (7 (c+d x))[phone] b^3 \cosh (9 (c+d x))-47338200 b^3 \cosh (11 (c+d x))+8011080 b^3 \cosh (13 (c+d x))-867867 b^3 \cosh (15 (c+d x))+45045 b^3 \cosh (17 (c+d x))}{50185175040 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(1531530*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*Cosh[c + d*x] - 2042040*(2560*a^3 + 8064*a^2*b +
7722*a*b^2 + 2431*b^3)*Cosh[3*(c + d*x)] + 627314688*a^3*Cosh[5*(c + d*x)] + 4234374144*a^2*b*Cosh[5*(c + d*x)
] + 5256210960*a*b^2*Cosh[5*(c + d*x)] + 1895421528*b^3*Cosh[5*(c + d*x)] - 756138240*a^2*b*Cosh[7*(c + d*x)]
- 1501774560*a*b^2*Cosh[7*(c + d*x)] - 676936260*b^3*Cosh[7*(c + d*x)] + 65345280*a^2*b*Cosh[9*(c + d*x)] + 31
8558240*a*b^2*Cosh[9*(c + d*x)] + 202502300*b^3*Cosh[9*(c + d*x)] - 43439760*a*b^2*Cosh[11*(c + d*x)] - 473382
00*b^3*Cosh[11*(c + d*x)] + 2827440*a*b^2*Cosh[13*(c + d*x)] + 8011080*b^3*Cosh[13*(c + d*x)] - 867867*b^3*Cos
h[15*(c + d*x)] + 45045*b^3*Cosh[17*(c + d*x)])/(50185175040*d)

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fricas [B]  time = 0.85, size = 1030, normalized size = 4.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/50185175040*(45045*b^3*cosh(d*x + c)^17 + 765765*b^3*cosh(d*x + c)*sinh(d*x + c)^16 - 867867*b^3*cosh(d*x +
c)^15 + 765765*(40*b^3*cosh(d*x + c)^3 - 17*b^3*cosh(d*x + c))*sinh(d*x + c)^14 + 471240*(6*a*b^2 + 17*b^3)*co
sh(d*x + c)^13 + 255255*(1092*b^3*cosh(d*x + c)^5 - 1547*b^3*cosh(d*x + c)^3 + 24*(6*a*b^2 + 17*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^12 - 556920*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^11 + 153153*(5720*b^3*cosh(d*x + c)^7 - 1701
7*b^3*cosh(d*x + c)^5 + 880*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^3 - 40*(78*a*b^2 + 85*b^3)*cosh(d*x + c))*sinh(d*
x + c)^10 + 340340*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^9 + 765765*(1430*b^3*cosh(d*x + c)^9 - 7293
*b^3*cosh(d*x + c)^7 + 792*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^5 - 120*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^3 + 4*(1
92*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 - 437580*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*c
osh(d*x + c)^7 + 255255*(2184*b^3*cosh(d*x + c)^11 - 17017*b^3*cosh(d*x + c)^9 + 3168*(6*a*b^2 + 17*b^3)*cosh(
d*x + c)^7 - 1008*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^5 + 112*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^3
- 12*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 1225224*(512*a^3 + 3456*a^2*b + 429
0*a*b^2 + 1547*b^3)*cosh(d*x + c)^5 + 765765*(140*b^3*cosh(d*x + c)^13 - 1547*b^3*cosh(d*x + c)^11 + 440*(6*a*
b^2 + 17*b^3)*cosh(d*x + c)^9 - 240*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^7 + 56*(192*a^2*b + 936*a*b^2 + 595*b^3)
*cosh(d*x + c)^5 - 20*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c)^3 + 8*(512*a^3 + 3456*a^2*b + 4290*a*
b^2 + 1547*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2042040*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*cosh(
d*x + c)^3 + 765765*(8*b^3*cosh(d*x + c)^15 - 119*b^3*cosh(d*x + c)^13 + 48*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^1
1 - 40*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^9 + 16*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^7 - 12*(1728*a
^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c)^5 + 16*(512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*cosh(d*x + c
)^3 - 8*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 1531530*(20480*a^3 +
48384*a^2*b + 41184*a*b^2 + 12155*b^3)*cosh(d*x + c))/d

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giac [B]  time = 0.39, size = 520, normalized size = 2.36 \[ \frac {b^{3} e^{\left (17 \, d x + 17 \, c\right )}}{2228224 \, d} - \frac {17 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )}}{1966080 \, d} - \frac {17 \, b^{3} e^{\left (-15 \, d x - 15 \, c\right )}}{1966080 \, d} + \frac {b^{3} e^{\left (-17 \, d x - 17 \, c\right )}}{2228224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (13 \, d x + 13 \, c\right )}}{212992 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (11 \, d x + 11 \, c\right )}}{180224 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{294912 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{229376 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{81920 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{49152 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (d x + c\right )}}{65536 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (-d x - c\right )}}{65536 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{49152 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{81920 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{229376 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{294912 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{180224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (-13 \, d x - 13 \, c\right )}}{212992 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/2228224*b^3*e^(17*d*x + 17*c)/d - 17/1966080*b^3*e^(15*d*x + 15*c)/d - 17/1966080*b^3*e^(-15*d*x - 15*c)/d +
1/2228224*b^3*e^(-17*d*x - 17*c)/d + 1/212992*(6*a*b^2 + 17*b^3)*e^(13*d*x + 13*c)/d - 1/180224*(78*a*b^2 + 8
5*b^3)*e^(11*d*x + 11*c)/d + 1/294912*(192*a^2*b + 936*a*b^2 + 595*b^3)*e^(9*d*x + 9*c)/d - 1/229376*(1728*a^2
*b + 3432*a*b^2 + 1547*b^3)*e^(7*d*x + 7*c)/d + 1/81920*(512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*e^(5*d*
x + 5*c)/d - 1/49152*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*e^(3*d*x + 3*c)/d + 1/65536*(20480*a^3 +
48384*a^2*b + 41184*a*b^2 + 12155*b^3)*e^(d*x + c)/d + 1/65536*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*
b^3)*e^(-d*x - c)/d - 1/49152*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*e^(-3*d*x - 3*c)/d + 1/81920*(51
2*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*e^(-5*d*x - 5*c)/d - 1/229376*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)
*e^(-7*d*x - 7*c)/d + 1/294912*(192*a^2*b + 936*a*b^2 + 595*b^3)*e^(-9*d*x - 9*c)/d - 1/180224*(78*a*b^2 + 85*
b^3)*e^(-11*d*x - 11*c)/d + 1/212992*(6*a*b^2 + 17*b^3)*e^(-13*d*x - 13*c)/d

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maple [A]  time = 0.17, size = 258, normalized size = 1.17 \[ \frac {b^{3} \left (\frac {32768}{109395}+\frac {\left (\sinh ^{16}\left (d x +c \right )\right )}{17}-\frac {16 \left (\sinh ^{14}\left (d x +c \right )\right )}{255}+\frac {224 \left (\sinh ^{12}\left (d x +c \right )\right )}{3315}-\frac {896 \left (\sinh ^{10}\left (d x +c \right )\right )}{12155}+\frac {1792 \left (\sinh ^{8}\left (d x +c \right )\right )}{21879}-\frac {2048 \left (\sinh ^{6}\left (d x +c \right )\right )}{21879}+\frac {4096 \left (\sinh ^{4}\left (d x +c \right )\right )}{36465}-\frac {16384 \left (\sinh ^{2}\left (d x +c \right )\right )}{109395}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {1024}{3003}+\frac {\left (\sinh ^{12}\left (d x +c \right )\right )}{13}-\frac {12 \left (\sinh ^{10}\left (d x +c \right )\right )}{143}+\frac {40 \left (\sinh ^{8}\left (d x +c \right )\right )}{429}-\frac {320 \left (\sinh ^{6}\left (d x +c \right )\right )}{3003}+\frac {128 \left (\sinh ^{4}\left (d x +c \right )\right )}{1001}-\frac {512 \left (\sinh ^{2}\left (d x +c \right )\right )}{3003}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {128}{315}+\frac {\left (\sinh ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{63}+\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{315}\right ) \cosh \left (d x +c \right )+a^{3} \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(b^3*(32768/109395+1/17*sinh(d*x+c)^16-16/255*sinh(d*x+c)^14+224/3315*sinh(d*x+c)^12-896/12155*sinh(d*x+c)
^10+1792/21879*sinh(d*x+c)^8-2048/21879*sinh(d*x+c)^6+4096/36465*sinh(d*x+c)^4-16384/109395*sinh(d*x+c)^2)*cos
h(d*x+c)+3*a*b^2*(1024/3003+1/13*sinh(d*x+c)^12-12/143*sinh(d*x+c)^10+40/429*sinh(d*x+c)^8-320/3003*sinh(d*x+c
)^6+128/1001*sinh(d*x+c)^4-512/3003*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*
x+c)^6+16/105*sinh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+c)+a^3*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*
cosh(d*x+c))

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maxima [B]  time = 0.34, size = 600, normalized size = 2.73 \[ -\frac {1}{14338621440} \, b^{3} {\left (\frac {{\left (123981 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1144440 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6762600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 28928900 \, e^{\left (-8 \, d x - 8 \, c\right )} + 96705180 \, e^{\left (-10 \, d x - 10 \, c\right )} - 270774504 \, e^{\left (-12 \, d x - 12 \, c\right )} + 709171320 \, e^{\left (-14 \, d x - 14 \, c\right )} - 2659392450 \, e^{\left (-16 \, d x - 16 \, c\right )} - 6435\right )} e^{\left (17 \, d x + 17 \, c\right )}}{d} - \frac {2659392450 \, e^{\left (-d x - c\right )} - 709171320 \, e^{\left (-3 \, d x - 3 \, c\right )} + 270774504 \, e^{\left (-5 \, d x - 5 \, c\right )} - 96705180 \, e^{\left (-7 \, d x - 7 \, c\right )} + 28928900 \, e^{\left (-9 \, d x - 9 \, c\right )} - 6762600 \, e^{\left (-11 \, d x - 11 \, c\right )} + 1144440 \, e^{\left (-13 \, d x - 13 \, c\right )} - 123981 \, e^{\left (-15 \, d x - 15 \, c\right )} + 6435 \, e^{\left (-17 \, d x - 17 \, c\right )}}{d}\right )} - \frac {1}{8200192} \, a b^{2} {\left (\frac {{\left (3549 \, e^{\left (-2 \, d x - 2 \, c\right )} - 26026 \, e^{\left (-4 \, d x - 4 \, c\right )} + 122694 \, e^{\left (-6 \, d x - 6 \, c\right )} - 429429 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1288287 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5153148 \, e^{\left (-12 \, d x - 12 \, c\right )} - 231\right )} e^{\left (13 \, d x + 13 \, c\right )}}{d} - \frac {5153148 \, e^{\left (-d x - c\right )} - 1288287 \, e^{\left (-3 \, d x - 3 \, c\right )} + 429429 \, e^{\left (-5 \, d x - 5 \, c\right )} - 122694 \, e^{\left (-7 \, d x - 7 \, c\right )} + 26026 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3549 \, e^{\left (-11 \, d x - 11 \, c\right )} + 231 \, e^{\left (-13 \, d x - 13 \, c\right )}}{d}\right )} - \frac {1}{53760} \, a^{2} b {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{480} \, a^{3} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/14338621440*b^3*((123981*e^(-2*d*x - 2*c) - 1144440*e^(-4*d*x - 4*c) + 6762600*e^(-6*d*x - 6*c) - 28928900*
e^(-8*d*x - 8*c) + 96705180*e^(-10*d*x - 10*c) - 270774504*e^(-12*d*x - 12*c) + 709171320*e^(-14*d*x - 14*c) -
2659392450*e^(-16*d*x - 16*c) - 6435)*e^(17*d*x + 17*c)/d - (2659392450*e^(-d*x - c) - 709171320*e^(-3*d*x -
3*c) + 270774504*e^(-5*d*x - 5*c) - 96705180*e^(-7*d*x - 7*c) + 28928900*e^(-9*d*x - 9*c) - 6762600*e^(-11*d*x
- 11*c) + 1144440*e^(-13*d*x - 13*c) - 123981*e^(-15*d*x - 15*c) + 6435*e^(-17*d*x - 17*c))/d) - 1/8200192*a*
b^2*((3549*e^(-2*d*x - 2*c) - 26026*e^(-4*d*x - 4*c) + 122694*e^(-6*d*x - 6*c) - 429429*e^(-8*d*x - 8*c) + 128
8287*e^(-10*d*x - 10*c) - 5153148*e^(-12*d*x - 12*c) - 231)*e^(13*d*x + 13*c)/d - (5153148*e^(-d*x - c) - 1288
287*e^(-3*d*x - 3*c) + 429429*e^(-5*d*x - 5*c) - 122694*e^(-7*d*x - 7*c) + 26026*e^(-9*d*x - 9*c) - 3549*e^(-1
1*d*x - 11*c) + 231*e^(-13*d*x - 13*c))/d) - 1/53760*a^2*b*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 88
20*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c) - 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x -
3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x - 7*c) + 35*e^(-9*d*x - 9*c))/d) + 1/480*a^3*(3*e^(5*d*x + 5*c)/d
- 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/
d)

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mupad [B]  time = 1.76, size = 319, normalized size = 1.45 \[ \frac {\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {2\,a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {12\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {18\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-4\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {18\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+5\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9-\frac {60\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+9\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-6\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{17}}{17}-\frac {8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{15}}{15}+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {56\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {70\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^5*(a + b*sinh(c + d*x)^4)^3,x)

[Out]

(a^3*cosh(c + d*x) + b^3*cosh(c + d*x) - (2*a^3*cosh(c + d*x)^3)/3 + (a^3*cosh(c + d*x)^5)/5 - (8*b^3*cosh(c +
d*x)^3)/3 + (28*b^3*cosh(c + d*x)^5)/5 - 8*b^3*cosh(c + d*x)^7 + (70*b^3*cosh(c + d*x)^9)/9 - (56*b^3*cosh(c
+ d*x)^11)/11 + (28*b^3*cosh(c + d*x)^13)/13 - (8*b^3*cosh(c + d*x)^15)/15 + (b^3*cosh(c + d*x)^17)/17 - 6*a*b
^2*cosh(c + d*x)^3 - 4*a^2*b*cosh(c + d*x)^3 + 9*a*b^2*cosh(c + d*x)^5 + (18*a^2*b*cosh(c + d*x)^5)/5 - (60*a*
b^2*cosh(c + d*x)^7)/7 - (12*a^2*b*cosh(c + d*x)^7)/7 + 5*a*b^2*cosh(c + d*x)^9 + (a^2*b*cosh(c + d*x)^9)/3 -
(18*a*b^2*cosh(c + d*x)^11)/11 + (3*a*b^2*cosh(c + d*x)^13)/13 + 3*a*b^2*cosh(c + d*x) + 3*a^2*b*cosh(c + d*x)
)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**5*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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