3.3.9 \(\int \sinh (c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [209]

Optimal. Leaf size=143 \[ \frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac {3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac {b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac {6 b^3 \cosh ^{11}(c+d x)}{11 d}+\frac {b^3 \cosh ^{13}(c+d x)}{13 d} \]

[Out]

(a+b)^3*cosh(d*x+c)/d-2*b*(a+b)^2*cosh(d*x+c)^3/d+3/5*b*(a+b)*(a+5*b)*cosh(d*x+c)^5/d-4/7*b^2*(3*a+5*b)*cosh(d
*x+c)^7/d+1/3*b^2*(a+5*b)*cosh(d*x+c)^9/d-6/11*b^3*cosh(d*x+c)^11/d+1/13*b^3*cosh(d*x+c)^13/d

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Rubi [A]
time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3294, 1104} \begin {gather*} \frac {b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac {4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac {3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac {2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^{13}(c+d x)}{13 d}-\frac {6 b^3 \cosh ^{11}(c+d x)}{11 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)^3*Cosh[c + d*x])/d - (2*b*(a + b)^2*Cosh[c + d*x]^3)/d + (3*b*(a + b)*(a + 5*b)*Cosh[c + d*x]^5)/(5*d
) - (4*b^2*(3*a + 5*b)*Cosh[c + d*x]^7)/(7*d) + (b^2*(a + 5*b)*Cosh[c + d*x]^9)/(3*d) - (6*b^3*Cosh[c + d*x]^1
1)/(11*d) + (b^3*Cosh[c + d*x]^13)/(13*d)

Rule 1104

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]
/; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]

Rule 3294

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

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Mathematica [A]
time = 0.74, size = 157, normalized size = 1.10 \begin {gather*} \frac {60060 \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) \cosh (c+d x)-15015 b \left (1280 a^2+1344 a b+429 b^2\right ) \cosh (3 (c+d x))+3003 b \left (768 a^2+1728 a b+715 b^2\right ) \cosh (5 (c+d x))-4290 b^2 (216 a+143 b) \cosh (7 (c+d x))+10010 b^2 (8 a+13 b) \cosh (9 (c+d x))-17745 b^3 \cosh (11 (c+d x))+1155 b^3 \cosh (13 (c+d x))}{61501440 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*Cosh[c + d*x] - 15015*b*(1280*a^2 + 1344*a*b + 429*b^2)*
Cosh[3*(c + d*x)] + 3003*b*(768*a^2 + 1728*a*b + 715*b^2)*Cosh[5*(c + d*x)] - 4290*b^2*(216*a + 143*b)*Cosh[7*
(c + d*x)] + 10010*b^2*(8*a + 13*b)*Cosh[9*(c + d*x)] - 17745*b^3*Cosh[11*(c + d*x)] + 1155*b^3*Cosh[13*(c + d
*x)])/(61501440*d)

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Maple [A]
time = 0.96, size = 183, normalized size = 1.28

method	result	size
default	\(\frac {\left (-\frac {143}{2048} b^{3}-\frac {27}{256} a \,b^{2}\right ) \cosh \left (7 d x +7 c \right )}{7 d}+\frac {\left (\frac {39}{2048} b^{3}+\frac {3}{256} a \,b^{2}\right ) \cosh \left (9 d x +9 c \right )}{9 d}+\frac {\left (-\frac {1287}{4096} b^{3}-\frac {63}{64} a \,b^{2}-\frac {15}{16} a^{2} b \right ) \cosh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {715}{4096} b^{3}+\frac {27}{64} a \,b^{2}+\frac {3}{16} a^{2} b \right ) \cosh \left (5 d x +5 c \right )}{5 d}+\frac {\left (\frac {429}{1024} b^{3}+\frac {189}{128} a \,b^{2}+\frac {15}{8} a^{2} b +a^{3}\right ) \cosh \left (d x +c \right )}{d}-\frac {13 b^{3} \cosh \left (11 d x +11 c \right )}{45056 d}+\frac {b^{3} \cosh \left (13 d x +13 c \right )}{53248 d}\)	\(183\)
risch	\(\frac {27 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}-\frac {21 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{128 d}+\frac {15 b \,{\mathrm e}^{d x +c} a^{2}}{16 d}+\frac {13 b^{3} {\mathrm e}^{9 d x +9 c}}{12288 d}+\frac {143 \,{\mathrm e}^{5 d x +5 c} b^{3}}{8192 d}-\frac {429 \,{\mathrm e}^{3 d x +3 c} b^{3}}{8192 d}+\frac {b^{3} {\mathrm e}^{13 d x +13 c}}{106496 d}-\frac {13 b^{3} {\mathrm e}^{11 d x +11 c}}{90112 d}+\frac {a^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {13 b^{3} {\mathrm e}^{-9 d x -9 c}}{12288 d}-\frac {13 b^{3} {\mathrm e}^{-11 d x -11 c}}{90112 d}+\frac {b^{3} {\mathrm e}^{-13 d x -13 c}}{106496 d}+\frac {189 a \,{\mathrm e}^{-d x -c} b^{2}}{256 d}-\frac {21 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{128 d}+\frac {429 b^{3} {\mathrm e}^{d x +c}}{2048 d}+\frac {429 b^{3} {\mathrm e}^{-d x -c}}{2048 d}-\frac {429 b^{3} {\mathrm e}^{-3 d x -3 c}}{8192 d}-\frac {143 b^{3} {\mathrm e}^{7 d x +7 c}}{28672 d}+\frac {27 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}+\frac {143 \,{\mathrm e}^{-5 d x -5 c} b^{3}}{8192 d}-\frac {143 b^{3} {\mathrm e}^{-7 d x -7 c}}{28672 d}+\frac {189 a \,{\mathrm e}^{d x +c} b^{2}}{256 d}+\frac {15 \,{\mathrm e}^{-d x -c} a^{2} b}{16 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{32 d}+\frac {3 \,{\mathrm e}^{-5 d x -5 c} a^{2} b}{160 d}+\frac {b^{2} {\mathrm e}^{9 d x +9 c} a}{1536 d}-\frac {27 b^{2} {\mathrm e}^{7 d x +7 c} a}{3584 d}+\frac {3 \,{\mathrm e}^{5 d x +5 c} a^{2} b}{160 d}-\frac {27 b^{2} {\mathrm e}^{-7 d x -7 c} a}{3584 d}+\frac {b^{2} {\mathrm e}^{-9 d x -9 c} a}{1536 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{32 d}\)	\(550\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(-143/2048*b^3-27/256*a*b^2)/d*cosh(7*d*x+7*c)+1/9*(39/2048*b^3+3/256*a*b^2)/d*cosh(9*d*x+9*c)+1/3*(-1287/
4096*b^3-63/64*a*b^2-15/16*a^2*b)/d*cosh(3*d*x+3*c)+1/5*(715/4096*b^3+27/64*a*b^2+3/16*a^2*b)/d*cosh(5*d*x+5*c
)+(429/1024*b^3+189/128*a*b^2+15/8*a^2*b+a^3)/d*cosh(d*x+c)-13/45056*b^3/d*cosh(11*d*x+11*c)+1/53248*b^3/d*cos
h(13*d*x+13*c)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (133) = 266\).
time = 0.27, size = 399, normalized size = 2.79 \begin {gather*} -\frac {1}{24600576} \, b^{3} {\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 b^{2} {\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}{160} \, a^{2} b {\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 )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24600576*b^3*((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) + 1288287*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) - 1288287*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^(-11*d*x - 11*c) + 231*e^(-13*d*x - 13*c))/d) - 1/53760*a*b^2*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x
- 4*c) + 8820*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) + 1/160*a^2*b*(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) + a^3*cosh(d*x + c)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (133) = 266\).
time = 0.39, size = 594, normalized size = 4.15 \begin {gather*} \frac {1155 \, b^{3} \cosh \left (d x + c\right )^{13} + 15015 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{12} - 17745 \, b^{3} \cosh \left (d x + c\right )^{11} + 15015 \, {\left (22 \, b^{3} \cosh \left (d x + c\right )^{3} - 13 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{10} + 10010 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 45045 \, {\left (33 \, b^{3} \cosh \left (d x + c\right )^{5} - 65 \, b^{3} \cosh \left (d x + c\right )^{3} + 2 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} - 4290 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 30030 \, {\left (66 \, b^{3} \cosh \left (d x + c\right )^{7} - 273 \, b^{3} \cosh \left (d x + c\right )^{5} + 28 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 3003 \, {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 15015 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{9} - 390 \, b^{3} \cosh \left (d x + c\right )^{7} + 84 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 15015 \, {\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 15015 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{11} - 65 \, b^{3} \cosh \left (d x + c\right )^{9} + 24 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 6 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 2 \, {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 60060 \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )}{61501440 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/61501440*(1155*b^3*cosh(d*x + c)^13 + 15015*b^3*cosh(d*x + c)*sinh(d*x + c)^12 - 17745*b^3*cosh(d*x + c)^11
+ 15015*(22*b^3*cosh(d*x + c)^3 - 13*b^3*cosh(d*x + c))*sinh(d*x + c)^10 + 10010*(8*a*b^2 + 13*b^3)*cosh(d*x +
c)^9 + 45045*(33*b^3*cosh(d*x + c)^5 - 65*b^3*cosh(d*x + c)^3 + 2*(8*a*b^2 + 13*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^8 - 4290*(216*a*b^2 + 143*b^3)*cosh(d*x + c)^7 + 30030*(66*b^3*cosh(d*x + c)^7 - 273*b^3*cosh(d*x + c)^5
+ 28*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^3 - (216*a*b^2 + 143*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 3003*(768*a^2
*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c)^5 + 15015*(55*b^3*cosh(d*x + c)^9 - 390*b^3*cosh(d*x + c)^7 + 84*(8*a
*b^2 + 13*b^3)*cosh(d*x + c)^5 - 10*(216*a*b^2 + 143*b^3)*cosh(d*x + c)^3 + (768*a^2*b + 1728*a*b^2 + 715*b^3)
*cosh(d*x + c))*sinh(d*x + c)^4 - 15015*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*cosh(d*x + c)^3 + 15015*(6*b^3*cos
h(d*x + c)^11 - 65*b^3*cosh(d*x + c)^9 + 24*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^7 - 6*(216*a*b^2 + 143*b^3)*cosh(
d*x + c)^5 + 2*(768*a^2*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c)^3 - 3*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*cosh
(d*x + c))*sinh(d*x + c)^2 + 60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*cosh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (131) = 262\).
time = 5.43, size = 377, normalized size = 2.64 \begin {gather*} \begin {cases} \frac {a^{3} \cosh {\left (c + d x \right )}}{d} + \frac {3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {48 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {192 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 a b^{2} \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac {b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {64 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{21 d} - \frac {512 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{231 d} + \frac {1024 b^{3} \cosh ^{13}{\left (c + d x \right )}}{3003 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*cosh(c + d*x)/d + 3*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 4*a**2*b*sinh(c + d*x)**2*cosh(c
+ d*x)**3/d + 8*a**2*b*cosh(c + d*x)**5/(5*d) + 3*a*b**2*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*a*b**2*sinh(c +
d*x)**6*cosh(c + d*x)**3/d + 48*a*b**2*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 192*a*b**2*sinh(c + d*x)**2*
cosh(c + d*x)**7/(35*d) + 128*a*b**2*cosh(c + d*x)**9/(105*d) + b**3*sinh(c + d*x)**12*cosh(c + d*x)/d - 4*b**
3*sinh(c + d*x)**10*cosh(c + d*x)**3/d + 8*b**3*sinh(c + d*x)**8*cosh(c + d*x)**5/d - 64*b**3*sinh(c + d*x)**6
*cosh(c + d*x)**7/(7*d) + 128*b**3*sinh(c + d*x)**4*cosh(c + d*x)**9/(21*d) - 512*b**3*sinh(c + d*x)**2*cosh(c
+ d*x)**11/(231*d) + 1024*b**3*cosh(c + d*x)**13/(3003*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**3*sinh(c), True)
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (133) = 266\).
time = 0.51, size = 372, normalized size = 2.60 \begin {gather*} \frac {b^{3} e^{\left (13 \, d x + 13 \, c\right )}}{106496 \, d} - \frac {13 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )}}{90112 \, d} - \frac {13 \, b^{3} e^{\left (-11 \, d x - 11 \, c\right )}}{90112 \, d} + \frac {b^{3} e^{\left (-13 \, d x - 13 \, c\right )}}{106496 \, d} + \frac {{\left (8 \, a b^{2} + 13 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{12288 \, d} - \frac {{\left (216 \, a b^{2} + 143 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{28672 \, d} + \frac {{\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{40960 \, d} - \frac {{\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{8192 \, d} + \frac {{\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (d x + c\right )}}{2048 \, d} + \frac {{\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (-d x - c\right )}}{2048 \, d} - \frac {{\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{8192 \, d} + \frac {{\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{40960 \, d} - \frac {{\left (216 \, a b^{2} + 143 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{28672 \, d} + \frac {{\left (8 \, a b^{2} + 13 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{12288 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/106496*b^3*e^(13*d*x + 13*c)/d - 13/90112*b^3*e^(11*d*x + 11*c)/d - 13/90112*b^3*e^(-11*d*x - 11*c)/d + 1/10
6496*b^3*e^(-13*d*x - 13*c)/d + 1/12288*(8*a*b^2 + 13*b^3)*e^(9*d*x + 9*c)/d - 1/28672*(216*a*b^2 + 143*b^3)*e
^(7*d*x + 7*c)/d + 1/40960*(768*a^2*b + 1728*a*b^2 + 715*b^3)*e^(5*d*x + 5*c)/d - 1/8192*(1280*a^2*b + 1344*a*
b^2 + 429*b^3)*e^(3*d*x + 3*c)/d + 1/2048*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*e^(d*x + c)/d + 1/204
8*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*e^(-d*x - c)/d - 1/8192*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*e
^(-3*d*x - 3*c)/d + 1/40960*(768*a^2*b + 1728*a*b^2 + 715*b^3)*e^(-5*d*x - 5*c)/d - 1/28672*(216*a*b^2 + 143*b
^3)*e^(-7*d*x - 7*c)/d + 1/12288*(8*a*b^2 + 13*b^3)*e^(-9*d*x - 9*c)/d

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Mupad [B]
time = 1.07, size = 211, normalized size = 1.48 \begin {gather*} \frac {a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {12\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {18\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-4\,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 )}^{13}}{13}-\frac {6\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {5\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {20\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+3\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5-2\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*cosh(c + d*x) + b^3*cosh(c + d*x) - 2*b^3*cosh(c + d*x)^3 + 3*b^3*cosh(c + d*x)^5 - (20*b^3*cosh(c + d*x)
^7)/7 + (5*b^3*cosh(c + d*x)^9)/3 - (6*b^3*cosh(c + d*x)^11)/11 + (b^3*cosh(c + d*x)^13)/13 - 4*a*b^2*cosh(c +
d*x)^3 - 2*a^2*b*cosh(c + d*x)^3 + (18*a*b^2*cosh(c + d*x)^5)/5 + (3*a^2*b*cosh(c + d*x)^5)/5 - (12*a*b^2*cos
h(c + d*x)^7)/7 + (a*b^2*cosh(c + d*x)^9)/3 + 3*a*b^2*cosh(c + d*x) + 3*a^2*b*cosh(c + d*x))/d

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