3.99 \(\int x \sinh (a+b \sqrt [3]{c+d x}) \, dx\)

Optimal. Leaf size=261 \[ -\frac {360 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {360 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

[Out]

-6*c*cosh(a+b*(d*x+c)^(1/3))/b^3/d^2+360*(d*x+c)^(1/3)*cosh(a+b*(d*x+c)^(1/3))/b^5/d^2-3*c*(d*x+c)^(2/3)*cosh(
a+b*(d*x+c)^(1/3))/b/d^2+60*(d*x+c)*cosh(a+b*(d*x+c)^(1/3))/b^3/d^2+3*(d*x+c)^(5/3)*cosh(a+b*(d*x+c)^(1/3))/b/
d^2-360*sinh(a+b*(d*x+c)^(1/3))/b^6/d^2+6*c*(d*x+c)^(1/3)*sinh(a+b*(d*x+c)^(1/3))/b^2/d^2-180*(d*x+c)^(2/3)*si
nh(a+b*(d*x+c)^(1/3))/b^4/d^2-15*(d*x+c)^(4/3)*sinh(a+b*(d*x+c)^(1/3))/b^2/d^2

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Rubi [A]  time = 0.31, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5364, 5286, 3296, 2638, 2637} \[ -\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {360 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sinh[a + b*(c + d*x)^(1/3)],x]

[Out]

(-6*c*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (360*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^2) - (
3*c*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^2) + (60*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^2
) + (3*(c + d*x)^(5/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^2) - (360*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^2) + (6
*c*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/
(b^4*d^2) - (15*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^2)

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5286

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegra
nd[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rule 5364

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int x \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (-c+x) \sinh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {3 \operatorname {Subst}\left (\int x^2 \left (-c+x^3\right ) \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac {3 \operatorname {Subst}\left (\int \left (-c x^2 \sinh (a+b x)+x^5 \sinh (a+b x)\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac {3 \operatorname {Subst}\left (\int x^5 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}-\frac {(3 c) \operatorname {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {15 \operatorname {Subst}\left (\int x^4 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac {(6 c) \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}\\ &=-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {60 \operatorname {Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {(6 c) \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}\\ &=-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 \operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^2}\\ &=-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {360 \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^2}\\ &=-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {360 \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^2}\\ &=-\frac {6 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {360 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {6 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {15 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 118, normalized size = 0.45 \[ \frac {3 b \left (b^4 d x (c+d x)^{2/3}+2 b^2 (9 c+10 d x)+120 \sqrt [3]{c+d x}\right ) \cosh \left (a+b \sqrt [3]{c+d x}\right )-3 \left (b^4 \sqrt [3]{c+d x} (3 c+5 d x)+60 b^2 (c+d x)^{2/3}+120\right ) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sinh[a + b*(c + d*x)^(1/3)],x]

[Out]

(3*b*(120*(c + d*x)^(1/3) + b^4*d*x*(c + d*x)^(2/3) + 2*b^2*(9*c + 10*d*x))*Cosh[a + b*(c + d*x)^(1/3)] - 3*(1
20 + 60*b^2*(c + d*x)^(2/3) + b^4*(c + d*x)^(1/3)*(3*c + 5*d*x))*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^2)

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fricas [A]  time = 0.52, size = 109, normalized size = 0.42 \[ \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{5} d x + 20 \, b^{3} d x + 18 \, b^{3} c + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (5 \, b^{4} d x + 3 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} + 120\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

3*(((d*x + c)^(2/3)*b^5*d*x + 20*b^3*d*x + 18*b^3*c + 120*(d*x + c)^(1/3)*b)*cosh((d*x + c)^(1/3)*b + a) - (60
*(d*x + c)^(2/3)*b^2 + (5*b^4*d*x + 3*b^4*c)*(d*x + c)^(1/3) + 120)*sinh((d*x + c)^(1/3)*b + a))/(b^6*d^2)

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giac [B]  time = 0.15, size = 706, normalized size = 2.70 \[ -\frac {3 \, {\left (\frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c + 2 \, a b^{3} c + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a + 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} + 5 \, a^{4} + 2 \, b^{3} c - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} + 20 \, a^{3} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + 60 \, a^{2} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 120\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{b^{5} d} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} + 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a - 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 2 \, b^{3} c - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} + 20 \, a^{3} - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a - 60 \, a^{2} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 120\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{5} d}\right )}}{2 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

-3/2*((((d*x + c)^(1/3)*b + a)^2*b^3*c - 2*((d*x + c)^(1/3)*b + a)*a*b^3*c + a^2*b^3*c - ((d*x + c)^(1/3)*b +
a)^5 + 5*((d*x + c)^(1/3)*b + a)^4*a - 10*((d*x + c)^(1/3)*b + a)^3*a^2 + 10*((d*x + c)^(1/3)*b + a)^2*a^3 - 5
*((d*x + c)^(1/3)*b + a)*a^4 + a^5 - 2*((d*x + c)^(1/3)*b + a)*b^3*c + 2*a*b^3*c + 5*((d*x + c)^(1/3)*b + a)^4
- 20*((d*x + c)^(1/3)*b + a)^3*a + 30*((d*x + c)^(1/3)*b + a)^2*a^2 - 20*((d*x + c)^(1/3)*b + a)*a^3 + 5*a^4
+ 2*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3 + 60*((d*x + c)^(1/3)*b + a)^2*a - 60*((d*x + c)^(1/3)*b + a)*a^2 + 2
0*a^3 + 60*((d*x + c)^(1/3)*b + a)^2 - 120*((d*x + c)^(1/3)*b + a)*a + 60*a^2 - 120*(d*x + c)^(1/3)*b + 120)*e
^((d*x + c)^(1/3)*b + a)/(b^5*d) + (((d*x + c)^(1/3)*b + a)^2*b^3*c - 2*((d*x + c)^(1/3)*b + a)*a*b^3*c + a^2*
b^3*c - ((d*x + c)^(1/3)*b + a)^5 + 5*((d*x + c)^(1/3)*b + a)^4*a - 10*((d*x + c)^(1/3)*b + a)^3*a^2 + 10*((d*
x + c)^(1/3)*b + a)^2*a^3 - 5*((d*x + c)^(1/3)*b + a)*a^4 + a^5 + 2*((d*x + c)^(1/3)*b + a)*b^3*c - 2*a*b^3*c
- 5*((d*x + c)^(1/3)*b + a)^4 + 20*((d*x + c)^(1/3)*b + a)^3*a - 30*((d*x + c)^(1/3)*b + a)^2*a^2 + 20*((d*x +
c)^(1/3)*b + a)*a^3 - 5*a^4 + 2*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3 + 60*((d*x + c)^(1/3)*b + a)^2*a - 60*((
d*x + c)^(1/3)*b + a)*a^2 + 20*a^3 - 60*((d*x + c)^(1/3)*b + a)^2 + 120*((d*x + c)^(1/3)*b + a)*a - 60*a^2 - 1
20*(d*x + c)^(1/3)*b - 120)*e^(-(d*x + c)^(1/3)*b - a)/(b^5*d))/(b*d)

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maple [B]  time = 0.02, size = 659, normalized size = 2.52 \[ \frac {\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2}+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-120 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-3 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2}+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {15 a^{4} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {3 a^{5} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+6 c a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 c \,a^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(a+b*(d*x+c)^(1/3)),x)

[Out]

3/d^2/b^3*(1/b^3*((a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c)^(1/3))^4*sinh(a+b*(d*x+c)^(1/3)
)+20*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))-60*sinh(a+b*(d*x+c)^(1/3))*(a+b*(d*x+c)^(1/3))^2+120*(a+b*(
d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-120*sinh(a+b*(d*x+c)^(1/3)))-5/b^3*a*((a+b*(d*x+c)^(1/3))^4*cosh(a+b*(d*
x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2
4*sinh(a+b*(d*x+c)^(1/3))*(a+b*(d*x+c)^(1/3))+24*cosh(a+b*(d*x+c)^(1/3)))+10/b^3*a^2*((a+b*(d*x+c)^(1/3))^3*co
sh(a+b*(d*x+c)^(1/3))-3*sinh(a+b*(d*x+c)^(1/3))*(a+b*(d*x+c)^(1/3))^2+6*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(
1/3))-6*sinh(a+b*(d*x+c)^(1/3)))-10/b^3*a^3*((a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*sinh(a+b*(d*x+c)^
(1/3))*(a+b*(d*x+c)^(1/3))+2*cosh(a+b*(d*x+c)^(1/3)))+5/b^3*a^4*((a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-s
inh(a+b*(d*x+c)^(1/3)))-1/b^3*a^5*cosh(a+b*(d*x+c)^(1/3))-c*((a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*s
inh(a+b*(d*x+c)^(1/3))*(a+b*(d*x+c)^(1/3))+2*cosh(a+b*(d*x+c)^(1/3)))+2*c*a*((a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x
+c)^(1/3))-sinh(a+b*(d*x+c)^(1/3)))-c*a^2*cosh(a+b*(d*x+c)^(1/3)))

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maxima [A]  time = 0.32, size = 371, normalized size = 1.42 \[ \frac {2 \, d^{2} x^{2} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (\frac {c^{2} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{b} - \frac {c^{2} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b} - \frac {2 \, {\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} c e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} + \frac {2 \, {\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} c e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} - 120 \, {\left (d x + c\right )} b^{3} e^{a} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} - 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} + 720 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{2} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 120 \, {\left (d x + c\right )} b^{3} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 720\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{7}}\right )} b}{4 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

1/4*(2*d^2*x^2*sinh((d*x + c)^(1/3)*b + a) - (c^2*e^((d*x + c)^(1/3)*b + a)/b - c^2*e^(-(d*x + c)^(1/3)*b - a)
/b - 2*((d*x + c)*b^3*e^a - 3*(d*x + c)^(2/3)*b^2*e^a + 6*(d*x + c)^(1/3)*b*e^a - 6*e^a)*c*e^((d*x + c)^(1/3)*
b)/b^4 + 2*((d*x + c)*b^3 + 3*(d*x + c)^(2/3)*b^2 + 6*(d*x + c)^(1/3)*b + 6)*c*e^(-(d*x + c)^(1/3)*b - a)/b^4
+ ((d*x + c)^2*b^6*e^a - 6*(d*x + c)^(5/3)*b^5*e^a + 30*(d*x + c)^(4/3)*b^4*e^a - 120*(d*x + c)*b^3*e^a + 360*
(d*x + c)^(2/3)*b^2*e^a - 720*(d*x + c)^(1/3)*b*e^a + 720*e^a)*e^((d*x + c)^(1/3)*b)/b^7 - ((d*x + c)^2*b^6 +
6*(d*x + c)^(5/3)*b^5 + 30*(d*x + c)^(4/3)*b^4 + 120*(d*x + c)*b^3 + 360*(d*x + c)^(2/3)*b^2 + 720*(d*x + c)^(
1/3)*b + 720)*e^(-(d*x + c)^(1/3)*b - a)/b^7)*b)/d^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(a + b*(c + d*x)^(1/3)),x)

[Out]

int(x*sinh(a + b*(c + d*x)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sinh {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*(d*x+c)**(1/3)),x)

[Out]

Integral(x*sinh(a + b*(c + d*x)**(1/3)), x)

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