\(\int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx\) [57]

Optimal result

Integrand size = 18, antiderivative size = 246 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {3 \sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \]

[Out]

-3/4*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*b^(1/2)*Pi^(1/2)/d^(3/2)-3/4*exp(a-b*c/d)*erfi(b^(1/2)*(
d*x+c)^(1/2)/d^(1/2))*b^(1/2)*Pi^(1/2)/d^(3/2)+1/4*exp(-3*a+3*b*c/d)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)
)*b^(1/2)*3^(1/2)*Pi^(1/2)/d^(3/2)+1/4*exp(3*a-3*b*c/d)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*b^(1/2)*3^
(1/2)*Pi^(1/2)/d^(3/2)-2*sinh(b*x+a)^3/d/(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3394, 3388, 2211, 2235, 2236} \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {3 \sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \]

[In]

Int[Sinh[a + b*x]^3/(c + d*x)^(3/2),x]

[Out]

(-3*Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(-3*a + (
3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (3*Sqrt[b]*E^(a - (b*c)/d)*Sq
rt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt
[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (2*Sinh[a + b*x]^3)/(d*Sqrt[c + d*x])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(6 b) \int \left (\frac {\cosh (a+b x)}{4 \sqrt {c+d x}}-\frac {\cosh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d} \\ & = -\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(3 b) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{2 d}+\frac {(3 b) \int \frac {\cosh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{2 d} \\ & = -\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(3 b) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}-\frac {(3 b) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}+\frac {(3 b) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d}+\frac {(3 b) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d} \\ & = -\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {(3 b) \text {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}-\frac {(3 b) \text {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}-\frac {(3 b) \text {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}+\frac {(3 b) \text {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2} \\ & = -\frac {3 \sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-3 \left (a+b \left (\frac {c}{d}+x\right )\right )} \left (\sqrt {3} e^{6 a+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )-3 e^{4 a+\frac {2 b c}{d}+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {3 b c}{d}} \left (\left (-1+e^{2 (a+b x)}\right )^3-3 e^{2 a+\frac {b c}{d}+3 b x} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )+\sqrt {3} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{4 d \sqrt {c+d x}} \]

[In]

Integrate[Sinh[a + b*x]^3/(c + d*x)^(3/2),x]

[Out]

(Sqrt[3]*E^(6*a + 3*b*x)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] - 3*E^(4*a + (2*b*c)/d + 3*b*
x)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)] - E^((3*b*c)/d)*((-1 + E^(2*(a + b*x)))^3 - 3*E^(2*
a + (b*c)/d + 3*b*x)*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, b*(c/d + x)] + Sqrt[3]*E^((3*b*(c + d*x))/d)*Sqrt[(b*(c
+ d*x))/d]*Gamma[1/2, (3*b*(c + d*x))/d]))/(4*d*E^(3*(a + b*(c/d + x)))*Sqrt[c + d*x])

Maple [F]

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

[In]

int(sinh(b*x+a)^3/(d*x+c)^(3/2),x)

[Out]

int(sinh(b*x+a)^3/(d*x+c)^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1346 vs. \(2 (182) = 364\).

Time = 0.26 (sec) , antiderivative size = 1346, normalized size of antiderivative = 5.47 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

1/4*(sqrt(3)*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^3*sinh(-3*(b
*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c - a*d)/d) - (d*x + c)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x
+ c)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 +
3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(
b*x + a))*sqrt(b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d)) - sqrt(3)*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-3
*(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c - a*d)/d) + (d*x
+ c)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + c)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) + (d*x + c)*c
osh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + (
d*x + c)*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d
)) - 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^3*sinh(-(b*c - a*d)/
d) + ((d*x + c)*cosh(-(b*c - a*d)/d) - (d*x + c)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + c)*cosh(b*x
+ a)*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh
(b*x + a)^2*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*er
f(sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x +
a)^3*sinh(-(b*c - a*d)/d) + ((d*x + c)*cosh(-(b*c - a*d)/d) + (d*x + c)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3
+ 3*((d*x + c)*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x +
a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sin
h(b*x + a))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) - (cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + si
nh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - 3*cosh(
b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2
+ 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*sqrt(d*x + c))/((d^2*x + c*d)*cos
h(b*x + a)^3 + 3*(d^2*x + c*d)*cosh(b*x + a)^2*sinh(b*x + a) + 3*(d^2*x + c*d)*cosh(b*x + a)*sinh(b*x + a)^2 +
(d^2*x + c*d)*sinh(b*x + a)^3)

Sympy [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(sinh(b*x+a)**3/(d*x+c)**(3/2),x)

[Out]

Integral(sinh(a + b*x)**3/(c + d*x)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {3} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {\sqrt {3} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {3 \, \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}}{8 \, d} \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

1/8*(sqrt(3)*sqrt((d*x + c)*b/d)*e^(3*(b*c - a*d)/d)*gamma(-1/2, 3*(d*x + c)*b/d)/sqrt(d*x + c) - sqrt(3)*sqrt
(-(d*x + c)*b/d)*e^(-3*(b*c - a*d)/d)*gamma(-1/2, -3*(d*x + c)*b/d)/sqrt(d*x + c) - 3*sqrt((d*x + c)*b/d)*e^(-
a + b*c/d)*gamma(-1/2, (d*x + c)*b/d)/sqrt(d*x + c) + 3*sqrt(-(d*x + c)*b/d)*e^(a - b*c/d)*gamma(-1/2, -(d*x +
c)*b/d)/sqrt(d*x + c))/d

Giac [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(b*x + a)^3/(d*x + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

int(sinh(a + b*x)^3/(c + d*x)^(3/2),x)

[Out]

int(sinh(a + b*x)^3/(c + d*x)^(3/2), x)