\(\int (c+d x)^{3/2} \sinh (a+b x) \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 146 \[ \int (c+d x)^{3/2} \sinh (a+b x) \, dx=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2} \] Output:

(d*x+c)^(3/2)*cosh(b*x+a)/b-3/8*d^(3/2)*exp(-a+b*c/d)*Pi^(1/2)*erf(b^(1/2) 
*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)+3/8*d^(3/2)*exp(a-b*c/d)*Pi^(1/2)*erfi(b^( 
1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-3/2*d*(d*x+c)^(1/2)*sinh(b*x+a)/b^2
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.73 \[ \int (c+d x)^{3/2} \sinh (a+b x) \, dx=\frac {d e^{-a-\frac {b c}{d}} \sqrt {c+d x} \left (-\frac {e^{2 a} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )}{\sqrt {-\frac {b (c+d x)}{d}}}+\frac {e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b (c+d x)}{d}}}\right )}{2 b^2} \] Input:

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

Output:

(d*E^(-a - (b*c)/d)*Sqrt[c + d*x]*(-((E^(2*a)*Gamma[5/2, -((b*(c + d*x))/d 
)])/Sqrt[-((b*(c + d*x))/d)]) + (E^((2*b*c)/d)*Gamma[5/2, (b*(c + d*x))/d] 
)/Sqrt[(b*(c + d*x))/d]))/(2*b^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3789, 2611, 2633, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^{3/2} \sinh (a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -i (c+d x)^{3/2} \sin (i a+i b x)dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int (c+d x)^{3/2} \sin (i a+i b x)dx\)

\(\Big \downarrow \) 3777

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \int \sqrt {c+d x} \cosh (a+b x)dx}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \int \sqrt {c+d x} \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{2 b}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {i d \int -\frac {i \sinh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {d \int \frac {\sinh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {d \int -\frac {i \sin (i a+i b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \int \frac {\sin (i a+i b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {1}{2} i \int \frac {e^{a+b x}}{\sqrt {c+d x}}dx-\frac {1}{2} i \int \frac {e^{-a-b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \int e^{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d}-\frac {i \int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle -i \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )\)

Input:

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

Output:

(-I)*((I*(c + d*x)^(3/2)*Cosh[a + b*x])/b - (((3*I)/2)*d*(((I/2)*d*(((-1/2 
*I)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[ 
b]*Sqrt[d]) + ((I/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x]) 
/Sqrt[d]])/(Sqrt[b]*Sqrt[d])))/b + (Sqrt[c + d*x]*Sinh[a + b*x])/b))/b)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2633

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 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3777

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

rule 3789

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

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (110) = 220\).

Time = 0.11 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.64 \[ \int (c+d x)^{3/2} \sinh (a+b x) \, dx=-\frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c + {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \sinh \left (b x + a\right )^{2} + 3 \, b d\right )} \sqrt {d x + c}}{8 \, {\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \] Input:

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

Output:

-1/8*(3*sqrt(pi)*(d^2*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - d^2*cosh(b*x + 
a)*sinh(-(b*c - a*d)/d) + (d^2*cosh(-(b*c - a*d)/d) - d^2*sinh(-(b*c - a*d 
)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(pi)*( 
d^2*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + d^2*cosh(b*x + a)*sinh(-(b*c - a* 
d)/d) + (d^2*cosh(-(b*c - a*d)/d) + d^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a 
))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) - 2*(2*b^2*d*x + 2*b^2*c + (2* 
b^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a)^2 + 2*(2*b^2*d*x + 2*b^2*c - 3*b* 
d)*cosh(b*x + a)*sinh(b*x + a) + (2*b^2*d*x + 2*b^2*c - 3*b*d)*sinh(b*x + 
a)^2 + 3*b*d)*sqrt(d*x + c))/(b^3*cosh(b*x + a) + b^3*sinh(b*x + a))
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (110) = 220\).

Time = 0.05 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.84 \[ \int (c+d x)^{3/2} \sinh (a+b x) \, dx=\frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} \sinh \left (b x + a\right ) + \frac {{\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} - \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {b c}{d}\right )} + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {b c}{d}\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{a} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{a} + 15 \, \sqrt {d x + c} d^{3} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{3}}\right )} b}{d}}{40 \, d} \] Input:

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

Output:

1/40*(16*(d*x + c)^(5/2)*sinh(b*x + a) + (15*sqrt(pi)*d^3*erf(sqrt(d*x + c 
)*sqrt(-b/d))*e^(a - b*c/d)/(b^3*sqrt(-b/d)) - 15*sqrt(pi)*d^3*erf(sqrt(d* 
x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b^3*sqrt(b/d)) + 2*(4*(d*x + c)^(5/2)*b^ 
2*d*e^(b*c/d) + 10*(d*x + c)^(3/2)*b*d^2*e^(b*c/d) + 15*sqrt(d*x + c)*d^3* 
e^(b*c/d))*e^(-a - (d*x + c)*b/d)/b^3 - 2*(4*(d*x + c)^(5/2)*b^2*d*e^a - 1 
0*(d*x + c)^(3/2)*b*d^2*e^a + 15*sqrt(d*x + c)*d^3*e^a)*e^((d*x + c)*b/d - 
 b*c/d)/b^3)*b/d)/d
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.38 \[ \int (c+d x)^{3/2} \sinh (a+b x) \, dx=\frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )}}{\sqrt {b d} b^{2}} - \frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )}}{\sqrt {-b d} b^{2}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d + 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (-\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}}}{8 \, d} \] Input:

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

Output:

1/8*(3*sqrt(pi)*d^3*erf(-sqrt(b*d)*sqrt(d*x + c)/d)*e^((b*c - a*d)/d)/(sqr 
t(b*d)*b^2) - 3*sqrt(pi)*d^3*erf(-sqrt(-b*d)*sqrt(d*x + c)/d)*e^(-(b*c - a 
*d)/d)/(sqrt(-b*d)*b^2) + 2*(2*(d*x + c)^(3/2)*b*d - 3*sqrt(d*x + c)*d^2)* 
e^(((d*x + c)*b - b*c + a*d)/d)/b^2 + 2*(2*(d*x + c)^(3/2)*b*d + 3*sqrt(d* 
x + c)*d^2)*e^(-((d*x + c)*b - b*c + a*d)/d)/b^2)/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(c + d*x)*sinh(a + b*x)*x,x)*d + int(sqrt(c + d*x)*sinh(a + b*x),x 
)*c