\(\int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx\) [60]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-3 \left (a+b \left (\frac {c}{d}+x\right )\right )} \left (\sqrt {3} e^{6 a+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )+3 e^{4 a+\frac {2 b c}{d}+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {3 b c}{d}} \left (-\left (1+e^{2 (a+b x)}\right )^3+3 e^{2 a+\frac {b c}{d}+3 b x} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )+\sqrt {3} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{4 d \sqrt {c+d x}} \] Input:

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

Output:

(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])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3794, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3794

\(\displaystyle -\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {6 i b \int \left (-\frac {i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {6 i b \left (\frac {i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {i \sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )}{d}\)

Input:

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

Output:

(-2*Cosh[a + b*x]^3)/(d*Sqrt[c + d*x]) + ((6*I)*b*(((I/8)*E^(-a + (b*c)/d) 
*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + ((I/8) 
*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[ 
d]])/(Sqrt[b]*Sqrt[d]) - ((I/8)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqr 
t[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) - ((I/8)*E^(3*a - (3*b*c)/d)*Sqrt[ 
Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d])))/d
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[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]
 

Maple [F]

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

Input:

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

Output:

int(cosh(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. 1344 vs. \(2 (182) = 364\).

Time = 0.11 (sec) , antiderivative size = 1344, normalized size of antiderivative = 5.46 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-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)*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(sq 
rt(3)*sqrt(d*x + c)*sqrt(-b/d)) + 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*co 
sh(-(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))*si 
nh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(pi)*((d*x ...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.80 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\frac {\sqrt {3} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {\sqrt {3} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}}{8 \, d} \] Input:

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

Output:

-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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(cosh(a + b*x)**3/(sqrt(c + d*x)*c + sqrt(c + d*x)*d*x),x)