\(\int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx\) [46]

Optimal result

Integrand size = 16, antiderivative size = 149 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}+\frac {2 b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 b \sinh (a+b x)}{3 d^2 \sqrt {c+d x}} \] Output:

-2/3*cosh(b*x+a)/d/(d*x+c)^(3/2)+2/3*b^(3/2)*exp(-a+b*c/d)*Pi^(1/2)*erf(b^ 
(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^(5/2)+2/3*b^(3/2)*exp(a-b*c/d)*Pi^(1/2)*erf 
i(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^(5/2)-4/3*b*sinh(b*x+a)/d^2/(d*x+c)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\frac {e^{-a} \left (-e^{-b x} \left (d \left (1+e^{2 (a+b x)}\right )+2 b \left (-1+e^{2 (a+b x)}\right ) (c+d x)+2 d e^{b \left (\frac {c}{d}+x\right )} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )\right )-2 d e^{2 a-\frac {b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )\right )}{3 d^2 (c+d x)^{3/2}} \] Input:

Integrate[Cosh[a + b*x]/(c + d*x)^(5/2),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3778, 26, 3042, 26, 3778, 3042, 3788, 26, 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.

\(\displaystyle \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{(c+d x)^{5/2}}dx\)

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

Int[Cosh[a + b*x]/(c + d*x)^(5/2),x]
 

Output:

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

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 3778

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

rule 3788

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

Maple [F]

\[\int \frac {\cosh \left (b x +a \right )}{\left (d x +c \right )^{\frac {5}{2}}}d x\]

Input:

int(cosh(b*x+a)/(d*x+c)^(5/2),x)
 

Output:

int(cosh(b*x+a)/(d*x+c)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (111) = 222\).

Time = 0.13 (sec) , antiderivative size = 534, normalized size of antiderivative = 3.58 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {\pi } {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 2 \, \sqrt {\pi } {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + {\left (2 \, b d x - {\left (2 \, b d x + 2 \, b c + d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (2 \, b d x + 2 \, b c + d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (2 \, b d x + 2 \, b c + d\right )} \sinh \left (b x + a\right )^{2} + 2 \, b c - d\right )} \sqrt {d x + c}}{3 \, {\left ({\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \cosh \left (b x + a\right ) + {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \sinh \left (b x + a\right )\right )}} \] Input:

integrate(cosh(b*x+a)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

1/3*(2*sqrt(pi)*((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh(b*x + a)*cosh(-(b*c 
- a*d)/d) - (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh(b*x + a)*sinh(-(b*c - a*d 
)/d) + ((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh(-(b*c - a*d)/d) - (b*d^2*x^2 
+ 2*b*c*d*x + b*c^2)*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sq 
rt(d*x + c)*sqrt(b/d)) - 2*sqrt(pi)*((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh( 
b*x + a)*cosh(-(b*c - a*d)/d) + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh(b*x + 
 a)*sinh(-(b*c - a*d)/d) + ((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cosh(-(b*c - a 
*d)/d) + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sinh(-(b*c - a*d)/d))*sinh(b*x + 
a))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) + (2*b*d*x - (2*b*d*x + 2*b*c 
 + d)*cosh(b*x + a)^2 - 2*(2*b*d*x + 2*b*c + d)*cosh(b*x + a)*sinh(b*x + a 
) - (2*b*d*x + 2*b*c + d)*sinh(b*x + a)^2 + 2*b*c - d)*sqrt(d*x + c))/((d^ 
4*x^2 + 2*c*d^3*x + c^2*d^2)*cosh(b*x + a) + (d^4*x^2 + 2*c*d^3*x + c^2*d^ 
2)*sinh(b*x + a))
 

Sympy [F]

\[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cosh {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cosh(b*x+a)/(d*x+c)**(5/2),x)
 

Output:

Integral(cosh(a + b*x)/(c + d*x)**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.77 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\frac {\frac {{\left (\frac {\sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {\sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}\right )} b}{d} - \frac {2 \, \cosh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {3}{2}}}}{3 \, d} \] Input:

integrate(cosh(b*x+a)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\cosh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cosh(b*x+a)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

integrate(cosh(b*x + a)/(d*x + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(cosh(a + b*x)/(c + d*x)^(5/2),x)
 

Output:

int(cosh(a + b*x)/(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cosh \left (b x +a \right )}{\sqrt {d x +c}\, c^{2}+2 \sqrt {d x +c}\, c d x +\sqrt {d x +c}\, d^{2} x^{2}}d x \] Input:

int(cosh(b*x+a)/(d*x+c)^(5/2),x)
 

Output:

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