\(\int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx\) [45]

Optimal result

Integrand size = 16, antiderivative size = 119 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}}-\frac {\sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}} \]

[Out]

-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)+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)-2*cosh(b*x+a)/d/(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3378, 3389, 2211, 2235, 2236} \[ \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}} \]

[In]

Int[Cosh[a + b*x]/(c + d*x)^(3/2),x]

[Out]

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

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 3378

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] - Dist[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 3389

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

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-a} \left (-e^{-b x} \left (1+e^{2 (a+b x)}\right )+e^{\frac {b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )+e^{2 a-\frac {b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )\right )}{d \sqrt {c+d x}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (91) = 182\).

Time = 0.27 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.84 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + \sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + \sqrt {d x + c} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )}}{{\left (d^{2} x + c d\right )} \cosh \left (b x + a\right ) + {\left (d^{2} x + c d\right )} \sinh \left (b x + a\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\frac {{\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{\sqrt {\frac {b}{d}}}\right )} b}{d} - \frac {2 \, \cosh \left (b x + a\right )}{\sqrt {d x + c}}}{d} \]

[In]

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

[Out]

((sqrt(pi)*erf(sqrt(d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/sqrt(-b/d) - sqrt(pi)*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-
a + b*c/d)/sqrt(b/d))*b/d - 2*cosh(b*x + a)/sqrt(d*x + c))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cosh(a + b*x)/(c + d*x)^(3/2),x)

[Out]

int(cosh(a + b*x)/(c + d*x)^(3/2), x)