\(\int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx\) [52]

Optimal result

Integrand size = 18, antiderivative size = 142 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {2 \cosh ^2(a+b x)}{d \sqrt {c+d x}}-\frac {\sqrt {b} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {b} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3394, 12, 3389, 2211, 2235, 2236} \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh ^2(a+b x)}{d \sqrt {c+d x}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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]

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.07 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-2 \left (a+b \left (\frac {c}{d}+x\right )\right )} \left (\sqrt {2} e^{4 a+2 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {2 b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \left (-\left (1+e^{2 (a+b x)}\right )^2+\sqrt {2} e^{\frac {2 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {2 b (c+d x)}{d}\right )\right )\right )}{2 d \sqrt {c+d x}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (109) = 218\).

Time = 0.28 (sec) , antiderivative size = 569, normalized size of antiderivative = 4.01 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (d x + c\right )} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (d x + c\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d x + c\right )} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d x + c\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \sqrt {d x + c}}{2 \, {\left ({\left (d^{2} x + c d\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (d^{2} x + c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (d^{2} x + c d\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {\frac {\sqrt {2} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {2 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {\sqrt {2} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {4}{\sqrt {d x + c}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(sqrt(2)*sqrt((d*x + c)*b/d)*e^(2*(b*c - a*d)/d)*gamma(-1/2, 2*(d*x + c)*b/d)/sqrt(d*x + c) + sqrt(2)*sqr
t(-(d*x + c)*b/d)*e^(-2*(b*c - a*d)/d)*gamma(-1/2, -2*(d*x + c)*b/d)/sqrt(d*x + c) + 4/sqrt(d*x + c))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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