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

Optimal. Leaf size=142 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh ^2(a+b x)}{d \sqrt {c+d x}} \]

[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)

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Rubi [A]  time = 0.23, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3313, 12, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh ^2(a+b x)}{d \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 3308

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 3313

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*} \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 {(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) \operatorname {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) \operatorname {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*}

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Mathematica [B]  time = 2.97, size = 570, normalized size = 4.01 \[ \frac {e^{-\frac {2 b (c+d x)}{d}} \left (-\sqrt {2 \pi } \sqrt {b} \cosh (2 a) \sqrt {c+d x} e^{\frac {2 b (c+d x)}{d}} \sinh \left (\frac {2 b c}{d}\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )-\sqrt {2 \pi } \sqrt {b} \cosh (2 a) \sqrt {c+d x} e^{\frac {2 b (c+d x)}{d}} \sinh \left (\frac {2 b c}{d}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+\sqrt {d} \sinh (2 a) e^{\frac {4 b (c+d x)}{d}} \sinh \left (\frac {2 b c}{d}\right )-\sqrt {d} \cosh (2 a) e^{\frac {4 b (c+d x)}{d}} \cosh \left (\frac {2 b c}{d}\right )-\sqrt {d} \sinh (2 a) e^{\frac {4 b (c+d x)}{d}} \cosh \left (\frac {2 b c}{d}\right )+\sqrt {d} \cosh (2 a) e^{\frac {4 b (c+d x)}{d}} \sinh \left (\frac {2 b c}{d}\right )+\sqrt {2} \sqrt {d} e^{\frac {2 b (c+d x)}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac {2 b c}{d}\right )+\sinh (2 a) \cosh \left (\frac {2 b c}{d}\right )\right )+\sqrt {2} \sqrt {d} e^{\frac {2 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {2 b (c+d x)}{d}\right ) \left (\cosh (2 a) \cosh \left (\frac {2 b c}{d}\right )-\sinh (2 a) \left (\sinh \left (\frac {2 b c}{d}\right )+\cosh \left (\frac {2 b c}{d}\right )\right )\right )+\sqrt {d} \sinh (2 a) \sinh \left (\frac {2 b c}{d}\right )-\sqrt {d} \cosh (2 a) \cosh \left (\frac {2 b c}{d}\right )+\sqrt {d} \sinh (2 a) \cosh \left (\frac {2 b c}{d}\right )-\sqrt {d} \cosh (2 a) \sinh \left (\frac {2 b c}{d}\right )-2 \sqrt {d} e^{\frac {2 b (c+d x)}{d}}\right )}{2 d^{3/2} \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d]*E^((2*b*(c + d*x))/d) - Sqrt[d]*Cosh[2*a]*Cosh[(2*b*c)/d] - Sqrt[d]*E^((4*b*(c + d*x))/d)*Cosh[2*a
]*Cosh[(2*b*c)/d] + Sqrt[d]*Cosh[(2*b*c)/d]*Sinh[2*a] - Sqrt[d]*E^((4*b*(c + d*x))/d)*Cosh[(2*b*c)/d]*Sinh[2*a
] + Sqrt[2]*Sqrt[d]*E^((2*b*(c + d*x))/d)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-2*b*(c + d*x))/d]*(Cosh[2*a -
(2*b*c)/d] + Cosh[(2*b*c)/d]*Sinh[2*a]) - Sqrt[d]*Cosh[2*a]*Sinh[(2*b*c)/d] + Sqrt[d]*E^((4*b*(c + d*x))/d)*Co
sh[2*a]*Sinh[(2*b*c)/d] - Sqrt[b]*E^((2*b*(c + d*x))/d)*Sqrt[2*Pi]*Sqrt[c + d*x]*Cosh[2*a]*Erf[(Sqrt[2]*Sqrt[b
]*Sqrt[c + d*x])/Sqrt[d]]*Sinh[(2*b*c)/d] - Sqrt[b]*E^((2*b*(c + d*x))/d)*Sqrt[2*Pi]*Sqrt[c + d*x]*Cosh[2*a]*E
rfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*Sinh[(2*b*c)/d] + Sqrt[d]*Sinh[2*a]*Sinh[(2*b*c)/d] + Sqrt[d]*E^(
(4*b*(c + d*x))/d)*Sinh[2*a]*Sinh[(2*b*c)/d] + Sqrt[2]*Sqrt[d]*E^((2*b*(c + d*x))/d)*Sqrt[(b*(c + d*x))/d]*Gam
ma[1/2, (2*b*(c + d*x))/d]*(Cosh[2*a]*Cosh[(2*b*c)/d] - Sinh[2*a]*(Cosh[(2*b*c)/d] + Sinh[(2*b*c)/d])))/(2*d^(
3/2)*E^((2*b*(c + d*x))/d)*Sqrt[c + d*x])

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fricas [B]  time = 0.54, size = 569, normalized size = 4.01 \[ -\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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}\left (b x +a \right )}{\left (d x +c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 1.28, size = 116, normalized size = 0.82 \[ -\frac {\frac {\sqrt {2} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {2 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {\sqrt {2} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {4}{\sqrt {d x + c}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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