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3.62 \(\int \frac {\cosh ^3(a+b x)}{(c+d x)^{7/2}} \, dx\)

Optimal. Leaf size=331 \[ -\frac {\sqrt {\pi } b^{5/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {3 \sqrt {3 \pi } b^{5/2} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {\sqrt {\pi } b^{5/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {3 \sqrt {3 \pi } b^{5/2} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \sinh (a+b x) \cosh ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}} \]

[Out]

-2/5*cosh(b*x+a)^3/d/(d*x+c)^(5/2)-4/5*b*cosh(b*x+a)^2*sinh(b*x+a)/d^2/(d*x+c)^(3/2)-1/5*b^(5/2)*exp(-a+b*c/d)
*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/d^(7/2)+1/5*b^(5/2)*exp(a-b*c/d)*erfi(b^(1/2)*(d*x+c)^(1/2)/d^(1/
2))*Pi^(1/2)/d^(7/2)-3/5*b^(5/2)*exp(-3*a+3*b*c/d)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)
/d^(7/2)+3/5*b^(5/2)*exp(3*a-3*b*c/d)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)/d^(7/2)+16/
5*b^2*cosh(b*x+a)/d^3/(d*x+c)^(1/2)-24/5*b^2*cosh(b*x+a)^3/d^3/(d*x+c)^(1/2)

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Rubi [A]  time = 0.68, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3314, 3297, 3308, 2180, 2204, 2205, 3313} \[ -\frac {\sqrt {\pi } b^{5/2} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {3 \sqrt {3 \pi } b^{5/2} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {\sqrt {\pi } b^{5/2} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {3 \sqrt {3 \pi } b^{5/2} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \sinh (a+b x) \cosh ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3/(c + d*x)^(7/2),x]

[Out]

(16*b^2*Cosh[a + b*x])/(5*d^3*Sqrt[c + d*x]) - (2*Cosh[a + b*x]^3)/(5*d*(c + d*x)^(5/2)) - (24*b^2*Cosh[a + b*
x]^3)/(5*d^3*Sqrt[c + d*x]) - (b^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7
/2)) - (3*b^(5/2)*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) +
(b^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (3*b^(5/2)*E^(3*a - (3*
b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (4*b*Cosh[a + b*x]^2*Sinh[a +
b*x])/(5*d^2*(c + d*x)^(3/2))

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 3297

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 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]

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rubi steps

\begin {align*} \int \frac {\cosh ^3(a+b x)}{(c+d x)^{7/2}} \, dx &=-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {\left (8 b^2\right ) \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}+\frac {\left (12 b^2\right ) \int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}\\ &=\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (72 i b^3\right ) \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}{5 d^3}-\frac {\left (16 b^3\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {\left (8 b^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (8 b^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3\right ) \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (16 b^3\right ) \operatorname {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 )}{5 d^4}-\frac {\left (16 b^3\right ) \operatorname {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 )}{5 d^4}+\frac {\left (9 b^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (9 b^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (9 b^3\right ) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (9 b^3\right ) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {8 b^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {8 b^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {\left (18 b^3\right ) \operatorname {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{5 d^4}-\frac {\left (18 b^3\right ) \operatorname {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 )}{5 d^4}+\frac {\left (18 b^3\right ) \operatorname {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 )}{5 d^4}+\frac {\left (18 b^3\right ) \operatorname {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{5 d^4}\\ &=\frac {16 b^2 \cosh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \cosh ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {b^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {3 b^{5/2} 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 )}{5 d^{7/2}}+\frac {b^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {3 b^{5/2} 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 )}{5 d^{7/2}}-\frac {4 b \cosh ^2(a+b x) \sinh (a+b x)}{5 d^2 (c+d x)^{3/2}}\\ \end {align*}

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Mathematica [B]  time = 6.35, size = 3211, normalized size = 9.70 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^3/(c + d*x)^(7/2),x]

[Out]

(3*(Sinh[a]*(-1/30*((-2*E^((b*(c + d*x))/d)*(3*d^2 + 2*b*d*(c + d*x) + 4*b^2*(c + d*x)^2) + 8*d^2*(-((b*(c + d
*x))/d))^(5/2)*Gamma[1/2, -((b*(c + d*x))/d)] + (-6*d^2 + 4*b*d*(c + d*x) - 8*b^2*(c + d*x)^2 + 8*b*d*E^((b*(c
 + d*x))/d)*(c + d*x)*((b*(c + d*x))/d)^(3/2)*Gamma[1/2, (b*(c + d*x))/d])/E^((b*(c + d*x))/d))*Sinh[(b*c)/d])
/(d^3*(c + d*x)^(5/2)) + (2*Cosh[(b*c)/d]*(-1/2*(b*(c + d*x)*(2*E^((b*(c + d*x))/d)*(d + 2*b*(c + d*x)) + 4*d*
(-((b*(c + d*x))/d))^(3/2)*Gamma[1/2, -((b*(c + d*x))/d)] + (2*(d - 2*b*(c + d*x) + 2*d*E^((b*(c + d*x))/d)*((
b*(c + d*x))/d)^(3/2)*Gamma[1/2, (b*(c + d*x))/d]))/E^((b*(c + d*x))/d))) - 3*d^2*Sinh[(b*(c + d*x))/d]))/(15*
d^3*(c + d*x)^(5/2))) + Cosh[a]*((Cosh[(b*c)/d]*(-2*E^((b*(c + d*x))/d)*(3*d^2 + 2*b*d*(c + d*x) + 4*b^2*(c +
d*x)^2) + 8*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, -((b*(c + d*x))/d)] + (-6*d^2 + 4*b*d*(c + d*x) - 8*b^2*
(c + d*x)^2 + 8*b*d*E^((b*(c + d*x))/d)*(c + d*x)*((b*(c + d*x))/d)^(3/2)*Gamma[1/2, (b*(c + d*x))/d])/E^((b*(
c + d*x))/d)))/(30*d^3*(c + d*x)^(5/2)) - (2*Sinh[(b*c)/d]*(-1/2*(b*(c + d*x)*(2*E^((b*(c + d*x))/d)*(d + 2*b*
(c + d*x)) + 4*d*(-((b*(c + d*x))/d))^(3/2)*Gamma[1/2, -((b*(c + d*x))/d)] + (2*(d - 2*b*(c + d*x) + 2*d*E^((b
*(c + d*x))/d)*((b*(c + d*x))/d)^(3/2)*Gamma[1/2, (b*(c + d*x))/d]))/E^((b*(c + d*x))/d))) - 3*d^2*Sinh[(b*(c
+ d*x))/d]))/(15*d^3*(c + d*x)^(5/2)))))/4 + (Sinh[3*a]*(-1/10*((1 + 2*Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*x))
/d)*(d^2 + 2*b*d*(c + d*x) + 12*b^2*(c + d*x)^2) + 24*Sqrt[3]*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3*b*
(c + d*x))/d] + (-2*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c + d*x)^2 + 24*Sqrt[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(c +
 d*x))/d)^(5/2)*Gamma[1/2, (3*b*(c + d*x))/d])/E^((3*b*(c + d*x))/d))*Sinh[(b*c)/d])/(d^3*(c + d*x)^(5/2)) - (
2*Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c +
d*x])/Sqrt[d]] - 6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Sqrt[d]*
(2*b*d*(c + d*x)*Cosh[(3*b*(c + d*x))/d] + (d^2 + 12*b^2*(c + d*x)^2)*Sinh[(3*b*(c + d*x))/d])))/(5*d^(7/2)*(c
 + d*x)^(5/2))) + Cosh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*x))/d)*(d^2 + 2*b*d*(c
 + d*x) + 12*b^2*(c + d*x)^2) + 24*Sqrt[3]*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3*b*(c + d*x))/d] + (-2
*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c + d*x)^2 + 24*Sqrt[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(c + d*x))/d)^(5/2)*Gam
ma[1/2, (3*b*(c + d*x))/d])/E^((3*b*(c + d*x))/d)))/(10*d^3*(c + d*x)^(5/2)) + (2*(1 + 2*Cosh[(2*b*c)/d])*Sinh
[(b*c)/d]*(-6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] - 6*b^(5/2)*Sqrt
[3*Pi]*(c + d*x)^(5/2)*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Sqrt[d]*(2*b*d*(c + d*x)*Cosh[(3*b*(c +
 d*x))/d] + (d^2 + 12*b^2*(c + d*x)^2)*Sinh[(3*b*(c + d*x))/d])))/(5*d^(7/2)*(c + d*x)^(5/2))))/4 + (-(Cosh[3*
a]*(-1/10*((1 + 2*Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*x))/d)*(d^2 + 2*b*d*(c + d*x) + 12*b^2*(c + d*x)^2) + 24
*Sqrt[3]*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3*b*(c + d*x))/d] + (-2*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c
 + d*x)^2 + 24*Sqrt[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(c + d*x))/d)^(5/2)*Gamma[1/2, (3*b*(c + d*x))/d])/E^((3*
b*(c + d*x))/d))*Sinh[(b*c)/d])/(d^3*(c + d*x)^(5/2)) - (2*Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-6*b^(5/2)*
Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] - 6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)
*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Sqrt[d]*(2*b*d*(c + d*x)*Cosh[(3*b*(c + d*x))/d] + (d^2 + 12*
b^2*(c + d*x)^2)*Sinh[(3*b*(c + d*x))/d])))/(5*d^(7/2)*(c + d*x)^(5/2)))) - Sinh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*
Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*x))/d)*(d^2 + 2*b*d*(c + d*x) + 12*b^2*(c + d*x)^2) + 24*Sqrt[3]*d^2*(-((b
*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3*b*(c + d*x))/d] + (-2*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c + d*x)^2 + 24*Sqr
t[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(c + d*x))/d)^(5/2)*Gamma[1/2, (3*b*(c + d*x))/d])/E^((3*b*(c + d*x))/d)))/
(10*d^3*(c + d*x)^(5/2)) + (2*(1 + 2*Cosh[(2*b*c)/d])*Sinh[(b*c)/d]*(-6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf
[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] - 6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c
+ d*x])/Sqrt[d]] + Sqrt[d]*(2*b*d*(c + d*x)*Cosh[(3*b*(c + d*x))/d] + (d^2 + 12*b^2*(c + d*x)^2)*Sinh[(3*b*(c
+ d*x))/d])))/(5*d^(7/2)*(c + d*x)^(5/2))))/8 + (Cosh[3*a]*(-1/10*((1 + 2*Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*
x))/d)*(d^2 + 2*b*d*(c + d*x) + 12*b^2*(c + d*x)^2) + 24*Sqrt[3]*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3
*b*(c + d*x))/d] + (-2*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c + d*x)^2 + 24*Sqrt[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(
c + d*x))/d)^(5/2)*Gamma[1/2, (3*b*(c + d*x))/d])/E^((3*b*(c + d*x))/d))*Sinh[(b*c)/d])/(d^3*(c + d*x)^(5/2))
- (2*Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c
 + d*x])/Sqrt[d]] - 6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Sqrt[
d]*(2*b*d*(c + d*x)*Cosh[(3*b*(c + d*x))/d] + (d^2 + 12*b^2*(c + d*x)^2)*Sinh[(3*b*(c + d*x))/d])))/(5*d^(7/2)
*(c + d*x)^(5/2))) + Sinh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-2*E^((3*b*(c + d*x))/d)*(d^2 + 2*b*d
*(c + d*x) + 12*b^2*(c + d*x)^2) + 24*Sqrt[3]*d^2*(-((b*(c + d*x))/d))^(5/2)*Gamma[1/2, (-3*b*(c + d*x))/d] +
(-2*d^2 + 4*b*d*(c + d*x) - 24*b^2*(c + d*x)^2 + 24*Sqrt[3]*d^2*E^((3*b*(c + d*x))/d)*((b*(c + d*x))/d)^(5/2)*
Gamma[1/2, (3*b*(c + d*x))/d])/E^((3*b*(c + d*x))/d)))/(10*d^3*(c + d*x)^(5/2)) + (2*(1 + 2*Cosh[(2*b*c)/d])*S
inh[(b*c)/d]*(-6*b^(5/2)*Sqrt[3*Pi]*(c + d*x)^(5/2)*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] - 6*b^(5/2)*S
qrt[3*Pi]*(c + d*x)^(5/2)*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Sqrt[d]*(2*b*d*(c + d*x)*Cosh[(3*b*(
c + d*x))/d] + (d^2 + 12*b^2*(c + d*x)^2)*Sinh[(3*b*(c + d*x))/d])))/(5*d^(7/2)*(c + d*x)^(5/2))))/8

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fricas [B]  time = 0.62, size = 3280, normalized size = 9.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

-1/20*(12*sqrt(3)*sqrt(pi)*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*cosh(-3*
(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)
/d) + ((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(-3*(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2
*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*
x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c
^2*d*x + b^2*c^3)*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 +
3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d
*x + b^2*c^3)*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)) + 12*sqrt(3)*sqrt(pi)*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*cosh(-3
*(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*sinh(-3*(b*c - a*d
)/d) + ((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(-3*(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^
2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2
*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*
c^2*d*x + b^2*c^3)*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 +
 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*
d*x + b^2*c^3)*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)*sqr
t(-b/d)) + 4*sqrt(pi)*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*cosh(-(b*c -
a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + ((b
^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(-(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2
+ 3*b^2*c^2*d*x + b^2*c^3)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c
^2*d*x + b^2*c^3)*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^
3)*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b
^2*c^3)*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) - (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(
b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) + 4*sqrt(pi)*((b^2*d^3*
x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c
*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + ((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3
*b^2*c^2*d*x + b^2*c^3)*cosh(-(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sinh(
-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)*
cosh(-(b*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)*sinh(-(b*c - a*
d)/d))*sinh(b*x + a)^2 + 3*((b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^2*cosh(-(b
*c - a*d)/d) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))
*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) + ((12*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(
12*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^6 + 6*(12*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12*b^2*c*d + b*d^
2)*x)*cosh(b*x + a)*sinh(b*x + a)^5 + (12*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12*b^2*c*d + b*d^2)*x)
*sinh(b*x + a)^6 + 12*b^2*d^2*x^2 + (4*b^2*d^2*x^2 + 4*b^2*c^2 + 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d + b*d^2)*x)*co
sh(b*x + a)^4 + (4*b^2*d^2*x^2 + 4*b^2*c^2 + 2*b*c*d + 15*(12*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12
*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^2 + 3*d^2 + 2*(4*b^2*c*d + b*d^2)*x)*sinh(b*x + a)^4 + 12*b^2*c^2 + 4*(5*(1
2*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^3 + (4*b^2*d^2*x^2 + 4*b^
2*c^2 + 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d + b*d^2)*x)*cosh(b*x + a))*sinh(b*x + a)^3 - 2*b*c*d + (4*b^2*d^2*x^2 +
 4*b^2*c^2 - 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d - b*d^2)*x)*cosh(b*x + a)^2 + (4*b^2*d^2*x^2 + 15*(12*b^2*d^2*x^2
+ 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^4 + 4*b^2*c^2 - 2*b*c*d + 6*(4*b^2*d^2*
x^2 + 4*b^2*c^2 + 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^2 + 3*d^2 + 2*(4*b^2*c*d - b*d^2)*x
)*sinh(b*x + a)^2 + d^2 + 2*(12*b^2*c*d - b*d^2)*x + 2*(3*(12*b^2*d^2*x^2 + 12*b^2*c^2 + 2*b*c*d + d^2 + 2*(12
*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^5 + 2*(4*b^2*d^2*x^2 + 4*b^2*c^2 + 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d + b*d^2)*
x)*cosh(b*x + a)^3 + (4*b^2*d^2*x^2 + 4*b^2*c^2 - 2*b*c*d + 3*d^2 + 2*(4*b^2*c*d - b*d^2)*x)*cosh(b*x + a))*si
nh(b*x + a))*sqrt(d*x + c))/((d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)*cosh(b*x + a)^3 + 3*(d^6*x^3 + 3*
c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)*cosh(b*x + a)^2*sinh(b*x + a) + 3*(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^
3*d^3)*cosh(b*x + a)*sinh(b*x + a)^2 + (d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)*sinh(b*x + a)^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^3/(d*x + c)^(7/2), x)

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maple [F]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}\left (b x +a \right )}{\left (d x +c \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3/(d*x+c)^(7/2),x)

[Out]

int(cosh(b*x+a)^3/(d*x+c)^(7/2),x)

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maxima [A]  time = 0.73, size = 196, normalized size = 0.59 \[ -\frac {3 \, {\left (\frac {3 \, \sqrt {3} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {3 \, \sqrt {3} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {\left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {\left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}}\right )}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

-3/8*(3*sqrt(3)*((d*x + c)*b/d)^(5/2)*e^(3*(b*c - a*d)/d)*gamma(-5/2, 3*(d*x + c)*b/d)/(d*x + c)^(5/2) + 3*sqr
t(3)*(-(d*x + c)*b/d)^(5/2)*e^(-3*(b*c - a*d)/d)*gamma(-5/2, -3*(d*x + c)*b/d)/(d*x + c)^(5/2) + ((d*x + c)*b/
d)^(5/2)*e^(-a + b*c/d)*gamma(-5/2, (d*x + c)*b/d)/(d*x + c)^(5/2) + (-(d*x + c)*b/d)^(5/2)*e^(a - b*c/d)*gamm
a(-5/2, -(d*x + c)*b/d)/(d*x + c)^(5/2))/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3/(c + d*x)^(7/2),x)

[Out]

int(cosh(a + b*x)^3/(c + d*x)^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3/(d*x+c)**(7/2),x)

[Out]

Integral(cosh(a + b*x)**3/(c + d*x)**(7/2), x)

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