3.1.51 \(\int \frac {\sinh ^2(a+b x)}{(c+d x)^{7/2}} \, dx\) [51]

Optimal. Leaf size=220 \[ -\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b^{5/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 b^{5/2} e^{2 a-\frac {2 b c}{d}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}} \]

[Out]

-8/15*b*cosh(b*x+a)*sinh(b*x+a)/d^2/(d*x+c)^(3/2)-2/5*sinh(b*x+a)^2/d/(d*x+c)^(5/2)-8/15*b^(5/2)*exp(-2*a+2*b*
c/d)*erf(2^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(7/2)+8/15*b^(5/2)*exp(2*a-2*b*c/d)*erfi(2^
(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(7/2)-16/15*b^2/d^3/(d*x+c)^(1/2)-32/15*b^2*sinh(b*x+a
)^2/d^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3395, 32, 3394, 12, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {8 \sqrt {2 \pi } b^{5/2} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 \sqrt {2 \pi } b^{5/2} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \sinh (a+b x) \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {16 b^2}{15 d^3 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b^2)/(15*d^3*Sqrt[c + d*x]) - (8*b^(5/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*
x])/Sqrt[d]])/(15*d^(7/2)) + (8*b^(5/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sq
rt[d]])/(15*d^(7/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(15*d^2*(c + d*x)^(3/2)) - (2*Sinh[a + b*x]^2)/(5*d*(
c + d*x)^(5/2)) - (32*b^2*Sinh[a + b*x]^2)/(15*d^3*Sqrt[c + d*x])

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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]

Rule 3395

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 {\sinh ^2(a+b x)}{(c+d x)^{7/2}} \, dx &=-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{(c+d x)^{3/2}} \, dx}{15 d^2}+\frac {\left (16 b^2\right ) \int \frac {\sinh ^2(a+b x)}{(c+d x)^{3/2}} \, dx}{15 d^2}\\ &=-\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {\left (64 i b^3\right ) \int \frac {i \sinh (2 a+2 b x)}{2 \sqrt {c+d x}} \, dx}{15 d^3}\\ &=-\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}+\frac {\left (32 b^3\right ) \int \frac {\sinh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{15 d^3}\\ &=-\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}+\frac {\left (16 b^3\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3}-\frac {\left (16 b^3\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3}\\ &=-\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {\left (32 b^3\right ) \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 )}{15 d^4}+\frac {\left (32 b^3\right ) \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 )}{15 d^4}\\ &=-\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {8 b^{5/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 b^{5/2} e^{2 a-\frac {2 b c}{d}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \sinh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(825\) vs. \(2(220)=440\).
time = 6.08, size = 825, normalized size = 3.75 \begin {gather*} \frac {e^{-\frac {2 b (c+d x)}{d}} \left (6 d^2 e^{\frac {2 b (c+d x)}{d}}-16 b^2 c^2 \cosh \left (2 a-\frac {2 b c}{d}\right )+4 b c d \cosh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 c^2 e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-4 b c d e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d x \cosh \left (2 a-\frac {2 b c}{d}\right )+4 b d^2 x \cosh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d e^{\frac {4 b (c+d x)}{d}} x \cosh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 e^{\frac {4 b (c+d x)}{d}} x \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 x^2 \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 e^{\frac {4 b (c+d x)}{d}} x^2 \cosh \left (2 a-\frac {2 b c}{d}\right )+16 \sqrt {2} d^2 e^{\frac {2 b (c+d x)}{d}} \left (\frac {b (c+d x)}{d}\right )^{5/2} \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 \left (2 a-\frac {2 b c}{d}\right )\right )+16 b^2 c^2 \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b c d \sinh \left (2 a-\frac {2 b c}{d}\right )+3 d^2 \sinh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 c^2 e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b c d e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )+32 b^2 c d x \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 x \sinh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d e^{\frac {4 b (c+d x)}{d}} x \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 e^{\frac {4 b (c+d x)}{d}} x \sinh \left (2 a-\frac {2 b c}{d}\right )+16 b^2 d^2 x^2 \sinh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 e^{\frac {4 b (c+d x)}{d}} x^2 \sinh \left (2 a-\frac {2 b c}{d}\right )+16 \sqrt {2} d^2 e^{\frac {2 b (c+d x)}{d}} \left (-\frac {b (c+d x)}{d}\right )^{5/2} \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 \left (2 a-\frac {2 b c}{d}\right )\right )\right )}{30 d^3 (c+d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*d^2*E^((2*b*(c + d*x))/d) - 16*b^2*c^2*Cosh[2*a - (2*b*c)/d] + 4*b*c*d*Cosh[2*a - (2*b*c)/d] - 3*d^2*Cosh[2
*a - (2*b*c)/d] - 16*b^2*c^2*E^((4*b*(c + d*x))/d)*Cosh[2*a - (2*b*c)/d] - 4*b*c*d*E^((4*b*(c + d*x))/d)*Cosh[
2*a - (2*b*c)/d] - 3*d^2*E^((4*b*(c + d*x))/d)*Cosh[2*a - (2*b*c)/d] - 32*b^2*c*d*x*Cosh[2*a - (2*b*c)/d] + 4*
b*d^2*x*Cosh[2*a - (2*b*c)/d] - 32*b^2*c*d*E^((4*b*(c + d*x))/d)*x*Cosh[2*a - (2*b*c)/d] - 4*b*d^2*E^((4*b*(c
+ d*x))/d)*x*Cosh[2*a - (2*b*c)/d] - 16*b^2*d^2*x^2*Cosh[2*a - (2*b*c)/d] - 16*b^2*d^2*E^((4*b*(c + d*x))/d)*x
^2*Cosh[2*a - (2*b*c)/d] + 16*Sqrt[2]*d^2*E^((2*b*(c + d*x))/d)*((b*(c + d*x))/d)^(5/2)*Gamma[1/2, (2*b*(c + d
*x))/d]*(Cosh[2*a - (2*b*c)/d] - Sinh[2*a - (2*b*c)/d]) + 16*b^2*c^2*Sinh[2*a - (2*b*c)/d] - 4*b*c*d*Sinh[2*a
- (2*b*c)/d] + 3*d^2*Sinh[2*a - (2*b*c)/d] - 16*b^2*c^2*E^((4*b*(c + d*x))/d)*Sinh[2*a - (2*b*c)/d] - 4*b*c*d*
E^((4*b*(c + d*x))/d)*Sinh[2*a - (2*b*c)/d] - 3*d^2*E^((4*b*(c + d*x))/d)*Sinh[2*a - (2*b*c)/d] + 32*b^2*c*d*x
*Sinh[2*a - (2*b*c)/d] - 4*b*d^2*x*Sinh[2*a - (2*b*c)/d] - 32*b^2*c*d*E^((4*b*(c + d*x))/d)*x*Sinh[2*a - (2*b*
c)/d] - 4*b*d^2*E^((4*b*(c + d*x))/d)*x*Sinh[2*a - (2*b*c)/d] + 16*b^2*d^2*x^2*Sinh[2*a - (2*b*c)/d] - 16*b^2*
d^2*E^((4*b*(c + d*x))/d)*x^2*Sinh[2*a - (2*b*c)/d] + 16*Sqrt[2]*d^2*E^((2*b*(c + d*x))/d)*(-((b*(c + d*x))/d)
)^(5/2)*Gamma[1/2, (-2*b*(c + d*x))/d]*(Cosh[2*a - (2*b*c)/d] + Sinh[2*a - (2*b*c)/d]))/(30*d^3*E^((2*b*(c + d
*x))/d)*(c + d*x)^(5/2))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (b x +a \right )}{\left (d x +c \right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 118, normalized size = 0.54 \begin {gather*} -\frac {\frac {5 \, \sqrt {2} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {5 \, \sqrt {2} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} - \frac {1}{{\left (d x + c\right )}^{\frac {5}{2}}}}{5 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(5*sqrt(2)*((d*x + c)*b/d)^(5/2)*e^(2*(b*c - a*d)/d)*gamma(-5/2, 2*(d*x + c)*b/d)/(d*x + c)^(5/2) + 5*sqr
t(2)*(-(d*x + c)*b/d)^(5/2)*e^(-2*(b*c - a*d)/d)*gamma(-5/2, -2*(d*x + c)*b/d)/(d*x + c)^(5/2) - 1/(d*x + c)^(
5/2))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (172) = 344\).
time = 0.46, size = 1352, normalized size = 6.15 \begin {gather*} -\frac {16 \, \sqrt {2} \sqrt {\pi } {\left ({\left (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}\right )} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (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}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\left (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}\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (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}\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (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}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (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}\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 ) + 16 \, \sqrt {2} \sqrt {\pi } {\left ({\left (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}\right )} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (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}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\left (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}\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (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}\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (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}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (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}\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 (16 \, b^{2} d^{2} x^{2} + {\left (16 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c^{2} + 4 \, b c d + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (16 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c^{2} + 4 \, b c d + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (16 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c^{2} + 4 \, b c d + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d + b d^{2}\right )} x\right )} \sinh \left (b x + a\right )^{4} + 16 \, b^{2} c^{2} - 6 \, d^{2} \cosh \left (b x + a\right )^{2} - 4 \, b c d + 6 \, {\left ({\left (16 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c^{2} + 4 \, b c d + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right )^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2} + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d - b d^{2}\right )} x + 4 \, {\left ({\left (16 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c^{2} + 4 \, b c d + 3 \, d^{2} + 4 \, {\left (8 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right )^{3} - 3 \, d^{2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {d x + c}}{30 \, {\left ({\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(16*sqrt(2)*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)^2*cosh(-2*
(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(-2*(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(-2*(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(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*((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(-2*(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(-2*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(2)*sqrt(d*x + c)*sqr
t(b/d)) + 16*sqrt(2)*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)^2*cosh(
-2*(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(-2*(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(-2*(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(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*((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(-2*(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(-2*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(2)*sqrt(d*x + c)
*sqrt(-b/d)) + (16*b^2*d^2*x^2 + (16*b^2*d^2*x^2 + 16*b^2*c^2 + 4*b*c*d + 3*d^2 + 4*(8*b^2*c*d + b*d^2)*x)*cos
h(b*x + a)^4 + 4*(16*b^2*d^2*x^2 + 16*b^2*c^2 + 4*b*c*d + 3*d^2 + 4*(8*b^2*c*d + b*d^2)*x)*cosh(b*x + a)*sinh(
b*x + a)^3 + (16*b^2*d^2*x^2 + 16*b^2*c^2 + 4*b*c*d + 3*d^2 + 4*(8*b^2*c*d + b*d^2)*x)*sinh(b*x + a)^4 + 16*b^
2*c^2 - 6*d^2*cosh(b*x + a)^2 - 4*b*c*d + 6*((16*b^2*d^2*x^2 + 16*b^2*c^2 + 4*b*c*d + 3*d^2 + 4*(8*b^2*c*d + b
*d^2)*x)*cosh(b*x + a)^2 - d^2)*sinh(b*x + a)^2 + 3*d^2 + 4*(8*b^2*c*d - b*d^2)*x + 4*((16*b^2*d^2*x^2 + 16*b^
2*c^2 + 4*b*c*d + 3*d^2 + 4*(8*b^2*c*d + b*d^2)*x)*cosh(b*x + a)^3 - 3*d^2*cosh(b*x + a))*sinh(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)^2 + 2*(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) + (d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)*sinh(b*x + a)^2
)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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