\(\int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx\) [46]

Optimal result

Integrand size = 18, antiderivative size = 211 \[ \int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx=-\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d}+\frac {3 d^{3/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2} \]

[Out]

-1/5*(d*x+c)^(5/2)/d+1/2*(d*x+c)^(3/2)*cosh(b*x+a)*sinh(b*x+a)/b+3/128*d^(3/2)*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)/b^(5/2)+3/128*d^(3/2)*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)/b^(5/2)-3/16*d*(d*x+c)^(1/2)/b^2-3/8*d*sinh(b*x+a)^2*(d*x+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3392, 32, 3393, 3388, 2211, 2235, 2236} \[ \int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d} \]

[In]

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

[Out]

(-3*d*Sqrt[c + d*x])/(16*b^2) - (c + d*x)^(5/2)/(5*d) + (3*d^(3/2)*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2
]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + (3*d^(3/2)*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt
[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d*Sqrt[c
+ d*x]*Sinh[a + b*x]^2)/(8*b^2)

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 3388

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

Rule 3392

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

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}-\frac {1}{2} \int (c+d x)^{3/2} \, dx+\frac {\left (3 d^2\right ) \int \frac {\sinh ^2(a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2} \\ & = -\frac {(c+d x)^{5/2}}{5 d}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}-\frac {\left (3 d^2\right ) \int \left (\frac {1}{2 \sqrt {c+d x}}-\frac {\cosh (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx}{16 b^2} \\ & = -\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}+\frac {\left (3 d^2\right ) \int \frac {\cosh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = -\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}+\frac {\left (3 d^2\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{64 b^2}+\frac {\left (3 d^2\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{64 b^2} \\ & = -\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2}+\frac {(3 d) \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 )}{32 b^2}+\frac {(3 d) \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 )}{32 b^2} \\ & = -\frac {3 d \sqrt {c+d x}}{16 b^2}-\frac {(c+d x)^{5/2}}{5 d}+\frac {3 d^{3/2} 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 )}{64 b^{5/2}}+\frac {3 d^{3/2} 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 )}{64 b^{5/2}}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d \sqrt {c+d x} \sinh ^2(a+b x)}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65 \[ \int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx=\frac {\sqrt {c+d x} \left (-32 (c+d x)^2-\frac {5 \sqrt {2} d^2 e^{2 a-\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},-\frac {2 b (c+d x)}{d}\right )}{b^2 \sqrt {-\frac {b (c+d x)}{d}}}-\frac {5 \sqrt {2} d^2 e^{-2 a+\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {2 b (c+d x)}{d}\right )}{b^2 \sqrt {\frac {b (c+d x)}{d}}}\right )}{160 d} \]

[In]

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

[Out]

(Sqrt[c + d*x]*(-32*(c + d*x)^2 - (5*Sqrt[2]*d^2*E^(2*a - (2*b*c)/d)*Gamma[5/2, (-2*b*(c + d*x))/d])/(b^2*Sqrt
[-((b*(c + d*x))/d)]) - (5*Sqrt[2]*d^2*E^(-2*a + (2*b*c)/d)*Gamma[5/2, (2*b*(c + d*x))/d])/(b^2*Sqrt[(b*(c + d
*x))/d])))/(160*d)

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (159) = 318\).

Time = 0.27 (sec) , antiderivative size = 755, normalized size of antiderivative = 3.58 \[ \int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx=\frac {15 \, \sqrt {2} \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{3} \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 ) - 15 \, \sqrt {2} \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{3} \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 ) - 4 \, {\left (20 \, b^{2} d^{2} x - 5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{4} - 20 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \sinh \left (b x + a\right )^{4} + 20 \, b^{2} c d + 15 \, b d^{2} + 32 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (16 \, b^{3} d^{2} x^{2} + 32 \, b^{3} c d x + 16 \, b^{3} c^{2} - 15 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (5 \, {\left (4 \, b^{2} d^{2} x + 4 \, b^{2} c d - 3 \, b d^{2}\right )} \cosh \left (b x + a\right )^{3} - 16 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {d x + c}}{640 \, {\left (b^{3} d \cosh \left (b x + a\right )^{2} + 2 \, b^{3} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} d \sinh \left (b x + a\right )^{2}\right )}} \]

[In]

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

[Out]

1/640*(15*sqrt(2)*sqrt(pi)*(d^3*cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) - d^3*cosh(b*x + a)^2*sinh(-2*(b*c - a*
d)/d) + (d^3*cosh(-2*(b*c - a*d)/d) - d^3*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*(d^3*cosh(b*x + a)*cosh(
-2*(b*c - a*d)/d) - d^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)) - 15*sqrt(2)*sqrt(pi)*(d^3*cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) + d^3*cosh(b*x + a)^2*sinh(-2*
(b*c - a*d)/d) + (d^3*cosh(-2*(b*c - a*d)/d) + d^3*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*(d^3*cosh(b*x +
a)*cosh(-2*(b*c - a*d)/d) + d^3*cosh(b*x + a)*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(2)*s
qrt(d*x + c)*sqrt(-b/d)) - 4*(20*b^2*d^2*x - 5*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)^4 - 20*(4*b^2
*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)*sinh(b*x + a)^3 - 5*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*sinh(b*x +
a)^4 + 20*b^2*c*d + 15*b*d^2 + 32*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*cosh(b*x + a)^2 + 2*(16*b^3*d^2*x^2 +
32*b^3*c*d*x + 16*b^3*c^2 - 15*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 4*(5*(4
*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)^3 - 16*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*cosh(b*x + a))*si
nh(b*x + a))*sqrt(d*x + c))/(b^3*d*cosh(b*x + a)^2 + 2*b^3*d*cosh(b*x + a)*sinh(b*x + a) + b^3*d*sinh(b*x + a)
^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.13 \[ \int (c+d x)^{3/2} \sinh ^2(a+b x) \, dx=-\frac {128 \, {\left (d x + c\right )}^{\frac {5}{2}} - \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} - \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} + \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {2 \, b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} - \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (2 \, a\right )} - 3 \, \sqrt {d x + c} d^{2} e^{\left (2 \, a\right )}\right )} e^{\left (\frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b^{2}}}{640 \, d} \]

[In]

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

[Out]

-1/640*(128*(d*x + c)^(5/2) - 15*sqrt(2)*sqrt(pi)*d^2*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-b/d))*e^(2*a - 2*b*c/d)/
(b^2*sqrt(-b/d)) - 15*sqrt(2)*sqrt(pi)*d^2*erf(sqrt(2)*sqrt(d*x + c)*sqrt(b/d))*e^(-2*a + 2*b*c/d)/(b^2*sqrt(b
/d)) + 20*(4*(d*x + c)^(3/2)*b*d*e^(2*b*c/d) + 3*sqrt(d*x + c)*d^2*e^(2*b*c/d))*e^(-2*a - 2*(d*x + c)*b/d)/b^2
- 20*(4*(d*x + c)^(3/2)*b*d*e^(2*a) - 3*sqrt(d*x + c)*d^2*e^(2*a))*e^(2*(d*x + c)*b/d - 2*b*c/d)/b^2)/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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