3.67 \(\int (a+i a \sinh (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{8 i a^2 \cosh (c+d x)}{3 d \sqrt{a+i a \sinh (c+d x)}}+\frac{2 i a \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{3 d} \]

[Out]

(((8*I)/3)*a^2*Cosh[c + d*x])/(d*Sqrt[a + I*a*Sinh[c + d*x]]) + (((2*I)/3)*a*Cosh[c + d*x]*Sqrt[a + I*a*Sinh[c
+ d*x]])/d

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Rubi [A]  time = 0.031358, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2647, 2646} \[ \frac{8 i a^2 \cosh (c+d x)}{3 d \sqrt{a+i a \sinh (c+d x)}}+\frac{2 i a \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Sinh[c + d*x])^(3/2),x]

[Out]

(((8*I)/3)*a^2*Cosh[c + d*x])/(d*Sqrt[a + I*a*Sinh[c + d*x]]) + (((2*I)/3)*a*Cosh[c + d*x]*Sqrt[a + I*a*Sinh[c
+ d*x]])/d

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int (a+i a \sinh (c+d x))^{3/2} \, dx &=\frac{2 i a \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a+i a \sinh (c+d x)} \, dx\\ &=\frac{8 i a^2 \cosh (c+d x)}{3 d \sqrt{a+i a \sinh (c+d x)}}+\frac{2 i a \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.205685, size = 113, normalized size = 1.64 \[ -\frac{a (\sinh (c+d x)-i) \sqrt{a+i a \sinh (c+d x)} \left (-9 i \sinh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{3}{2} (c+d x)\right )+9 \cosh \left (\frac{1}{2} (c+d x)\right )+\cosh \left (\frac{3}{2} (c+d x)\right )\right )}{3 d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(-I + Sinh[c + d*x])*Sqrt[a + I*a*Sinh[c + d*x]]*(9*Cosh[(c + d*x)/2] + Cosh[(3*(c + d*x))/2] - (9*I)*Sinh
[(c + d*x)/2] + I*Sinh[(3*(c + d*x))/2]))/(3*d*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^3)

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\sinh \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.98399, size = 259, normalized size = 3.75 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (i \, a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 9 i \, a e^{\left (d x + c\right )} + a\right )} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{3 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - i \, d e^{\left (d x + c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(1/2)*(I*a*e^(3*d*x + 3*c) + 9*a*e^(2*d*x + 2*c) + 9*I*a*e^(d*x + c) + a)*sqrt(I*a*e^(2*d*x + 2*c) + 2
*a*e^(d*x + c) - I*a)*e^(-1/2*d*x - 1/2*c)/(d*e^(2*d*x + 2*c) - I*d*e^(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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