3.171 \(\int x^3 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=393 \[ \frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^2}+\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\frac{3 c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^4}-\frac{6 c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^4}-\frac{3 c x \sin (e+f x) \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^3}-\frac{6 c x \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{3 c x \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^3}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c \sin (e+f x)+c)^{5/2}}{2 c f}-\frac{3 c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f} \]

[Out]

(-6*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4 + (3*c*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Si
n[e + f*x]])/f^2 + (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^3) - (3*c*x^3*S
ec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) - (3*c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*
x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^4) + (3*c*x^2*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]
])/(4*f^2) + (x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (6*c*x*Sqrt[a -
a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (3*c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt
[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)

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Rubi [A]  time = 0.375268, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4604, 4422, 3317, 3296, 2638, 3311, 30, 2635, 8} \[ \frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^2}+\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\frac{3 c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^4}-\frac{6 c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^4}-\frac{3 c x \sin (e+f x) \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^3}-\frac{6 c x \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{3 c x \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^3}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c \sin (e+f x)+c)^{5/2}}{2 c f}-\frac{3 c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]

[Out]

(-6*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4 + (3*c*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Si
n[e + f*x]])/f^2 + (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^3) - (3*c*x^3*S
ec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) - (3*c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*
x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^4) + (3*c*x^2*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]
])/(4*f^2) + (x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (6*c*x*Sqrt[a -
a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (3*c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt
[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)

Rule 4604

Int[((g_.) + (h_.)*(x_))^(p_.)*((a_) + (b_.)*Sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*Sin[(e_.) + (f_.)*(x_
)])^(n_), x_Symbol] :> Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^F
racPart[m])/Cos[e + f*x]^(2*FracPart[m]), Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x],
x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ
[2*m] && IGeQ[n - m, 0]

Rule 4422

Int[Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol]
:> Simp[((e + f*x)^m*(a + b*Sin[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x
)^(m - 1)*(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3311

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x^3 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x^3 \cos (e+f x) (c+c \sin (e+f x)) \, dx\\ &=\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{\left (3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x^2 (c+c \sin (e+f x))^2 \, dx}{2 c f}\\ &=\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{\left (3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \left (c^2 x^2+2 c^2 x^2 \sin (e+f x)+c^2 x^2 \sin ^2(e+f x)\right ) \, dx}{2 c f}\\ &=-\frac{c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 f}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{\left (3 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x^2 \sin ^2(e+f x) \, dx}{2 f}-\frac{\left (3 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x^2 \sin (e+f x) \, dx}{f}\\ &=\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{f^2}-\frac{c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 f}+\frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{4 f^2}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{3 c x \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{4 f^3}+\frac{\left (3 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \sin ^2(e+f x) \, dx}{4 f^3}-\frac{\left (6 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x \cos (e+f x) \, dx}{f^2}-\frac{\left (3 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int x^2 \, dx}{4 f}\\ &=\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{f^2}-\frac{3 c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{4 f}-\frac{3 c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{8 f^4}+\frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{4 f^2}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{6 c x \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{f^3}-\frac{3 c x \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{4 f^3}+\frac{\left (3 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int 1 \, dx}{8 f^3}+\frac{\left (6 c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \sin (e+f x) \, dx}{f^3}\\ &=-\frac{6 c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{f^4}+\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{f^2}+\frac{3 c x \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{8 f^3}-\frac{3 c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{4 f}-\frac{3 c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{8 f^4}+\frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{4 f^2}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{5/2}}{2 c f}-\frac{6 c x \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{f^3}-\frac{3 c x \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{4 f^3}\\ \end{align*}

Mathematica [A]  time = 1.15899, size = 113, normalized size = 0.29 \[ \frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left (\left (6 f^2 x^2-3\right ) \sin (e+f x)+8 \left (f x \left (f^2 x^2-6\right ) \tan (e+f x)+3 f^2 x^2-6\right )-f x \left (2 f^2 x^2-3\right ) \cos (2 (e+f x)) \sec (e+f x)\right )}{8 f^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]

[Out]

(c*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(-(f*x*(-3 + 2*f^2*x^2)*Cos[2*(e + f*x)]*Sec[e + f*x])
+ (-3 + 6*f^2*x^2)*Sin[e + f*x] + 8*(-6 + 3*f^2*x^2 + f*x*(-6 + f^2*x^2)*Tan[e + f*x])))/(8*f^4)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a-a\sin \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),x)

[Out]

int(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a*sin(f*x + e) + a)*(c*sin(f*x + e) + c)^(3/2)*x^3, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(x**3*(c+c*sin(f*x+e))**(3/2)*(a-a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError