3.329 \(\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=233 \[ -\frac{38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt{a \sin (c+d x)+a}}-\frac{1724 a^2 \cos (c+d x)}{6435 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac{6 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac{3448 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{45045 d} \]

[Out]

(-1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]]) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqr
t[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[
c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143
*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c +
d*x])^(3/2))/(13*d)

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Rubi [A]  time = 0.72037, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2879, 2976, 2981, 2770, 2759, 2751, 2646} \[ -\frac{38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt{a \sin (c+d x)+a}}-\frac{1724 a^2 \cos (c+d x)}{6435 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac{6 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac{3448 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{45045 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]]) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqr
t[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[
c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143
*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c +
d*x])^(3/2))/(13*d)

Rule 2879

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

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2770

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d)
)/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2759

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*
Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 2751

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

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 \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\int \sin ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2}\\ &=\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{2 \int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac{5 a^2}{2}-\frac{3}{2} a^2 \sin (c+d x)\right ) \, dx}{13 a^2}\\ &=\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{4 \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{31 a^3}{4}+\frac{19}{4} a^3 \sin (c+d x)\right ) \, dx}{143 a^2}\\ &=-\frac{38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt{a+a \sin (c+d x)}}+\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{(431 a) \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{1287}\\ &=-\frac{862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt{a+a \sin (c+d x)}}-\frac{38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt{a+a \sin (c+d x)}}+\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{(862 a) \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{3003}\\ &=-\frac{862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt{a+a \sin (c+d x)}}-\frac{38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt{a+a \sin (c+d x)}}+\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{1724 \int \left (\frac{3 a}{2}-a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{15015}\\ &=-\frac{862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt{a+a \sin (c+d x)}}-\frac{38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt{a+a \sin (c+d x)}}+\frac{3448 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{45045 d}+\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{(862 a) \int \sqrt{a+a \sin (c+d x)} \, dx}{6435}\\ &=-\frac{1724 a^2 \cos (c+d x)}{6435 d \sqrt{a+a \sin (c+d x)}}-\frac{862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt{a+a \sin (c+d x)}}-\frac{38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt{a+a \sin (c+d x)}}+\frac{3448 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{45045 d}+\frac{6 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}\\ \end{align*}

Mathematica [A]  time = 3.76826, size = 120, normalized size = 0.52 \[ -\frac{a \sqrt{a (\sin (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (381174 \sin (c+d x)-77665 \sin (3 (c+d x))+3465 \sin (5 (c+d x))-194160 \cos (2 (c+d x))+22680 \cos (4 (c+d x))+281816)}{360360 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(281816 - 194160*Cos[2*(c + d*x)] + 226
80*Cos[4*(c + d*x)] + 381174*Sin[c + d*x] - 77665*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)]))/(360360*d*(Cos[(c
+ d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.723, size = 97, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 3465\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}+11340\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}+15085\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+12930\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+10344\,\sin \left ( dx+c \right ) +6896 \right ) }{45045\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x)

[Out]

-2/45045*(1+sin(d*x+c))*a^2*(sin(d*x+c)-1)^2*(3465*sin(d*x+c)^5+11340*sin(d*x+c)^4+15085*sin(d*x+c)^3+12930*si
n(d*x+c)^2+10344*sin(d*x+c)+6896)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^2*sin(d*x + c)^3, x)

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Fricas [A]  time = 1.65028, size = 562, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (3465 \, a \cos \left (d x + c\right )^{7} - 4410 \, a \cos \left (d x + c\right )^{6} - 14140 \, a \cos \left (d x + c\right )^{5} + 7330 \, a \cos \left (d x + c\right )^{4} + 15299 \, a \cos \left (d x + c\right )^{3} - 568 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) -{\left (3465 \, a \cos \left (d x + c\right )^{6} + 7875 \, a \cos \left (d x + c\right )^{5} - 6265 \, a \cos \left (d x + c\right )^{4} - 13595 \, a \cos \left (d x + c\right )^{3} + 1704 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) + 4544 \, a\right )} \sin \left (d x + c\right ) + 4544 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{45045 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45045*(3465*a*cos(d*x + c)^7 - 4410*a*cos(d*x + c)^6 - 14140*a*cos(d*x + c)^5 + 7330*a*cos(d*x + c)^4 + 152
99*a*cos(d*x + c)^3 - 568*a*cos(d*x + c)^2 + 2272*a*cos(d*x + c) - (3465*a*cos(d*x + c)^6 + 7875*a*cos(d*x + c
)^5 - 6265*a*cos(d*x + c)^4 - 13595*a*cos(d*x + c)^3 + 1704*a*cos(d*x + c)^2 + 2272*a*cos(d*x + c) + 4544*a)*s
in(d*x + c) + 4544*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)

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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(cos(d*x+c)**2*sin(d*x+c)**3*(a+a*sin(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 (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^2*sin(d*x + c)^3, x)