3.938 \(\int \sin (c+d x) (a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)) \, dx\)

Optimal. Leaf size=61 \[ -\frac{a \cos (c+d x)}{d}+\frac{2 b E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d}-\frac{c \sin (c+d x) \cos (c+d x)}{2 d}+\frac{c x}{2} \]

[Out]

(c*x)/2 - (a*Cos[c + d*x])/d + (2*b*EllipticE[(c - Pi/2 + d*x)/2, 2])/d - (c*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.289499, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {4395, 4401, 2639, 2638, 2635, 8} \[ -\frac{a \cos (c+d x)}{d}+\frac{2 b E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d}-\frac{c \sin (c+d x) \cos (c+d x)}{2 d}+\frac{c x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x]),x]

[Out]

(c*x)/2 - (a*Cos[c + d*x])/d + (2*b*EllipticE[(c - Pi/2 + d*x)/2, 2])/d - (c*Cos[c + d*x]*Sin[c + d*x])/(2*d)

Rule 4395

Int[(u_)*((a_) + (b_.)*(F_)[(d_.) + (e_.)*(x_)]^(p_.) + (c_.)*(F_)[(d_.) + (e_.)*(x_)]^(q_.))^(n_.), x_Symbol]
:> Int[ActivateTrig[u*F[d + e*x]^(n*p)*(b + a/F[d + e*x]^p + c*F[d + e*x]^(q - p))^n], x] /; FreeQ[{a, b, c,
d, e, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && NegQ[p]

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2638

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

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 \sin (c+d x) \left (a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)\right ) \, dx &=\int \sqrt{\sin (c+d x)} \left (b+a \sqrt{\sin (c+d x)}+c \sin ^{\frac{3}{2}}(c+d x)\right ) \, dx\\ &=\int \left (b \sqrt{\sin (c+d x)}+a \sin (c+d x)+c \sin ^2(c+d x)\right ) \, dx\\ &=a \int \sin (c+d x) \, dx+b \int \sqrt{\sin (c+d x)} \, dx+c \int \sin ^2(c+d x) \, dx\\ &=-\frac{a \cos (c+d x)}{d}+\frac{2 b E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{d}-\frac{c \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} c \int 1 \, dx\\ &=\frac{c x}{2}-\frac{a \cos (c+d x)}{d}+\frac{2 b E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{d}-\frac{c \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.180645, size = 55, normalized size = 0.9 \[ \frac{-4 a \cos (c+d x)-8 b E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )+c (-\sin (2 (c+d x))+2 c+2 d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x]),x]

[Out]

(-4*a*Cos[c + d*x] - 8*b*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + c*(2*c + 2*d*x - Sin[2*(c + d*x)]))/(4*d)

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Maple [A]  time = 0.924, size = 136, normalized size = 2.2 \begin{align*} cx-{\frac{a\cos \left ( dx+c \right ) }{d}}-{\frac{c}{d} \left ({\frac{\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{b}{d\cos \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) +1}\sqrt{-2\,\sin \left ( dx+c \right ) +2}\sqrt{-\sin \left ( dx+c \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{\sin \left ( dx+c \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)^(1/2)),x)

[Out]

c*x-a*cos(d*x+c)/d-c/d*(1/2*sin(d*x+c)*cos(d*x+c)+1/2*d*x+1/2*c)-b*(sin(d*x+c)+1)^(1/2)*(-2*sin(d*x+c)+2)^(1/2
)*(-sin(d*x+c))^(1/2)*(2*EllipticE((sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-EllipticF((sin(d*x+c)+1)^(1/2),1/2*2^(1/2
)))/cos(d*x+c)/sin(d*x+c)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

1/4*(2*c*d*x - 4*a*cos(d*x + c) + 2*d*integrate(-(((b*cos(3/2*d*x + 3/2*c) - b*cos(1/2*d*x + 1/2*c) - b*sin(3/
2*d*x + 3/2*c) - b*sin(1/2*d*x + 1/2*c))*cos(1/2*arctan2(sin(d*x + c), -cos(d*x + c) + 1)) - (b*cos(3/2*d*x +
3/2*c) - b*cos(1/2*d*x + 1/2*c) + b*sin(3/2*d*x + 3/2*c) + b*sin(1/2*d*x + 1/2*c))*sin(1/2*arctan2(sin(d*x + c
), -cos(d*x + c) + 1)))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c) + 1)) + ((b*cos(3/2*d*x + 3/2*c) - b*cos(1/
2*d*x + 1/2*c) + b*sin(3/2*d*x + 3/2*c) + b*sin(1/2*d*x + 1/2*c))*cos(1/2*arctan2(sin(d*x + c), -cos(d*x + c)
+ 1)) + (b*cos(3/2*d*x + 3/2*c) - b*cos(1/2*d*x + 1/2*c) - b*sin(3/2*d*x + 3/2*c) - b*sin(1/2*d*x + 1/2*c))*si
n(1/2*arctan2(sin(d*x + c), -cos(d*x + c) + 1)))*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c) + 1)))/((cos(d*x +
c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1)^(1/4)*(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*cos(d*x + c) + 1)^(1/4
)), x) - c*sin(2*d*x + 2*c))/d

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-c \cos \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + b \sqrt{\sin \left (d x + c\right )} + c, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

integral(-c*cos(d*x + c)^2 + a*sin(d*x + c) + b*sqrt(sin(d*x + c)) + c, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)**(1/2)),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c \sin \left (d x + c\right ) + a + \frac{b}{\sqrt{\sin \left (d x + c\right )}}\right )} \sin \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

integrate((c*sin(d*x + c) + a + b/sqrt(sin(d*x + c)))*sin(d*x + c), x)