3.455 \(\int \frac{(c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=62 \[ -\frac{(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{d x (2 c-d)}{a}-\frac{d^2 \cos (e+f x)}{a f} \]

[Out]

((2*c - d)*d*x)/a - (d^2*Cos[e + f*x])/(a*f) - ((c - d)^2*Cos[e + f*x])/(a*f*(1 + Sin[e + f*x]))

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Rubi [A]  time = 0.136563, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2746, 2735, 2648} \[ -\frac{(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{d x (2 c-d)}{a}-\frac{d^2 \cos (e+f x)}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c - d)*d*x)/a - (d^2*Cos[e + f*x])/(a*f) - ((c - d)^2*Cos[e + f*x])/(a*f*(1 + Sin[e + f*x]))

Rule 2746

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

Rule 2735

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

Rule 2648

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

Rubi steps

\begin{align*} \int \frac{(c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx &=-\frac{d^2 \cos (e+f x)}{a f}+\frac{\int \frac{a c^2+a (2 c-d) d \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac{(2 c-d) d x}{a}-\frac{d^2 \cos (e+f x)}{a f}+(c-d)^2 \int \frac{1}{a+a \sin (e+f x)} \, dx\\ &=\frac{(2 c-d) d x}{a}-\frac{d^2 \cos (e+f x)}{a f}-\frac{(c-d)^2 \cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.448811, size = 122, normalized size = 1.97 \[ -\frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right ) \left (-2 c^2-2 c d (e+f x-2)+d^2 (e+f x-2)+d^2 \cos (e+f x)\right )+d \cos \left (\frac{1}{2} (e+f x)\right ) (d \cos (e+f x)-(2 c-d) (e+f x))\right )}{a f (\sin (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*Cos[(e + f*x)/2]*(-((2*c - d)*(e + f*x)) + d*Cos[e + f*x]) + (-2*c
^2 - 2*c*d*(-2 + e + f*x) + d^2*(-2 + e + f*x) + d^2*Cos[e + f*x])*Sin[(e + f*x)/2]))/(a*f*(1 + Sin[e + f*x]))
)

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Maple [B]  time = 0.059, size = 140, normalized size = 2.3 \begin{align*} -2\,{\frac{{d}^{2}}{af \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }}+4\,{\frac{d\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) c}{af}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ){d}^{2}}{af}}-2\,{\frac{{c}^{2}}{af \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }}+4\,{\frac{cd}{af \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }}-2\,{\frac{{d}^{2}}{af \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e)),x)

[Out]

-2/a/f*d^2/(1+tan(1/2*f*x+1/2*e)^2)+4/a/f*d*arctan(tan(1/2*f*x+1/2*e))*c-2/a/f*arctan(tan(1/2*f*x+1/2*e))*d^2-
2/a/f/(tan(1/2*f*x+1/2*e)+1)*c^2+4/a/f/(tan(1/2*f*x+1/2*e)+1)*c*d-2/a/f/(tan(1/2*f*x+1/2*e)+1)*d^2

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Maxima [B]  time = 1.71011, size = 282, normalized size = 4.55 \begin{align*} -\frac{2 \,{\left (d^{2}{\left (\frac{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2}{a + \frac{a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{\arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 2 \, c d{\left (\frac{\arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac{1}{a + \frac{a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + \frac{c^{2}}{a + \frac{a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d^2*((sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a + a*sin(f*x + e)/(cos(
f*x + e) + 1) + a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + arctan(sin(f*
x + e)/(cos(f*x + e) + 1))/a) - 2*c*d*(arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a + 1/(a + a*sin(f*x + e)/(cos(
f*x + e) + 1))) + c^2/(a + a*sin(f*x + e)/(cos(f*x + e) + 1)))/f

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Fricas [B]  time = 1.58977, size = 321, normalized size = 5.18 \begin{align*} -\frac{d^{2} \cos \left (f x + e\right )^{2} -{\left (2 \, c d - d^{2}\right )} f x + c^{2} - 2 \, c d + d^{2} -{\left ({\left (2 \, c d - d^{2}\right )} f x - c^{2} + 2 \, c d - 2 \, d^{2}\right )} \cos \left (f x + e\right ) -{\left ({\left (2 \, c d - d^{2}\right )} f x - d^{2} \cos \left (f x + e\right ) + c^{2} - 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d^2*cos(f*x + e)^2 - (2*c*d - d^2)*f*x + c^2 - 2*c*d + d^2 - ((2*c*d - d^2)*f*x - c^2 + 2*c*d - 2*d^2)*cos(f
*x + e) - ((2*c*d - d^2)*f*x - d^2*cos(f*x + e) + c^2 - 2*c*d + d^2)*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin
(f*x + e) + a*f)

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Sympy [A]  time = 4.12448, size = 877, normalized size = 14.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**2/(a+a*sin(f*x+e)),x)

[Out]

Piecewise((-2*c**2*tan(e/2 + f*x/2)**2/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/
2) + a*f) - 2*c**2/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 2*c*d*f*
x*tan(e/2 + f*x/2)**3/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 2*c*d
*f*x*tan(e/2 + f*x/2)**2/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 2*
c*d*f*x*tan(e/2 + f*x/2)/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 2*
c*d*f*x/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 4*c*d*tan(e/2 + f*x
/2)**2/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 4*c*d/(a*f*tan(e/2 +
f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) - d**2*f*x*tan(e/2 + f*x/2)**3/(a*f*tan(e/2
+ f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) - d**2*f*x*tan(e/2 + f*x/2)**2/(a*f*tan(e
/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) - d**2*f*x*tan(e/2 + f*x/2)/(a*f*tan(e/
2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) - d**2*f*x/(a*f*tan(e/2 + f*x/2)**3 + a*
f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) + 2*d**2*tan(e/2 + f*x/2)**3/(a*f*tan(e/2 + f*x/2)**3 + a*
f*tan(e/2 + f*x/2)**2 + a*f*tan(e/2 + f*x/2) + a*f) - 2*d**2/(a*f*tan(e/2 + f*x/2)**3 + a*f*tan(e/2 + f*x/2)**
2 + a*f*tan(e/2 + f*x/2) + a*f), Ne(f, 0)), (x*(c + d*sin(e))**2/(a*sin(e) + a), True))

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Giac [B]  time = 1.19535, size = 193, normalized size = 3.11 \begin{align*} \frac{\frac{{\left (2 \, c d - d^{2}\right )}{\left (f x + e\right )}}{a} - \frac{2 \,{\left (c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 2 \, c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + c^{2} - 2 \, c d + 2 \, d^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )} a}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*c*d - d^2)*(f*x + e)/a - 2*(c^2*tan(1/2*f*x + 1/2*e)^2 - 2*c*d*tan(1/2*f*x + 1/2*e)^2 + d^2*tan(1/2*f*x +
1/2*e)^2 + d^2*tan(1/2*f*x + 1/2*e) + c^2 - 2*c*d + 2*d^2)/((tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e)^2 +
tan(1/2*f*x + 1/2*e) + 1)*a))/f