\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx\) [309]

Optimal result

Integrand size = 35, antiderivative size = 130 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) (c-d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 (3 B c+3 A d-2 B d) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 a f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3047, 3102, 2830, 2728, 212} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) (c-d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 (3 A d+3 B c-2 B d) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B d \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 a f} \]

[In]

Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

-((Sqrt[2]*(A - B)*(c - d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) -
(2*(3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*B*d*Cos[e + f*x]*Sqrt[a + a*Sin[e
+ f*x]])/(3*a*f)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

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

Rule 2830

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*S
in[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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*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*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = -\frac {2 B d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 a f}+\frac {2 \int \frac {\frac {1}{2} a (3 A c+B d)+\frac {1}{2} a (3 B c+3 A d-2 B d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a} \\ & = -\frac {2 (3 B c+3 A d-2 B d) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 a f}+((A-B) (c-d)) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = -\frac {2 (3 B c+3 A d-2 B d) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 a f}-\frac {(2 (A-B) (c-d)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f} \\ & = -\frac {\sqrt {2} (A-B) (c-d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 (3 B c+3 A d-2 B d) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 B d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 a f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((-6-6 i) (-1)^{3/4} (A-B) (c-d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )+2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3 B c+3 A d-B d+B d \sin (e+f x))\right )}{3 f \sqrt {a (1+\sin (e+f x))}} \]

[In]

Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

-1/3*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-6 - 6*I)*(-1)^(3/4)*(A - B)*(c - d)*ArcTanh[(1/2 + I/2)*(-1)^(3
/4)*(-1 + Tan[(e + f*x)/4])] + 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*B*c + 3*A*d - B*d + B*d*Sin[e + f*x]
)))/(f*Sqrt[a*(1 + Sin[e + f*x])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(113)=226\).

Time = 2.19 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (3 A \,a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c -3 A \,a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) d -3 B \,a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c +3 B \,a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) d -2 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d +6 \sqrt {a -a \sin \left (f x +e \right )}\, A a d +6 \sqrt {a -a \sin \left (f x +e \right )}\, B a c \right )}{3 a^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(232\)
parts	\(-\frac {A c \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}-\frac {\left (d A +B c \right ) \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+2 \sqrt {a -a \sin \left (f x +e \right )}\right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {d B \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-3 a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+2 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}\right )}{3 a^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(277\)

[In]

int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(3*A*a^(3/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/
a^(1/2))*c-3*A*a^(3/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*d-3*B*a^(3/2)*2^(1/2)*arcta
nh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*c+3*B*a^(3/2)*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2
)/a^(1/2))*d-2*B*(a-a*sin(f*x+e))^(3/2)*d+6*(a-a*sin(f*x+e))^(1/2)*A*a*d+6*(a-a*sin(f*x+e))^(1/2)*B*a*c)/a^2/c
os(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (113) = 226\).

Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.33 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left ({\left (A - B\right )} a c - {\left (A - B\right )} a d + {\left ({\left (A - B\right )} a c - {\left (A - B\right )} a d\right )} \cos \left (f x + e\right ) + {\left ({\left (A - B\right )} a c - {\left (A - B\right )} a d\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {a}} - 4 \, {\left (B d \cos \left (f x + e\right )^{2} + 3 \, B c + {\left (3 \, A - 2 \, B\right )} d + {\left (3 \, B c + {\left (3 \, A - B\right )} d\right )} \cos \left (f x + e\right ) + {\left (B d \cos \left (f x + e\right ) - 3 \, B c - {\left (3 \, A - 2 \, B\right )} d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{6 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]

[In]

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

[Out]

1/6*(3*sqrt(2)*((A - B)*a*c - (A - B)*a*d + ((A - B)*a*c - (A - B)*a*d)*cos(f*x + e) + ((A - B)*a*c - (A - B)*
a*d)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)
*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x
+ e) - cos(f*x + e) - 2))/sqrt(a) - 4*(B*d*cos(f*x + e)^2 + 3*B*c + (3*A - 2*B)*d + (3*B*c + (3*A - B)*d)*cos(
f*x + e) + (B*d*cos(f*x + e) - 3*B*c - (3*A - 2*B)*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e
) + a*f*sin(f*x + e) + a*f)

Sympy [F]

\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))/(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral((A + B*sin(e + f*x))*(c + d*sin(e + f*x))/sqrt(a*(sin(e + f*x) + 1)), x)

Maxima [F]

\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

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

[Out]

integrate((B*sin(f*x + e) + A)*(d*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (A \sqrt {a} c - B \sqrt {a} c - A \sqrt {a} d + B \sqrt {a} d\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (A \sqrt {a} c - B \sqrt {a} c - A \sqrt {a} d + B \sqrt {a} d\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, \sqrt {2} {\left (2 \, B a^{\frac {5}{2}} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, B a^{\frac {5}{2}} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A a^{\frac {5}{2}} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{6 \, f} \]

[In]

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

[Out]

1/6*(3*sqrt(2)*(A*sqrt(a)*c - B*sqrt(a)*c - A*sqrt(a)*d + B*sqrt(a)*d)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)
/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 3*sqrt(2)*(A*sqrt(a)*c - B*sqrt(a)*c - A*sqrt(a)*d + B*sqrt(a)*d)*l
og(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 4*sqrt(2)*(2*B*a^(5/2)*d*sin
(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 3*B*a^(5/2)*c*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 3*A*a^(5/2)*d*sin(-1/4*pi + 1/2
*f*x + 1/2*e))/(a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,\sin \left (e+f\,x\right )\right )}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x)))/(a + a*sin(e + f*x))^(1/2),x)

[Out]

int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x)))/(a + a*sin(e + f*x))^(1/2), x)