3.1.36 \(\int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx\) [36]

Optimal. Leaf size=61 \[ -\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3022, 212} \begin {gather*} -\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt
[c]*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 3022

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 2.94, size = 367, normalized size = 6.02 \begin {gather*} -\frac {\left (\log \left (-\frac {(1+i) e^{\frac {i e}{2}} \left (\sqrt {2} c \left (-1+e^{i (e+f x)}\right )+i \sqrt {2} d \left (1+e^{i (e+f x)}\right )-2 i \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{\sqrt {c} \left (1+e^{i (e+f x)}\right )}\right )+\log \left (\frac {(1+i) e^{\frac {i e}{2}} \left (-i \sqrt {2} d \left (-1+e^{i (e+f x)}\right )+\sqrt {2} c \left (1+e^{i (e+f x)}\right )+2 \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{\sqrt {c} \left (-1+e^{i (e+f x)}\right )}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt {c} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((Log[((-1 - I)*E^((I/2)*e)*(Sqrt[2]*c*(-1 + E^(I*(e + f*x))) + I*Sqrt[2]*d*(1 + E^(I*(e + f*x))) - (2*I)*Sq
rt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(Sqrt[c]*(1 + E^(I*(e + f*x))))] + Log[((
1 + I)*E^((I/2)*e)*((-I)*Sqrt[2]*d*(-1 + E^(I*(e + f*x))) + Sqrt[2]*c*(1 + E^(I*(e + f*x))) + 2*Sqrt[c]*Sqrt[2
*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(Sqrt[c]*(-1 + E^(I*(e + f*x))))])*(Cos[(e + f*x)/2]
- I*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/(
Sqrt[c]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(49)=98\).
time = 20.06, size = 238, normalized size = 3.90

method	result	size
default	\(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \sqrt {2}\, \left (\ln \left (-\frac {-\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right ) c -d \sin \left (f x +e \right )-c}{\sin \left (f x +e \right ) \sqrt {c}}\right )-\ln \left (\frac {-2 \sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )+2 d \cos \left (f x +e \right )-2 c \sin \left (f x +e \right )-2 d}{-1+\cos \left (f x +e \right )}\right )\right ) \left (-1+\cos \left (f x +e \right )\right )}{f \sin \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {c}}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (52) = 104\).
time = 0.90, size = 1080, normalized size = 17.70 \begin {gather*} \left [\frac {\sqrt {\frac {a}{c}} \log \left (\frac {{\left (a c^{4} - 28 \, a c^{3} d + 70 \, a c^{2} d^{2} - 28 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left (31 \, a c^{4} - 196 \, a c^{3} d + 154 \, a c^{2} d^{2} - 4 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (81 \, a c^{4} - 252 \, a c^{3} d + 150 \, a c^{2} d^{2} - 28 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )^{3} + 2 \, {\left (79 \, a c^{4} - 100 \, a c^{3} d + 74 \, a c^{2} d^{2} - 4 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left ({\left (c^{4} - 7 \, c^{3} d + 7 \, c^{2} d^{2} - c d^{3}\right )} \cos \left (f x + e\right )^{4} + 51 \, c^{4} - 59 \, c^{3} d + 17 \, c^{2} d^{2} - c d^{3} - 2 \, {\left (5 \, c^{4} - 14 \, c^{3} d + 5 \, c^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (18 \, c^{4} - 33 \, c^{3} d + 12 \, c^{2} d^{2} - c d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (13 \, c^{4} - 14 \, c^{3} d + 5 \, c^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (51 \, c^{4} - 59 \, c^{3} d + 17 \, c^{2} d^{2} - c d^{3} - {\left (c^{4} - 7 \, c^{3} d + 7 \, c^{2} d^{2} - c d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (11 \, c^{4} - 35 \, c^{3} d + 17 \, c^{2} d^{2} - c d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (25 \, c^{4} - 31 \, c^{3} d + 7 \, c^{2} d^{2} - c d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{c}} + {\left (289 \, a c^{4} - 476 \, a c^{3} d + 230 \, a c^{2} d^{2} - 28 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right ) + {\left (a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + {\left (a c^{4} - 28 \, a c^{3} d + 70 \, a c^{2} d^{2} - 28 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )^{4} + 32 \, {\left (a c^{4} - 7 \, a c^{3} d + 7 \, a c^{2} d^{2} - a c d^{3}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (65 \, a c^{4} - 140 \, a c^{3} d + 38 \, a c^{2} d^{2} - 12 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )^{2} - 32 \, {\left (9 \, a c^{4} - 15 \, a c^{3} d + 7 \, a c^{2} d^{2} - a c d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{5} + \cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) + 1}\right )}{4 \, f}, \frac {\sqrt {-\frac {a}{c}} \arctan \left (-\frac {{\left ({\left (c^{2} - 6 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 9 \, c^{2} + 6 \, c d - d^{2} + 8 \, {\left (c^{2} - c d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{c}}}{4 \, {\left ({\left (a c d - a d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a c^{2} - 3 \, a c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (2 \, a c^{2} - a c d + a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(a/c)*log(((a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*
d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (31*a*c^4 - 196*a*c^3*d + 154*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e
)^4 - 2*(81*a*c^4 - 252*a*c^3*d + 150*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^3 + 2*(79*a*c^4 - 100*a*c^3
*d + 74*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e)^2 - 8*((c^4 - 7*c^3*d + 7*c^2*d^2 - c*d^3)*cos(f*x + e)^4
+ 51*c^4 - 59*c^3*d + 17*c^2*d^2 - c*d^3 - 2*(5*c^4 - 14*c^3*d + 5*c^2*d^2)*cos(f*x + e)^3 - 2*(18*c^4 - 33*c^
3*d + 12*c^2*d^2 - c*d^3)*cos(f*x + e)^2 + 2*(13*c^4 - 14*c^3*d + 5*c^2*d^2)*cos(f*x + e) - (51*c^4 - 59*c^3*d
+ 17*c^2*d^2 - c*d^3 - (c^4 - 7*c^3*d + 7*c^2*d^2 - c*d^3)*cos(f*x + e)^3 - (11*c^4 - 35*c^3*d + 17*c^2*d^2 -
c*d^3)*cos(f*x + e)^2 + (25*c^4 - 31*c^3*d + 7*c^2*d^2 - c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/c) + (289*a*c^4 - 476*a*c^3*d + 230*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*co
s(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + (a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*
c*d^3 + a*d^4)*cos(f*x + e)^4 + 32*(a*c^4 - 7*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e)^3 - 2*(65*a*c^4 -
140*a*c^3*d + 38*a*c^2*d^2 - 12*a*c*d^3 + a*d^4)*cos(f*x + e)^2 - 32*(9*a*c^4 - 15*a*c^3*d + 7*a*c^2*d^2 - a*c
*d^3)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^5 + cos(f*x + e)^4 - 2*cos(f*x + e)^3 - 2*cos(f*x + e)^2 + (co
s(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sin(f*x + e) + cos(f*x + e) + 1))/f, 1/2*sqrt(-a/c)*arctan(-1/4*((c^2 - 6
*c*d + d^2)*cos(f*x + e)^2 - 9*c^2 + 6*c*d - d^2 + 8*(c^2 - c*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d
*sin(f*x + e) + c)*sqrt(-a/c)/((a*c*d - a*d^2)*cos(f*x + e)^3 - (a*c^2 - 3*a*c*d)*cos(f*x + e)*sin(f*x + e) +
(2*a*c^2 - a*c*d + a*d^2)*cos(f*x + e)))/f]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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