\(\int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx\) [324]

Optimal result

Integrand size = 25, antiderivative size = 117 \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a (a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 b^{3/2} f}+\frac {(a+4 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f} \]

[Out]

1/8*a*(a+4*b)*arctanh(sin(f*x+e)*b^(1/2)/(a+b*sin(f*x+e)^2)^(1/2))/b^(3/2)/f-1/4*sin(f*x+e)*(a+b*sin(f*x+e)^2)
^(3/2)/b/f+1/8*(a+4*b)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/b/f

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 396, 201, 223, 212} \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a (a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 b^{3/2} f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f}+\frac {(a+4 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 b f} \]

[In]

Int[Cos[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

(a*(a + 4*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*b^(3/2)*f) + ((a + 4*b)*Sin[e + f*
x]*Sqrt[a + b*Sin[e + f*x]^2])/(8*b*f) - (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(4*b*f)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 3269

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1-x^2\right ) \sqrt {a+b x^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f}+\frac {(a+4 b) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\sin (e+f x)\right )}{4 b f} \\ & = \frac {(a+4 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f}+\frac {(a (a+4 b)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{8 b f} \\ & = \frac {(a+4 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f}+\frac {(a (a+4 b)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 b f} \\ & = \frac {a (a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 b^{3/2} f}+\frac {(a+4 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 b f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)} \left (\sqrt {a} (a+4 b) \text {arcsinh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a}}\right )-\sqrt {b} \sin (e+f x) \left (a-4 b+2 b \sin ^2(e+f x)\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right )}{8 b^{3/2} f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]

[In]

Integrate[Cos[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

(Sqrt[a + b*Sin[e + f*x]^2]*(Sqrt[a]*(a + 4*b)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]] - Sqrt[b]*Sin[e + f*x]*
(a - 4*b + 2*b*Sin[e + f*x]^2)*Sqrt[1 + (b*Sin[e + f*x]^2)/a]))/(8*b^(3/2)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2}+\frac {a \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{2 \sqrt {b}}-\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2}+\frac {a \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{2 \sqrt {b}}\right )}{4 b}}{f}\)	\(146\)
default	\(\frac {\frac {\sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2}+\frac {a \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{2 \sqrt {b}}-\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2}+\frac {a \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{2 \sqrt {b}}\right )}{4 b}}{f}\)	\(146\)

[In]

int(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/f*(1/2*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*sin(f*x+e)+(a+b*sin(f*x+e)^2)^(1/2))-1/4
*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2)/b+1/4*a/b*(1/2*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/
2)*sin(f*x+e)+(a+b*sin(f*x+e)^2)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 511, normalized size of antiderivative = 4.37 \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [\frac {{\left (a^{2} + 4 \, a b\right )} \sqrt {b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b} \sin \left (f x + e\right )\right ) + 8 \, {\left (2 \, b^{2} \cos \left (f x + e\right )^{2} - a b + 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{64 \, b^{2} f}, -\frac {{\left (a^{2} + 4 \, a b\right )} \sqrt {-b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) - 4 \, {\left (2 \, b^{2} \cos \left (f x + e\right )^{2} - a b + 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{32 \, b^{2} f}\right ] \]

[In]

integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/64*((a^2 + 4*a*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 +
24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b
^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 -
10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)
*sqrt(b)*sin(f*x + e)) + 8*(2*b^2*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/
(b^2*f), -1/32*((a^2 + 4*a*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2
+ 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 -
(3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*b^2*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(-b*cos(f*x + e)
^2 + a + b)*sin(f*x + e))/(b^2*f)]

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**3*(a+b*sin(f*x+e)**2)**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\frac {a^{2} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {4 \, a \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {b}} + 4 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} \sin \left (f x + e\right ) - \frac {2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )}{b} + \frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right )}{b}}{8 \, f} \]

[In]

integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

1/8*(a^2*arcsinh(b*sin(f*x + e)/sqrt(a*b))/b^(3/2) + 4*a*arcsinh(b*sin(f*x + e)/sqrt(a*b))/sqrt(b) + 4*sqrt(b*
sin(f*x + e)^2 + a)*sin(f*x + e) - 2*(b*sin(f*x + e)^2 + a)^(3/2)*sin(f*x + e)/b + sqrt(b*sin(f*x + e)^2 + a)*
a*sin(f*x + e)/b)/f

Giac [F]

\[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{3} \,d x } \]

[In]

integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sin(f*x + e)^2 + a)*cos(f*x + e)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int {\cos \left (e+f\,x\right )}^3\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]

[In]

int(cos(e + f*x)^3*(a + b*sin(e + f*x)^2)^(1/2),x)

[Out]

int(cos(e + f*x)^3*(a + b*sin(e + f*x)^2)^(1/2), x)