3.557 \(\int \frac {\tan (c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx\)

Optimal. Leaf size=51 \[ \frac {\tanh ^{-1}\left (\frac {a+b \sin ^2(c+d x)}{\sqrt {a+b} \sqrt {a+b \sin ^4(c+d x)}}\right )}{2 d \sqrt {a+b}} \]

[Out]

1/2*arctanh((a+b*sin(d*x+c)^2)/(a+b)^(1/2)/(a+b*sin(d*x+c)^4)^(1/2))/d/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3229, 725, 206} \[ \frac {\tanh ^{-1}\left (\frac {a+b \sin ^2(c+d x)}{\sqrt {a+b} \sqrt {a+b \sin ^4(c+d x)}}\right )}{2 d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]

[Out]

ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])]/(2*Sqrt[a + b]*d)

Rule 206

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 3229

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

Rubi steps

\begin {align*} \int \frac {\tan (c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a-b \sin ^2(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}}\right )}{2 d}\\ &=\frac {\tanh ^{-1}\left (\frac {a+b \sin ^2(c+d x)}{\sqrt {a+b} \sqrt {a+b \sin ^4(c+d x)}}\right )}{2 \sqrt {a+b} d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 65, normalized size = 1.27 \[ \frac {\tanh ^{-1}\left (\frac {a-b \cos ^2(c+d x)+b}{\sqrt {a+b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{2 d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]

[Out]

ArcTanh[(a + b - b*Cos[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])]/(2*Sqrt[
a + b]*d)

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 240, normalized size = 4.71 \[ \left [\frac {\log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} {\left (b \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {a + b} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}}{\cos \left (d x + c\right )^{4}}\right )}{4 \, \sqrt {a + b} d}, \frac {\sqrt {-a - b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} {\left (b \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right )}{2 \, {\left (a + b\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/4*log(((a*b + 2*b^2)*cos(d*x + c)^4 - 4*(a*b + b^2)*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x
+ c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt(a + b) + 2*a^2 + 4*a*b + 2*b^2)/cos(d*x + c)^4)/(sqrt(a + b)*d
), 1/2*sqrt(-a - b)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt
(-a - b)/((a*b + b^2)*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))/((a + b)*d)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right )^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)/sqrt(b*sin(d*x + c)^4 + a), x)

________________________________________________________________________________________

maple [A]  time = 0.51, size = 72, normalized size = 1.41 \[ \frac {\ln \left (\frac {2 a +2 b -2 b \left (\cos ^{2}\left (d x +c \right )\right )+2 \sqrt {a +b}\, \sqrt {a +b -2 b \left (\cos ^{2}\left (d x +c \right )\right )+b \left (\cos ^{4}\left (d x +c \right )\right )}}{\cos \left (d x +c \right )^{2}}\right )}{2 d \sqrt {a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x)

[Out]

1/2/d/(a+b)^(1/2)*ln((2*a+2*b-2*b*cos(d*x+c)^2+2*(a+b)^(1/2)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2))/cos(
d*x+c)^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right )^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(tan(d*x + c)/sqrt(b*sin(d*x + c)^4 + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {tan}\left (c+d\,x\right )}{\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)/(a + b*sin(c + d*x)^4)^(1/2),x)

[Out]

int(tan(c + d*x)/(a + b*sin(c + d*x)^4)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+b*sin(d*x+c)**4)**(1/2),x)

[Out]

Integral(tan(c + d*x)/sqrt(a + b*sin(c + d*x)**4), x)

________________________________________________________________________________________