3.348 \(\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx\)

Optimal. Leaf size=91 \[ \frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)} \]

[Out]

((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]
*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*(a + b)*f)

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Rubi [A]  time = 0.111315, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3190, 382, 377, 206} \[ \frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]
*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*(a + b)*f)

Rule 3190

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]

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 377

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

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])

Rubi steps

\begin{align*} \int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x)}{2 (a+b) f}+\frac{(a+2 b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 (a+b) f}\\ &=\frac{\sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x)}{2 (a+b) f}+\frac{(a+2 b) \operatorname{Subst}\left (\int \frac{1}{1-(a+b) x^2} \, dx,x,\frac{\sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 (a+b) f}\\ &=\frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 (a+b)^{3/2} f}+\frac{\sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x)}{2 (a+b) f}\\ \end{align*}

Mathematica [C]  time = 9.84732, size = 436, normalized size = 4.79 \[ \frac{\tan (e+f x) \sec ^3(e+f x) \left (\frac{b \sin ^2(e+f x)}{a}+1\right ) \left (-30 b \sin ^2(e+f x) \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left (a^2+a b \left (\sin ^2(e+f x)+1\right )+b^2 \sin ^2(e+f x)\right )}{a^2}}-45 a \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left (a^2+a b \left (\sin ^2(e+f x)+1\right )+b^2 \sin ^2(e+f x)\right )}{a^2}}+16 b \sin ^2(e+f x) \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right ) \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}+16 a \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right ) \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}+30 b \sin ^2(e+f x) \sin ^{-1}\left (\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right )+45 a \sin ^{-1}\left (\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right )\right )}{30 a f \sqrt{a+b \sin ^2(e+f x)} \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sec[e + f*x]^3*(1 + (b*Sin[e + f*x]^2)/a)*Tan[e + f*x]*(45*a*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] + 30
*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2 + 16*a*Hypergeometric2F1[2, 3, 7/2, -(((a + b)*T
an[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + 16*
b*Hypergeometric2F1[2, 3, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e
+ f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) - 45*a*Sqrt[-((Sec[e + f*x]^2*(a^2 + b^2*Sin[e + f*x]^2 +
a*b*(1 + Sin[e + f*x]^2))*Tan[e + f*x]^2)/a^2)] - 30*b*Sin[e + f*x]^2*Sqrt[-((Sec[e + f*x]^2*(a^2 + b^2*Sin[e
+ f*x]^2 + a*b*(1 + Sin[e + f*x]^2))*Tan[e + f*x]^2)/a^2)]))/(30*a*f*Sqrt[a + b*Sin[e + f*x]^2]*Sqrt[(Sec[e +
f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))

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Maple [B]  time = 2.79, size = 360, normalized size = 4. \begin{align*}{\frac{1}{4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}f} \left ( 2\,\sin \left ( fx+e \right ) \sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( a+b \right ) ^{3/2}- \left ( \ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ){a}^{2}+3\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ) ab+2\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ){b}^{2}-\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ){a}^{2}-3\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ) ab-2\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ){b}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( a+b \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sin(f*x+e)*(a+b-b*cos(f*x+e)^2)^(1/2)*(a+b)^(3/2)-(ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2
)^(1/2)-b*sin(f*x+e)+a))*a^2+3*ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)-b*sin(f*x+e)+a))*a*
b+2*ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)-b*sin(f*x+e)+a))*b^2-ln(2/(-1+sin(f*x+e))*((a+
b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)+b*sin(f*x+e)+a))*a^2-3*ln(2/(-1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)
^2)^(1/2)+b*sin(f*x+e)+a))*a*b-2*ln(2/(-1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)+b*sin(f*x+e)+a))
*b^2)*cos(f*x+e)^2)/(a+b)^(5/2)/cos(f*x+e)^2/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )^{3}}{\sqrt{b \sin \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.2708, size = 906, normalized size = 9.96 \begin{align*} \left [\frac{{\left (a + 2 \, b\right )} \sqrt{a + b} \cos \left (f x + e\right )^{2} \log \left (\frac{{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 8 \,{\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a + b} \sin \left (f x + e\right ) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) + 4 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} \sin \left (f x + e\right )}{8 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2}}, -\frac{{\left (a + 2 \, b\right )} \sqrt{-a - b} \arctan \left (\frac{{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a - b}}{2 \,{\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{2} - 2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} \sin \left (f x + e\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((a + 2*b)*sqrt(a + b)*cos(f*x + e)^2*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)
*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x
+ e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a^
2 + 2*a*b + b^2)*f*cos(f*x + e)^2), -1/4*((a + 2*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a -
2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x +
e)))*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x +
e)^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sqrt{a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(e + f*x)**3/sqrt(a + b*sin(e + f*x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )^{3}}{\sqrt{b \sin \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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