3.545 \(\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}} \]

[Out]

(2*b)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[
c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))

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Rubi [A]  time = 0.0716748, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2636, 2642, 2641} \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]

[Out]

(2*b)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[
c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+a \int \frac{1}{(e \cos (c+d x))^{5/2}} \, dx\\ &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{a \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{\left (a \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 e^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.145656, size = 55, normalized size = 0.57 \[ \frac{2 \left (a \sin (c+d x)+a \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+b\right )}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]

[Out]

(2*(b + a*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + a*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))

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Maple [A]  time = 1.668, size = 193, normalized size = 2. \begin{align*} -{\frac{2}{3\,d{e}^{2}} \left ( 2\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) a+2\,a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +b\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c))/(e*cos(d*x+c))^(5/2),x)

[Out]

-2/3/(2*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^2*(2*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*sin(1/2*d*x+1/2*c)^2-(si
n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a+2*a*sin(1/2
*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+b*sin(1/2*d*x+1/2*c))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)/(e*cos(d*x + c))^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))*(b*sin(d*x + c) + a)/(e^3*cos(d*x + c)^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)/(e*cos(d*x + c))^(5/2), x)