3.234 \(\int (a \cos (c+d x)+b \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=131 \[ \frac{2 \left (a^2+b^2\right ) \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} \text{EllipticF}\left (\frac{1}{2} \left (-\tan ^{-1}(a,b)+c+d x\right ),2\right )}{3 d \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x)) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0580733, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3073, 3078, 2641} \[ \frac{2 \left (a^2+b^2\right ) \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} F\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{3 d \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x)) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3073

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[((b*Cos[c + d*x]
- a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1))/(d*n), x] + Dist[((n - 1)*(a^2 + b^2))/n, Int[(a*
Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] &&  !IntegerQ[(n
- 1)/2] && GtQ[n, 1]

Rule 3078

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

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

Mathematica [C]  time = 1.32645, size = 143, normalized size = 1.09 \[ \frac{2 \left (\frac{\left (a^2+b^2\right ) \tan \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d x\right ) \sqrt{\cos ^2\left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d x\right )} \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d x\right )\right )}{\sqrt{b \sqrt{\frac{a^2}{b^2}+1} \sin \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d x\right )}}+\sqrt{a \cos (c+d x)+b \sin (c+d x)} (a \sin (c+d x)-b \cos (c+d x))\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-(b*Cos[c + d*x]) + a*Sin[c + d*x])*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]] + ((a^2 + b^2)*Sqrt[Cos[c + d*x
+ ArcTan[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[a/b]]^2]*Tan[c + d*x + ArcTan[a/b
]])/Sqrt[Sqrt[1 + a^2/b^2]*b*Sin[c + d*x + ArcTan[a/b]]]))/(3*d)

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Maple [A]  time = 1.221, size = 163, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}+{b}^{2}}{3\,\cos \left ( dx+c-\arctan \left ( -a,b \right ) \right ) d} \left ( \sqrt{1+\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) }\sqrt{-2\,\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) +2}\sqrt{-\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) }{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) +2\, \left ( \sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) \right ) ^{3}-2\,\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) \right ){\frac{1}{\sqrt{\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) \sqrt{{a}^{2}+{b}^{2}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(d*x+c)+b*sin(d*x+c))^(3/2),x)

[Out]

1/3*(a^2+b^2)*((1+sin(d*x+c-arctan(-a,b)))^(1/2)*(-2*sin(d*x+c-arctan(-a,b))+2)^(1/2)*(-sin(d*x+c-arctan(-a,b)
))^(1/2)*EllipticF((1+sin(d*x+c-arctan(-a,b)))^(1/2),1/2*2^(1/2))+2*sin(d*x+c-arctan(-a,b))^3-2*sin(d*x+c-arct
an(-a,b)))/cos(d*x+c-arctan(-a,b))/(sin(d*x+c-arctan(-a,b))*(a^2+b^2)^(1/2))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + b*sin(d*x + c))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

integral((a*cos(d*x + c) + b*sin(d*x + c))^(3/2), 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*cos(d*x+c)+b*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + b*sin(d*x + c))^(3/2), x)