\(\int (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx\) [438]

Optimal result

Integrand size = 34, antiderivative size = 130 \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \]

[Out]

8/3*(c*cos(e*x+d)-b*sin(e*x+d))*(b^2+c^2)^(1/2)/e/(b*cos(e*x+d)+c*sin(e*x+d)-(b^2+c^2)^(1/2))^(1/2)-2/3*(c*cos
(e*x+d)-b*sin(e*x+d))*(b*cos(e*x+d)+c*sin(e*x+d)-(b^2+c^2)^(1/2))^(1/2)/e

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3192, 3191} \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \]

[In]

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

[Out]

(8*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d +
e*x]]) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e)

Rule 3191

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[-2*((c*Cos[d
+ e*x] - b*Sin[d + e*x])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rule 3192

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

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 13.03 (sec) , antiderivative size = 5142, normalized size of antiderivative = 39.55 \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {2 \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \left (b^{2}+c^{2}\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-5\right )}{3 \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )-b^{2}-c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\)	\(130\)

[In]

int((b*cos(e*x+d)+c*sin(e*x+d)-(b^2+c^2)^(1/2))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(sin(e*x+d-arctan(-b,c))-1)*(b^2+c^2)*(sin(e*x+d-arctan(-b,c))+1)*(sin(e*x+d-arctan(-b,c))-5)/cos(e*x+d-ar
ctan(-b,c))/((b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-b^2-c^2)/(b^2+c^2)^(1/2))^(1/2)/e

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.98 \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\frac {2 \, {\left (2 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} - 5 \, b^{2} - 4 \, c^{2} - 4 \, \sqrt {b^{2} + c^{2}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right )\right )}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}}}{3 \, {\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \]

[In]

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

[Out]

2/3*(2*b*c*cos(e*x + d)*sin(e*x + d) + (b^2 - c^2)*cos(e*x + d)^2 - 5*b^2 - 4*c^2 - 4*sqrt(b^2 + c^2)*(b*cos(e
*x + d) + c*sin(e*x + d)))*sqrt(b*cos(e*x + d) + c*sin(e*x + d) - sqrt(b^2 + c^2))/(c*e*cos(e*x + d) - b*e*sin
(e*x + d))

Sympy [F]

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

[In]

integrate((b*cos(e*x+d)+c*sin(e*x+d)-(b**2+c**2)**(1/2))**(3/2),x)

[Out]

Integral((b*cos(d + e*x) + c*sin(d + e*x) - sqrt(b**2 + c**2))**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F(-2)]

Exception generated. \[ \int \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int((b*cos(d + e*x) + c*sin(d + e*x) - (b^2 + c^2)^(1/2))^(3/2),x)

[Out]

int((b*cos(d + e*x) + c*sin(d + e*x) - (b^2 + c^2)^(1/2))^(3/2), x)