Integrand size = 32, antiderivative size = 258 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=-\frac {256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e} \]
-24/35*(c*cos(e*x+d)-b*sin(e*x+d))*(b^2+c^2)^(1/2)*(b*cos(e*x+d)+c*sin(e*x +d)+(b^2+c^2)^(1/2))^(3/2)/e-2/7*(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))^(5/2)/e-256/35*(b^2+c^2)^(3/2)*(c*cos(e*x+d )-b*sin(e*x+d))/e/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(1/2)-64/35* (b^2+c^2)*(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
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 37.94 (sec) , antiderivative size = 5490, normalized size of antiderivative = 21.28 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Result too large to show} \]
Time = 0.76 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3592, 3042, 3592, 3042, 3592, 3042, 3591}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2}dx\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{5/2}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{5/2}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \left (\frac {8}{5} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{3/2}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \left (\frac {8}{5} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{3/2}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \left (\frac {8}{5} \sqrt {b^2+c^2} \left (\frac {4}{3} \sqrt {b^2+c^2} \int \sqrt {b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \left (\frac {8}{5} \sqrt {b^2+c^2} \left (\frac {4}{3} \sqrt {b^2+c^2} \int \sqrt {b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
\(\Big \downarrow \) 3591 |
\(\displaystyle \frac {12}{7} \sqrt {b^2+c^2} \left (\frac {8}{5} \sqrt {b^2+c^2} \left (-\frac {2 \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} (c \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\) |
(-2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2))/(7*e) + (12*Sqrt[b^2 + c^2]*((-2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) /(5*e) + (8*Sqrt[b^2 + c^2]*((-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)))/5))/7
3.5.30.3.1 Defintions of rubi rules used
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] && EqQ[a^ 2 - b^2 - c^2, 0]
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] + Simp[a*((2*n - 1)/n) Int[(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]
Time = 1.64 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {2 \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \left (5 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+10 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+5 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+27 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+54 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+27 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+71 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+142 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+71 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+177 b^{4}+354 b^{2} c^{2}+177 c^{4}\right )}{35 \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\) | \(306\) |
2/35*(sin(e*x+d-arctan(-b,c))+1)*(sin(e*x+d-arctan(-b,c))-1)*(5*b^4*sin(e* x+d-arctan(-b,c))^3+10*b^2*c^2*sin(e*x+d-arctan(-b,c))^3+5*c^4*sin(e*x+d-a rctan(-b,c))^3+27*b^4*sin(e*x+d-arctan(-b,c))^2+54*b^2*c^2*sin(e*x+d-arcta n(-b,c))^2+27*c^4*sin(e*x+d-arctan(-b,c))^2+71*b^4*sin(e*x+d-arctan(-b,c)) +142*b^2*c^2*sin(e*x+d-arctan(-b,c))+71*c^4*sin(e*x+d-arctan(-b,c))+177*b^ 4+354*b^2*c^2+177*c^4)/cos(e*x+d-arctan(-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
Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\frac {2 \, {\left (5 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} - 177 \, b^{4} - 310 \, b^{2} c^{2} - 128 \, c^{4} + 2 \, {\left (22 \, b^{4} + 15 \, b^{2} c^{2} - 27 \, c^{4}\right )} \cos \left (e x + d\right )^{2} + 4 \, {\left (5 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} + {\left (22 \, b^{3} c + 27 \, b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \, {\left (11 \, {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} + {\left (53 \, b^{3} + 86 \, b c^{2}\right )} \cos \left (e x + d\right ) + {\left (53 \, b^{2} c + 64 \, c^{3} + 11 \, {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{35 \, {\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \]
2/35*(5*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^4 - 177*b^4 - 310*b^2*c^2 - 1 28*c^4 + 2*(22*b^4 + 15*b^2*c^2 - 27*c^4)*cos(e*x + d)^2 + 4*(5*(b^3*c - b *c^3)*cos(e*x + d)^3 + (22*b^3*c + 27*b*c^3)*cos(e*x + d))*sin(e*x + d) + 2*(11*(b^3 - 3*b*c^2)*cos(e*x + d)^3 + (53*b^3 + 86*b*c^2)*cos(e*x + d) + (53*b^2*c + 64*c^3 + 11*(3*b^2*c - c^3)*cos(e*x + d)^2)*sin(e*x + d))*sqrt (b^2 + c^2))*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))
Timed out. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\int {\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}\right )}^{7/2} \,d x \]