Integrand size = 32, antiderivative size = 226 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{16 \sqrt {2} \left (b^2+c^2\right )^{5/4} e}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 \left (b^2+c^2\right ) e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}} \]
3/32*arctanh(1/2*(b^2+c^2)^(1/4)*sin(d+e*x-arctan(b,c))*2^(1/2)/((b^2+c^2) ^(1/2)+cos(d+e*x-arctan(b,c))*(b^2+c^2)^(1/2))^(1/2))/(b^2+c^2)^(5/4)/e*2^ (1/2)+1/4*(-c*cos(e*x+d)+b*sin(e*x+d))/e/(b^2+c^2)^(1/2)/(b*cos(e*x+d)+c*s in(e*x+d)+(b^2+c^2)^(1/2))^(5/2)-3/16*(c*cos(e*x+d)-b*sin(e*x+d))/(b^2+c^2 )/e/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(3/2)
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\text {\$Aborted} \]
Time = 0.74 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {3042, 3595, 3042, 3595, 3042, 3594, 3042, 3128, 219}
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.
\(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3595 |
\(\displaystyle \frac {3 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{3/2}}dx}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int \frac {1}{\left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^{3/2}}dx}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3595 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}}dx}{4 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}}}dx}{4 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3594 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )+\sqrt {b^2+c^2}}}dx}{4 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )+\sqrt {b^2+c^2}}}dx}{4 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {3 \left (-\frac {\int \frac {1}{2 \sqrt {b^2+c^2}-\frac {\left (b^2+c^2\right ) \sin ^2\left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )+\sqrt {b^2+c^2}}}d\left (-\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )+\sqrt {b^2+c^2}}}\right )}{2 e \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )+\sqrt {b^2+c^2}}}\right )}{2 \sqrt {2} e \left (b^2+c^2\right )^{3/4}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\right )}{8 \sqrt {b^2+c^2}}-\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}\) |
-1/4*(c*Cos[d + e*x] - b*Sin[d + e*x])/(Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + (3*(ArcTanh[((b^2 + c^2)^(1/4 )*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])]/(2*Sqrt[2]*(b^2 + c^2)^(3/4)*e) - (c*C os[d + e*x] - b*Sin[d + e*x])/(2*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Co s[d + e*x] + c*Sin[d + e*x])^(3/2))))/(8*Sqrt[b^2 + c^2])
3.5.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], 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/(a*e*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) 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] && LtQ[n, -1]
Time = 1.60 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.55
method | result | size |
default | \(\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) \left (b^{2}+c^{2}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) b^{2}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) c^{2}+2 \sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+\sqrt {b^{2}+c^{2}}}\, \left (b^{2}+c^{2}\right )^{\frac {3}{4}}\right ) \sqrt {-\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}}{4 \left (b^{2}+c^{2}\right )^{\frac {5}{4}} \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\) | \(350\) |
1/4*(sin(e*x+d-arctan(-b,c))*2^(1/2)*arctanh(1/2*(-(b^2+c^2)^(1/2)*sin(e*x +d-arctan(-b,c))+(b^2+c^2)^(1/2))^(1/2)*2^(1/2)/(b^2+c^2)^(1/4))*(b^2+c^2) +2^(1/2)*arctanh(1/2*(-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+(b^2+c^2)^( 1/2))^(1/2)*2^(1/2)/(b^2+c^2)^(1/4))*b^2+2^(1/2)*arctanh(1/2*(-(b^2+c^2)^( 1/2)*sin(e*x+d-arctan(-b,c))+(b^2+c^2)^(1/2))^(1/2)*2^(1/2)/(b^2+c^2)^(1/4 ))*c^2+2*(-(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+(b^2+c^2)^(1/2))^(1/2)* (b^2+c^2)^(3/4))*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1))^(1/2)/(b^2 +c^2)^(5/4)/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
Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (203) = 406\).
Time = 0.50 (sec) , antiderivative size = 895, normalized size of antiderivative = 3.96 \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/32*(3*sqrt(1/2)*(5*b^4*c*cos(e*x + d) + (5*b^4*c - 10*b^2*c^3 + c^5)*cos (e*x + d)^5 - 10*(b^4*c - b^2*c^3)*cos(e*x + d)^3 - (b^5 + (b^5 - 10*b^3*c ^2 + 5*b*c^4)*cos(e*x + d)^4 - 2*(b^5 - 5*b^3*c^2)*cos(e*x + d)^2)*sin(e*x + d))*log(((3*b^2*c - c^3)*cos(e*x + d)^3 + (b^2*c + 4*c^3)*cos(e*x + d) - (3*b^3 + 4*b*c^2 + (b^3 - 3*b*c^2)*cos(e*x + d)^2)*sin(e*x + d) + 4*sqrt (1/2)*(2*(b^3 + b*c^2)*cos(e*x + d) + 2*(b^2*c + c^3)*sin(e*x + d) - (2*b* c*cos(e*x + d)*sin(e*x + d) + (b^2 - c^2)*cos(e*x + d)^2 + b^2 + 2*c^2)*sq rt(b^2 + c^2))*sqrt(b*cos(e*x + d) + c*sin(e*x + d) + sqrt(b^2 + c^2))/(b^ 2 + c^2)^(1/4) - 4*(2*b*c*cos(e*x + d)^2 - (b^2 - c^2)*cos(e*x + d)*sin(e* x + d) - b*c)*sqrt(b^2 + c^2))/(3*b^2*c*cos(e*x + d) - (3*b^2*c - c^3)*cos (e*x + d)^3 - (b^3 - (b^3 - 3*b*c^2)*cos(e*x + d)^2)*sin(e*x + d)))/(b^2 + c^2)^(1/4) + 2*(3*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^4 - 7*b^4 - 26*b^2 *c^2 - 16*c^4 - 6*(2*b^4 - 3*b^2*c^2 - c^4)*cos(e*x + d)^2 + 12*((b^3*c - b*c^3)*cos(e*x + d)^3 - (2*b^3*c + b*c^3)*cos(e*x + d))*sin(e*x + d) - 2*( (b^3 - 3*b*c^2)*cos(e*x + d)^3 - 3*(3*b^3 + 2*b*c^2)*cos(e*x + d) - (9*b^2 *c + 8*c^3 - (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)))/((5*b^6*c - 5*b ^4*c^3 - 9*b^2*c^5 + c^7)*e*cos(e*x + d)^5 - 10*(b^6*c - b^2*c^5)*e*cos(e* x + d)^3 + 5*(b^6*c + b^4*c^3)*e*cos(e*x + d) - ((b^7 - 9*b^5*c^2 - 5*b^3* c^4 + 5*b*c^6)*e*cos(e*x + d)^4 - 2*(b^7 - 4*b^5*c^2 - 5*b^3*c^4)*e*cos...
\[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\int \frac {1}{\left (b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}\right )}^{5/2}} \,d x \]