Integrand size = 27, antiderivative size = 198 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {a \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e}-\frac {3 a \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))}+\frac {3 a \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))} \]
-a*(e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/e-3*a*arcsinh((e*cos(d*x+ c))^(1/2)/e^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+cos(d* x+c)+sin(d*x+c))/e^(1/2)+3*a*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2 )/(1+cos(d*x+c))^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+c os(d*x+c)+sin(d*x+c))/e^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.38 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {4\ 2^{3/4} \sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{d e (1+\sin (c+d x))^{7/4}} \]
(-4*2^(3/4)*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - Si n[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(d*e*(1 + Sin[c + d*x])^(7/4) )
Time = 0.77 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3157, 3042, 3156, 3042, 25, 3254, 216, 3312, 63, 222}
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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{\sqrt {e \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle \frac {3}{2} a \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} a \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {3}{2} a \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {3}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {3}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {3}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )-\frac {a \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{d e}\) |
-((a*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e)) + (3*a*((-2*Arc Sinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (2*ArcTan[(Sqrt[ e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + C os[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[ c + d*x]))))/2
3.3.83.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)] *(g_.)], x_Symbol] :> Simp[a*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x ]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])) Int[Sqrt[1 + Cos[e + f*x]]/Sqrt [g*Cos[e + f*x]], x], x] + Simp[b*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])) Int[Sin[e + f*x]/(Sqrt[g*C os[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] & & EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 6.87 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(-\frac {\left (-3 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-3 \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos ^{2}\left (d x +c \right )-3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\cos \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a}{d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}}\) | \(291\) |
-1/d*(-3*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*(-cos(d*x+c)/(1+cos(d* x+c)))^(1/2)*cos(d*x+c)-3*arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/( 1+cos(d*x+c)))^(1/2))*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)+cos(d* x+c)*sin(d*x+c)+cos(d*x+c)^2-3*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(( -cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-3*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ar ctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+cos(d* x+c))*(a*(1+sin(d*x+c)))^(1/2)*a/(1+cos(d*x+c)+sin(d*x+c))/(e*cos(d*x+c))^ (1/2)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 970, normalized size of antiderivative = 4.90 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=\text {Too large to display} \]
-1/4*(3*I*d*e*(-a^6/(d^4*e^2))^(1/4)*log(-27/2*(2*(a^4*sin(d*x + c) + (a*d ^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-a^6/(d^4*e^2)))*sqrt(e*cos(d*x + c))*sq rt(a*sin(d*x + c) + a) + (I*d^3*e^2*cos(d*x + c) + I*d^3*e^2 + (2*I*d^3*e^ 2*cos(d*x + c) + I*d^3*e^2)*sin(d*x + c))*(-a^6/(d^4*e^2))^(3/4) + (-2*I*a ^3*d*e*cos(d*x + c)^2 - I*a^3*d*e*cos(d*x + c) + I*a^3*d*e*sin(d*x + c) + I*a^3*d*e)*(-a^6/(d^4*e^2))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 3* I*d*e*(-a^6/(d^4*e^2))^(1/4)*log(-27/2*(2*(a^4*sin(d*x + c) + (a*d^2*e*cos (d*x + c) + a*d^2*e)*sqrt(-a^6/(d^4*e^2)))*sqrt(e*cos(d*x + c))*sqrt(a*sin (d*x + c) + a) + (-I*d^3*e^2*cos(d*x + c) - I*d^3*e^2 + (-2*I*d^3*e^2*cos( d*x + c) - I*d^3*e^2)*sin(d*x + c))*(-a^6/(d^4*e^2))^(3/4) + (2*I*a^3*d*e* cos(d*x + c)^2 + I*a^3*d*e*cos(d*x + c) - I*a^3*d*e*sin(d*x + c) - I*a^3*d *e)*(-a^6/(d^4*e^2))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 3*d*e*(-a ^6/(d^4*e^2))^(1/4)*log(-27/2*(2*(a^4*sin(d*x + c) - (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-a^6/(d^4*e^2)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (d^3*e^2*cos(d*x + c) + d^3*e^2 + (2*d^3*e^2*cos(d*x + c) + d^3*e^ 2)*sin(d*x + c))*(-a^6/(d^4*e^2))^(3/4) + (2*a^3*d*e*cos(d*x + c)^2 + a^3* d*e*cos(d*x + c) - a^3*d*e*sin(d*x + c) - a^3*d*e)*(-a^6/(d^4*e^2))^(1/4)) /(cos(d*x + c) + sin(d*x + c) + 1)) + 3*d*e*(-a^6/(d^4*e^2))^(1/4)*log(-27 /2*(2*(a^4*sin(d*x + c) - (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-a^6/(d^4* e^2)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) - (d^3*e^2*cos(d*x...
\[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {e \cos {\left (c + d x \right )}}}\, dx \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]