Integrand size = 19, antiderivative size = 108 \[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {5 d^3 \sin (a+b x)}{3 b \sqrt {d \tan (a+b x)}}-\frac {5 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{6 b}+\frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
5/3*d^3*sin(b*x+a)/b/(d*tan(b*x+a))^(1/2)+5/6*d^2*csc(b*x+a)*(sin(a+1/4*Pi +b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin( 2*b*x+2*a)^(1/2)*(d*tan(b*x+a))^(1/2)/b+2/3*d*sin(b*x+a)*(d*tan(b*x+a))^(3 /2)/b
Result contains complex when optimal does not.
Time = 1.77 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {\cos (2 (a+b x)) \csc (a+b x) \sqrt {\sec ^2(a+b x)} \left (10 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right )+(7+3 \cos (2 (a+b x))) \sqrt {\sec ^2(a+b x)} \sqrt {\tan (a+b x)}\right ) (d \tan (a+b x))^{5/2}}{6 b \tan ^{\frac {3}{2}}(a+b x) \left (-1+\tan ^2(a+b x)\right )} \]
-1/6*(Cos[2*(a + b*x)]*Csc[a + b*x]*Sqrt[Sec[a + b*x]^2]*(10*(-1)^(1/4)*El lipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1] + (7 + 3*Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(5/2))/(b *Tan[a + b*x]^(3/2)*(-1 + Tan[a + b*x]^2))
Time = 0.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 3074, 3042, 3078, 3042, 3081, 3042, 3053, 3042, 3120}
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.
\(\displaystyle \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x) (d \tan (a+b x))^{5/2}dx\) |
\(\Big \downarrow \) 3074 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \int \sin (a+b x) \sqrt {d \tan (a+b x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \int \sin (a+b x) \sqrt {d \tan (a+b x)}dx\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {1}{2} \int \csc (a+b x) \sqrt {d \tan (a+b x)}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {1}{2} \int \frac {\sqrt {d \tan (a+b x)}}{\sin (a+b x)}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\sin (a+b x)}}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\sin (a+b x)}}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {1}{2} \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {1}{2} \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac {5}{3} d^2 \left (\frac {\sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{2 b}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\right )\) |
(2*d*Sin[a + b*x]*(d*Tan[a + b*x])^(3/2))/(3*b) - (5*d^2*(-((d*Sin[a + b*x ])/(b*Sqrt[d*Tan[a + b*x]])) + (Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]* Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(2*b)))/3
3.1.79.3.1 Defintions of rubi rules used
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] - Simp[b^2*((m + n - 1)/(n - 1)) Int[(a*Sin[e + f*x])^m*(b*Ta n[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && In tegersQ[2*m, 2*n] && !(GtQ[m, 1] && !IntegerQ[(m - 1)/2])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( f*m)), x] + Simp[a^2*((m + n - 1)/m) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 ] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Time = 5.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {\sqrt {d \tan \left (b x +a \right )}\, \sin \left (b x +a \right ) \left (-5 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+3 \sin \left (b x +a \right ) \sqrt {2}\, \cos \left (b x +a \right )-5 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \tan \left (b x +a \right ) \sqrt {2}\right ) d^{2} \sqrt {2}}{6 b \left (\cos ^{2}\left (b x +a \right )-1\right )}\) | \(233\) |
-1/6/b*(d*tan(b*x+a))^(1/2)*sin(b*x+a)*(-5*(cot(b*x+a)-csc(b*x+a))^(1/2)*( -csc(b*x+a)+1+cot(b*x+a))^(1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*EllipticF( (1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)+3*sin(b*x+a)*2^(1/ 2)*cos(b*x+a)-5*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^( 1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^( 1/2),1/2*2^(1/2))+2*tan(b*x+a)*2^(1/2))*d^2/(cos(b*x+a)^2-1)*2^(1/2)
\[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \sin \left (b x + a\right ) \,d x } \]
Timed out. \[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \sin \left (b x + a\right ) \,d x } \]
Exception generated. \[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0,0 ]ext_reduce Error: Bad Argument TypeDone
Timed out. \[ \int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx=\int \sin \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2} \,d x \]