Integrand size = 21, antiderivative size = 110 \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac {7 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b} \]
7/2*d^2*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4* Pi+b*x),2^(1/2))*sin(b*x+a)/b/sin(2*b*x+2*a)^(1/2)/(d*tan(b*x+a))^(1/2)+2* d*sin(b*x+a)^3*(d*tan(b*x+a))^(1/2)/b+7/3*d^3*sin(b*x+a)^3/b/(d*tan(b*x+a) )^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {\left (-28 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sec (a+b x)+2 \cos (a+b x) (13+\cos (2 (a+b x))) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{12 b \sqrt {\sec ^2(a+b x)}} \]
((-28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x] + 2*C os[a + b*x]*(13 + Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x]) ^(3/2))/(12*b*Sqrt[Sec[a + b*x]^2])
Time = 0.64 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3074, 3042, 3078, 3042, 3081, 3042, 3052, 3042, 3119}
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.
\(\displaystyle \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^3 (d \tan (a+b x))^{3/2}dx\) |
\(\Big \downarrow \) 3074 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \int \frac {\sin ^3(a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \int \frac {\sin (a+b x)^3}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
(2*d*Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/b - 7*d^2*(-1/3*(d*Sin[a + b*x]^ 3)/(b*(d*Tan[a + b*x])^(3/2)) + (EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x] )/(2*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]]))
3.1.69.3.1 Defintions of rubi rules used
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(408\) vs. \(2(123)=246\).
Time = 0.82 (sec) , antiderivative size = 409, normalized size of antiderivative = 3.72
method | result | size |
default | \(-\frac {\sin \left (b x +a \right ) \left (42 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-21 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-2 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+42 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+11 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-21 \sqrt {2}\, \cos \left (b x +a \right )+12 \sqrt {2}\right ) \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {2}}{12 b \left (\cos ^{2}\left (b x +a \right )-1\right )}\) | \(409\) |
-1/12/b*sin(b*x+a)*(42*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot( b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b* x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)-21*(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+c sc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)-2*2^(1/2)*cos(b*x+a)^4 +42*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot( b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^( 1/2))-21*(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))+11*cos(b*x+a)^2*2^(1/2)-21*2^(1/2)*cos(b*x+a)+12*2^(1/2))*(d*ta n(b*x+a))^(1/2)*d/(cos(b*x+a)^2-1)*2^(1/2)
\[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sin \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sin \left (b x + a\right )^{3} \,d x } \]
Exception generated. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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 TypeThe choice was done assuming 0=[0,0]ex t_reduce
Timed out. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \]