Integrand size = 25, antiderivative size = 64 \[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\frac {6 \cos ^2(e+f x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {17}{12},\frac {29}{12},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2}}{17 d f} \] Output:
6/17*(cos(f*x+e)^2)^(3/4)*hypergeom([3/4, 17/12],[29/12],sin(f*x+e)^2)*(b* sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2)/d/f
Time = 11.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08 \[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\frac {3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {17}{12},\frac {29}{12},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3} \sin (2 (e+f x)) \sqrt {d \tan (e+f x)}}{17 f \sqrt [4]{\cos ^2(e+f x)}} \] Input:
Integrate[(b*Sin[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]],x]
Output:
(3*Hypergeometric2F1[3/4, 17/12, 29/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^( 4/3)*Sin[2*(e + f*x)]*Sqrt[d*Tan[e + f*x]])/(17*f*(Cos[e + f*x]^2)^(1/4))
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3082, 3042, 3057}
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.
\(\displaystyle \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)}dx\) |
\(\Big \downarrow \) 3082 |
\(\displaystyle \frac {b \cos ^{\frac {3}{2}}(e+f x) (d \tan (e+f x))^{3/2} \int \frac {(b \sin (e+f x))^{11/6}}{\sqrt {\cos (e+f x)}}dx}{d (b \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \cos ^{\frac {3}{2}}(e+f x) (d \tan (e+f x))^{3/2} \int \frac {(b \sin (e+f x))^{11/6}}{\sqrt {\cos (e+f x)}}dx}{d (b \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3057 |
\(\displaystyle \frac {6 \cos ^2(e+f x)^{3/4} (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {17}{12},\frac {29}{12},\sin ^2(e+f x)\right )}{17 d f}\) |
Input:
Int[(b*Sin[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]],x]
Output:
(6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[3/4, 17/12, 29/12, Sin[e + f*x ]^2]*(b*Sin[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2))/(17*d*f)
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a*Cos[e + f*x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b* (a*Sin[e + f*x])^(n + 1))) 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]
\[\int \left (b \sin \left (f x +e \right )\right )^{\frac {4}{3}} \sqrt {d \tan \left (f x +e \right )}d x\]
Input:
int((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x)
Output:
int((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x)
\[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\int { \left (b \sin \left (f x + e\right )\right )^{\frac {4}{3}} \sqrt {d \tan \left (f x + e\right )} \,d x } \] Input:
integrate((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
integral((b*sin(f*x + e))^(1/3)*sqrt(d*tan(f*x + e))*b*sin(f*x + e), x)
Timed out. \[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((b*sin(f*x+e))**(4/3)*(d*tan(f*x+e))**(1/2),x)
Output:
Timed out
\[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\int { \left (b \sin \left (f x + e\right )\right )^{\frac {4}{3}} \sqrt {d \tan \left (f x + e\right )} \,d x } \] Input:
integrate((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e))^(4/3)*sqrt(d*tan(f*x + e)), x)
Timed out. \[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\int {\left (b\,\sin \left (e+f\,x\right )\right )}^{4/3}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \] Input:
int((b*sin(e + f*x))^(4/3)*(d*tan(e + f*x))^(1/2),x)
Output:
int((b*sin(e + f*x))^(4/3)*(d*tan(e + f*x))^(1/2), x)
\[ \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx=\sqrt {d}\, b^{\frac {4}{3}} \left (\int \sqrt {\tan \left (f x +e \right )}\, \sin \left (f x +e \right )^{\frac {4}{3}}d x \right ) \] Input:
int((b*sin(f*x+e))^(4/3)*(d*tan(f*x+e))^(1/2),x)
Output:
sqrt(d)*b**(1/3)*int(sqrt(tan(e + f*x))*sin(e + f*x)**(1/3)*sin(e + f*x),x )*b