Integrand size = 25, antiderivative size = 88 \[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}+\frac {4 a^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{3 f \sqrt {a \sin (e+f x)}} \] Output:
-2/3*b*(a*sin(f*x+e))^(3/2)/f/(b*tan(f*x+e))^(1/2)+4/3*a^2*cos(f*x+e)^(1/2 )*InverseJacobiAM(1/2*f*x+1/2*e,2^(1/2))*(b*tan(f*x+e))^(1/2)/f/(a*sin(f*x +e))^(1/2)
Time = 6.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91 \[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=-\frac {2 a b \sqrt {a \sin (e+f x)} \left (-2 \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (\sin (e+f x)),2\right )+\sqrt [4]{\cos ^2(e+f x)} \sin (e+f x)\right )}{3 f \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)}} \] Input:
Integrate[(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]],x]
Output:
(-2*a*b*Sqrt[a*Sin[e + f*x]]*(-2*EllipticF[ArcSin[Sin[e + f*x]]/2, 2] + (C os[e + f*x]^2)^(1/4)*Sin[e + f*x]))/(3*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan [e + f*x]])
Time = 0.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3078, 3042, 3081, 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.
\(\displaystyle \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)}dx\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {2}{3} a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx}{3 \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {4 a^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{3 f \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\) |
Input:
Int[(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]],x]
Output:
(-2*b*(a*Sin[e + f*x])^(3/2))/(3*f*Sqrt[b*Tan[e + f*x]]) + (4*a^2*Sqrt[Cos [e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(3*f*Sqrt[a*Sin [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*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]
Result contains complex when optimal does not.
Time = 2.96 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {2 \sqrt {a \sin \left (f x +e \right )}\, a \sqrt {b \tan \left (f x +e \right )}\, \left (i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticF}\left (i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ), i\right ) \left (2 \cot \left (f x +e \right )+2 \csc \left (f x +e \right )\right )-\cos \left (f x +e \right )\right )}{3 f}\) | \(108\) |
Input:
int((a*sin(f*x+e))^(3/2)*(b*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3/f*(a*sin(f*x+e))^(1/2)*a*(b*tan(f*x+e))^(1/2)*(I*(1/(1+cos(f*x+e)))^(1 /2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(-csc(f*x+e)+cot(f*x+e)) ,I)*(2*cot(f*x+e)+2*csc(f*x+e))-cos(f*x+e))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=-\frac {2 \, {\left (\sqrt {a \sin \left (f x + e\right )} a \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) - 2 \, \sqrt {\frac {1}{2}} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {1}{2}} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{3 \, f} \] Input:
integrate((a*sin(f*x+e))^(3/2)*(b*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/3*(sqrt(a*sin(f*x + e))*a*sqrt(b*sin(f*x + e)/cos(f*x + e))*cos(f*x + e ) - 2*sqrt(1/2)*sqrt(-a*b)*a*weierstrassPInverse(-4, 0, cos(f*x + e) + I*s in(f*x + e)) - 2*sqrt(1/2)*sqrt(-a*b)*a*weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)))/f
Timed out. \[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((a*sin(f*x+e))**(3/2)*(b*tan(f*x+e))**(1/2),x)
Output:
Timed out
\[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )} \,d x } \] Input:
integrate((a*sin(f*x+e))^(3/2)*(b*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e))^(3/2)*sqrt(b*tan(f*x + e)), x)
\[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )} \,d x } \] Input:
integrate((a*sin(f*x+e))^(3/2)*(b*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e))^(3/2)*sqrt(b*tan(f*x + e)), x)
Timed out. \[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \] Input:
int((a*sin(e + f*x))^(3/2)*(b*tan(e + f*x))^(1/2),x)
Output:
int((a*sin(e + f*x))^(3/2)*(b*tan(e + f*x))^(1/2), x)
\[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\sqrt {b}\, \sqrt {a}\, \left (\int \sqrt {\tan \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}\, \sin \left (f x +e \right )d x \right ) a \] Input:
int((a*sin(f*x+e))^(3/2)*(b*tan(f*x+e))^(1/2),x)
Output:
sqrt(b)*sqrt(a)*int(sqrt(tan(e + f*x))*sqrt(sin(e + f*x))*sin(e + f*x),x)* a