Integrand size = 25, antiderivative size = 130 \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=-\frac {2 a^2 (a \sin (e+f x))^{3/2}}{21 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}+\frac {4 a^4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{21 b^2 f \sqrt {a \sin (e+f x)}} \]
-2/21*a^2*(a*sin(f*x+e))^(3/2)/b/f/(b*tan(f*x+e))^(1/2)+2/7*(a*sin(f*x+e)) ^(7/2)/b/f/(b*tan(f*x+e))^(1/2)+4/21*a^4*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos( 1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(b*t an(f*x+e))^(1/2)/b^2/f/(a*sin(f*x+e))^(1/2)
Time = 0.85 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.75 \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\frac {a^3 \sqrt {a \sin (e+f x)} \left (8 \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (\sin (e+f x)),2\right )+\sqrt [4]{\cos ^2(e+f x)} (5 \sin (e+f x)-3 \sin (3 (e+f x)))\right )}{42 b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)}} \]
(a^3*Sqrt[a*Sin[e + f*x]]*(8*EllipticF[ArcSin[Sin[e + f*x]]/2, 2] + (Cos[e + f*x]^2)^(1/4)*(5*Sin[e + f*x] - 3*Sin[3*(e + f*x)])))/(42*b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]])
Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3076, 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.
\(\displaystyle \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 3076 |
\(\displaystyle \frac {a^2 \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)}dx}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)}dx}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {a^2 \left (\frac {2}{3} a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\right )}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \left (\frac {2}{3} a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\right )}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {a^2 \left (\frac {2 a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx}{3 \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\right )}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \left (\frac {2 a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\right )}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {a^2 \left (\frac {4 a^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{3 f \sqrt {a \sin (e+f x)}}-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt {b \tan (e+f x)}}\right )}{7 b^2}+\frac {2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt {b \tan (e+f x)}}\) |
(2*(a*Sin[e + f*x])^(7/2))/(7*b*f*Sqrt[b*Tan[e + f*x]]) + (a^2*((-2*b*(a*S in[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]] )))/(7*b^2)
3.2.40.3.1 Defintions of rubi rules used
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*m)) , x] - Simp[a^2*((n + 1)/(b^2*m)) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
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 = 3.80 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.97
method | result | size |
default | \(\frac {\left (\frac {3}{1985}-\frac {i}{41685}\right ) \sec \left (f x +e \right ) \csc \left (f x +e \right ) \left (126 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sin \left (f x +e \right )+3 i \left (\cos ^{4}\left (f x +e \right )\right )-3 i \left (\cos ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sin \left (f x +e \right )+189 \left (\cos ^{4}\left (f x +e \right )\right )-2 i \left (\cos ^{2}\left (f x +e \right )\right )-189 \left (\cos ^{3}\left (f x +e \right )\right )+2 i \cos \left (f x +e \right )-126 \left (\cos ^{2}\left (f x +e \right )\right )+126 \cos \left (f x +e \right )\right ) \sqrt {\sin \left (f x +e \right ) a}\, a^{3} \left (\cos \left (f x +e \right )+1\right )}{f \sqrt {b \tan \left (f x +e \right )}\, b}\) | \(256\) |
(3/1985-1/41685*I)/f*sec(f*x+e)*csc(f*x+e)*(126*I*(1/(cos(f*x+e)+1))^(1/2) *(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)* sin(f*x+e)+3*I*cos(f*x+e)^4-3*I*cos(f*x+e)^3-2*(1/(cos(f*x+e)+1))^(1/2)*(c os(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)*sin (f*x+e)+189*cos(f*x+e)^4-2*I*cos(f*x+e)^2-189*cos(f*x+e)^3+2*I*cos(f*x+e)- 126*cos(f*x+e)^2+126*cos(f*x+e))*(sin(f*x+e)*a)^(1/2)*a^3*(cos(f*x+e)+1)/( b*tan(f*x+e))^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97 \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {-a b} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} \sqrt {-a b} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - {\left (3 \, a^{3} \cos \left (f x + e\right )^{3} - 2 \, a^{3} \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}\right )}}{21 \, b^{2} f} \]
2/21*(sqrt(2)*sqrt(-a*b)*a^3*weierstrassPInverse(-4, 0, cos(f*x + e) + I*s in(f*x + e)) + sqrt(2)*sqrt(-a*b)*a^3*weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) - (3*a^3*cos(f*x + e)^3 - 2*a^3*cos(f*x + e))*sqrt(a *sin(f*x + e))*sqrt(b*sin(f*x + e)/cos(f*x + e)))/(b^2*f)
Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]