Integrand size = 25, antiderivative size = 131 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(3 a+b) \cos (e+f x)}{3 (a-b)^2 f \sqrt {a-b+b \sec ^2(e+f x)}}+\frac {\cos ^3(e+f x)}{3 (a-b) f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {2 b (3 a+b) \sec (e+f x)}{3 (a-b)^3 f \sqrt {a-b+b \sec ^2(e+f x)}} \]
-1/3*(3*a+b)*cos(f*x+e)/(a-b)^2/f/(a-b+b*sec(f*x+e)^2)^(1/2)+1/3*cos(f*x+e )^3/(a-b)/f/(a-b+b*sec(f*x+e)^2)^(1/2)-2/3*b*(3*a+b)*sec(f*x+e)/(a-b)^3/f/ (a-b+b*sec(f*x+e)^2)^(1/2)
Time = 5.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\left (9 a^2+46 a b+9 b^2+8 \left (a^2-b^2\right ) \cos (2 (e+f x))-(a-b)^2 \cos (4 (e+f x))\right ) \sec (e+f x)}{12 \sqrt {2} (a-b)^3 f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \]
-1/12*((9*a^2 + 46*a*b + 9*b^2 + 8*(a^2 - b^2)*Cos[2*(e + f*x)] - (a - b)^ 2*Cos[4*(e + f*x)])*Sec[e + f*x])/(Sqrt[2]*(a - b)^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])
Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4147, 25, 359, 245, 208}
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.
\(\displaystyle \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (e+f x)^3}{\left (a+b \tan (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4147 |
\(\displaystyle \frac {\int -\frac {\cos ^4(e+f x) \left (1-\sec ^2(e+f x)\right )}{\left (b \sec ^2(e+f x)+a-b\right )^{3/2}}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\cos ^4(e+f x) \left (1-\sec ^2(e+f x)\right )}{\left (b \sec ^2(e+f x)+a-b\right )^{3/2}}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {(3 a+b) \int \frac {\cos ^2(e+f x)}{\left (b \sec ^2(e+f x)+a-b\right )^{3/2}}d\sec (e+f x)}{3 (a-b)}+\frac {\cos ^3(e+f x)}{3 (a-b) \sqrt {a+b \sec ^2(e+f x)-b}}}{f}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {\frac {(3 a+b) \left (-\frac {2 b \int \frac {1}{\left (b \sec ^2(e+f x)+a-b\right )^{3/2}}d\sec (e+f x)}{a-b}-\frac {\cos (e+f x)}{(a-b) \sqrt {a+b \sec ^2(e+f x)-b}}\right )}{3 (a-b)}+\frac {\cos ^3(e+f x)}{3 (a-b) \sqrt {a+b \sec ^2(e+f x)-b}}}{f}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {\cos ^3(e+f x)}{3 (a-b) \sqrt {a+b \sec ^2(e+f x)-b}}+\frac {(3 a+b) \left (-\frac {2 b \sec (e+f x)}{(a-b)^2 \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {\cos (e+f x)}{(a-b) \sqrt {a+b \sec ^2(e+f x)-b}}\right )}{3 (a-b)}}{f}\) |
(Cos[e + f*x]^3/(3*(a - b)*Sqrt[a - b + b*Sec[e + f*x]^2]) + ((3*a + b)*(- (Cos[e + f*x]/((a - b)*Sqrt[a - b + b*Sec[e + f*x]^2])) - (2*b*Sec[e + f*x ])/((a - b)^2*Sqrt[a - b + b*Sec[e + f*x]^2])))/(3*(a - b)))/f
3.2.29.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Time = 1.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.59
method | result | size |
default | \(-\frac {\left (a -b \right ) \left (\sin \left (f x +e \right )^{4} \cos \left (f x +e \right )^{2} b^{3}+3 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{4} a \,b^{2}-a^{3} \cos \left (f x +e \right )^{6}+3 a^{2} b \cos \left (f x +e \right )^{6}+2 \sin \left (f x +e \right )^{4} b^{3}+3 a^{3} \cos \left (f x +e \right )^{4}-6 a^{2} b \cos \left (f x +e \right )^{4}+6 \sin \left (f x +e \right )^{2} a \,b^{2}+9 a^{2} b \cos \left (f x +e \right )^{2}\right ) a^{4} \sec \left (f x +e \right )^{3}}{3 f \left (\sqrt {-b \left (a -b \right )}-a +b \right )^{4} \left (\sqrt {-b \left (a -b \right )}+a -b \right )^{4} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}\) | \(208\) |
-1/3/f*(a-b)/((-b*(a-b))^(1/2)-a+b)^4/((-b*(a-b))^(1/2)+a-b)^4*(sin(f*x+e) ^4*cos(f*x+e)^2*b^3+3*sin(f*x+e)^2*cos(f*x+e)^4*a*b^2-a^3*cos(f*x+e)^6+3*a ^2*b*cos(f*x+e)^6+2*sin(f*x+e)^4*b^3+3*a^3*cos(f*x+e)^4-6*a^2*b*cos(f*x+e) ^4+6*sin(f*x+e)^2*a*b^2+9*a^2*b*cos(f*x+e)^2)*a^4/(a+b*tan(f*x+e)^2)^(3/2) *sec(f*x+e)^3
Time = 0.35 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} f\right )}} \]
1/3*((a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - (3*a^2 - 2*a*b - b^2)*cos(f*x + e)^3 - 2*(3*a*b + b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos (f*x + e)^2)/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f)
Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.65 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 6 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {3 \, b^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )} + \frac {3 \, b}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}}{3 \, f} \]
-1/3*(3*sqrt(a - b + b/cos(f*x + e)^2)*cos(f*x + e)/(a^2 - 2*a*b + b^2) - ((a - b + b/cos(f*x + e)^2)^(3/2)*cos(f*x + e)^3 - 6*sqrt(a - b + b/cos(f* x + e)^2)*b*cos(f*x + e))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 3*b^2/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt(a - b + b/cos(f*x + e)^2)*cos(f*x + e)) + 3* b/((a^2 - 2*a*b + b^2)*sqrt(a - b + b/cos(f*x + e)^2)*cos(f*x + e)))/f
\[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{3}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]