Integrand size = 25, antiderivative size = 157 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {a^2 (a+6 b) \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f} \]
1/16*a^2*(a+6*b)*arctanh(sin(f*x+e)*b^(1/2)/(a+b*sin(f*x+e)^2)^(1/2))/b^(3 /2)/f+1/24*(a+6*b)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2)/b/f-1/6*sin(f*x+e)* (a+b*sin(f*x+e)^2)^(5/2)/b/f+1/16*a*(a+6*b)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^ (1/2)/b/f
Time = 0.88 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)} \left (3 a^{3/2} (a+6 b) \text {arcsinh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a}}\right )+\sqrt {b} \sin (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \left (-3 a (a-10 b)-2 (7 a-6 b) b \sin ^2(e+f x)-8 b^2 \sin ^4(e+f x)\right )\right )}{48 b^{3/2} f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]
(Sqrt[a + b*Sin[e + f*x]^2]*(3*a^(3/2)*(a + 6*b)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]] + Sqrt[b]*Sin[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a]*(-3*a *(a - 10*b) - 2*(7*a - 6*b)*b*Sin[e + f*x]^2 - 8*b^2*Sin[e + f*x]^4)))/(48 *b^(3/2)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3669, 299, 211, 211, 224, 219}
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.
| \(\displaystyle \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x)^3 \left (a+b \sin (e+f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right ) \left (b \sin ^2(e+f x)+a\right )^{3/2}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {(a+6 b) \int \left (b \sin ^2(e+f x)+a\right )^{3/2}d\sin (e+f x)}{6 b}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b}}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {(a+6 b) \left (\frac {3}{4} a \int \sqrt {b \sin ^2(e+f x)+a}d\sin (e+f x)+\frac {1}{4} \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}\right )}{6 b}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b}}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {(a+6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)+\frac {1}{2} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )+\frac {1}{4} \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}\right )}{6 b}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b}}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {(a+6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b \sin ^2(e+f x)}{b \sin ^2(e+f x)+a}}d\frac {\sin (e+f x)}{\sqrt {b \sin ^2(e+f x)+a}}+\frac {1}{2} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )+\frac {1}{4} \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}\right )}{6 b}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {(a+6 b) \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 \sqrt {b}}+\frac {1}{2} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right )+\frac {1}{4} \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}\right )}{6 b}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b}}{f}\) |
(-1/6*(Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(5/2))/b + ((a + 6*b)*((Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/4 + (3*a*((a*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*Sqrt[b]) + (Sin[e + f*x]*Sqrt[a + b* Sin[e + f*x]^2])/2))/4))/(6*b))/f
3.4.34.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.50 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 f}+\frac {3 a \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{8 f \sqrt {b}}-\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {5}{2}}}{6 b f}+\frac {a \sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{24 f b}+\frac {a^{2} \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{16 f b}+\frac {a^{3} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{16 f \,b^{\frac {3}{2}}}\) | \(215\) |
| default | \(\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 f}+\frac {3 a \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{8 f \sqrt {b}}-\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {5}{2}}}{6 b f}+\frac {a \sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{24 f b}+\frac {a^{2} \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{16 f b}+\frac {a^{3} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{16 f \,b^{\frac {3}{2}}}\) | \(215\) |
1/4*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2)/f+3/8*a*sin(f*x+e)*(a+b*sin(f*x+e) ^2)^(1/2)/f+3/8/f*a^2/b^(1/2)*ln(b^(1/2)*sin(f*x+e)+(a+b*sin(f*x+e)^2)^(1/ 2))-1/6*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(5/2)/b/f+1/24/f/b*a*sin(f*x+e)*(a+b *sin(f*x+e)^2)^(3/2)+1/16/f/b*a^2*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)+1/16 /f/b^(3/2)*a^3*ln(b^(1/2)*sin(f*x+e)+(a+b*sin(f*x+e)^2)^(1/2))
Time = 1.38 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.68 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, {\left (a^{3} + 6 \, a^{2} b\right )} \sqrt {b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b} \sin \left (f x + e\right )\right ) - 8 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left (7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{384 \, b^{2} f}, -\frac {3 \, {\left (a^{3} + 6 \, a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left (7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{192 \, b^{2} f}\right ] \]
[1/384*(3*(a^3 + 6*a^2*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^ 4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^ 2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24* (a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5 *a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b )*sqrt(b)*sin(f*x + e)) - 8*(8*b^3*cos(f*x + e)^4 + 3*a^2*b - 16*a*b^2 - 4 *b^3 - 2*(7*a*b^2 + 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b) *sin(f*x + e))/(b^2*f), -1/192*(3*(a^3 + 6*a^2*b)*sqrt(-b)*arctan(1/4*(8*b ^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)* sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(8* b^3*cos(f*x + e)^4 + 3*a^2*b - 16*a*b^2 - 4*b^3 - 2*(7*a*b^2 + 2*b^3)*cos( f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b^2*f)]
Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Time = 0.42 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.11 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\frac {3 \, a^{3} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {18 \, a^{2} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {b}} + 12 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right ) + 18 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right ) - \frac {8 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} \sin \left (f x + e\right )}{b} + \frac {2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )}{b} + \frac {3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2} \sin \left (f x + e\right )}{b}}{48 \, f} \]
1/48*(3*a^3*arcsinh(b*sin(f*x + e)/sqrt(a*b))/b^(3/2) + 18*a^2*arcsinh(b*s in(f*x + e)/sqrt(a*b))/sqrt(b) + 12*(b*sin(f*x + e)^2 + a)^(3/2)*sin(f*x + e) + 18*sqrt(b*sin(f*x + e)^2 + a)*a*sin(f*x + e) - 8*(b*sin(f*x + e)^2 + a)^(5/2)*sin(f*x + e)/b + 2*(b*sin(f*x + e)^2 + a)^(3/2)*a*sin(f*x + e)/b + 3*sqrt(b*sin(f*x + e)^2 + a)*a^2*sin(f*x + e)/b)/f
Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]