Integrand size = 21, antiderivative size = 112 \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3} \]
-2*cos(b*x+a)^3/b/d/(d*tan(b*x+a))^(1/2)+7/2*cos(b*x+a)*(sin(a+1/4*Pi+b*x) ^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*tan(b* x+a))^(1/2)/b/d^2/sin(2*b*x+2*a)^(1/2)-7/3*cos(b*x+a)^3*(d*tan(b*x+a))^(3/ 2)/b/d^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.64 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sin (a+b x) \left (-13+\cos (2 (a+b x))-14 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \tan ^2(a+b x)\right )}{6 b (d \tan (a+b x))^{3/2}} \]
(Sin[a + b*x]*(-13 + Cos[2*(a + b*x)] - 14*Hypergeometric2F1[3/4, 3/2, 7/4 , -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(6*b*(d*Tan[a + b *x])^(3/2))
Time = 0.62 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3089, 3042, 3092, 3042, 3095, 3042, 3052, 3042, 3119}
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.
| \(\displaystyle \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (a+b x)^3 (d \tan (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3089 |
| \(\displaystyle -\frac {7 \int \cos ^3(a+b x) \sqrt {d \tan (a+b x)}dx}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \int \frac {\sqrt {d \tan (a+b x)}}{\sec (a+b x)^3}dx}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3092 |
| \(\displaystyle -\frac {7 \left (\frac {1}{2} \int \cos (a+b x) \sqrt {d \tan (a+b x)}dx+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {1}{2} \int \frac {\sqrt {d \tan (a+b x)}}{\sec (a+b x)}dx+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3095 |
| \(\displaystyle -\frac {7 \left (\frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\sin (a+b x)}}+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\sin (a+b x)}}+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {7 \left (\frac {\cos (a+b x) \sqrt {d \tan (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 \left (\frac {\cos (a+b x) \sqrt {d \tan (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}+\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {7 \left (\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}+\frac {\cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b \sqrt {\sin (2 a+2 b x)}}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
(-2*Cos[a + b*x]^3)/(b*d*Sqrt[d*Tan[a + b*x]]) - (7*((Cos[a + b*x]*Ellipti cE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(2*b*Sqrt[Sin[2*a + 2*b*x]]) + (Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(3*b*d)))/d^2
3.3.67.3.1 Defintions of rubi rules used
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1))), x] - Simp[(m + n + 1)/(b^2*(n + 1)) Int[(a*Sec[e + f*x])^m*(b*Ta n[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && I ntegersQ[2*m, 2*n]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(a*Sec[e + f*x])^m)*((b*Tan[e + f*x])^(n + 1)/(b*f* m)), x] + Simp[(m + n + 1)/(a^2*m) Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1 ] && EqQ[n, -2^(-1)])) && IntegersQ[2*m, 2*n]
Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[ Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(125)=250\).
Time = 1.49 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.39
| method | result | size |
| default | \(\frac {\left (2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-21 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-21 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+42 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+7 \sqrt {2}\, \cos \left (b x +a \right )-21 \sqrt {2}\right ) \sqrt {2}}{12 b \sqrt {d \tan \left (b x +a \right )}\, d}\) | \(380\) |
1/12/b/(d*tan(b*x+a))^(1/2)/d*(2*cos(b*x+a)^3*2^(1/2)-21*(cot(b*x+a)-csc(b *x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1 /2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+42*(1+csc(b*x+a )-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a ))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-21*sec(b*x +a)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot( b*x+a)-csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^( 1/2))+42*sec(b*x+a)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x +a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a ))^(1/2),1/2*2^(1/2))+7*2^(1/2)*cos(b*x+a)-21*2^(1/2))*2^(1/2)
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]