Integrand size = 21, antiderivative size = 112 \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3} \] Output:
-2*cos(b*x+a)^3/b/d/(d*tan(b*x+a))^(1/2)+7/2*cos(b*x+a)*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 = 0.58 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sin (a+b x) \left (-13+\cos (2 (a+b x))-14 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \tan ^2(a+b x)\right )}{6 b (d \tan (a+b x))^{3/2}} \] Input:
Integrate[Cos[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]
Output:
(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 = 0.61 (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 |
| \(\displaystyle -\frac {7 \left (\frac {\cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d}+\frac {\cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b \sqrt {\sin (2 a+2 b x)}}\right )}{d^2}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}\) |
Input:
Int[Cos[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]
Output:
(-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
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. \(273\) vs. \(2(101)=202\).
Time = 1.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.45
| method | result | size |
| default | \(\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (42+42 \sec \left (b x +a \right )\right )+\sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \left (-21-21 \sec \left (b x +a \right )\right )+4 \cos \left (b x +a \right )^{3}+14 \cos \left (b x +a \right )-42\right ) \sqrt {2}}{24 b \sqrt {d \tan \left (b x +a \right )}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, d}\) | \(274\) |
Input:
int(cos(b*x+a)^3/(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/24/b*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(d*tan(b*x+a))^(1 /2)/(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/d*((-2*csc(b*x+a)+2*co t(b*x+a)+2)^(1/2)*(-csc(b*x+a)+cot(b*x+a))^(1/2)*(csc(b*x+a)-cot(b*x+a)+1) ^(1/2)*EllipticE((csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/2*2^(1/2))*(42+42*sec(b *x+a))+(-2*csc(b*x+a)+2*cot(b*x+a)+2)^(1/2)*(-csc(b*x+a)+cot(b*x+a))^(1/2) *EllipticF((csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/2*2^(1/2))*(csc(b*x+a)-cot(b* x+a)+1)^(1/2)*(-21-21*sec(b*x+a))+4*cos(b*x+a)^3+14*cos(b*x+a)-42)*2^(1/2)
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(b*x+a)^3/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(d*tan(b*x + a))*cos(b*x + a)^3/(d^2*tan(b*x + a)^2), x)
Timed out. \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**3/(d*tan(b*x+a))**(3/2),x)
Output:
Timed out
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(b*x+a)^3/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)^3/(d*tan(b*x + a))^(3/2), x)
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(b*x+a)^3/(d*tan(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate(cos(b*x + a)^3/(d*tan(b*x + a))^(3/2), x)
Timed out. \[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \] Input:
int(cos(a + b*x)^3/(d*tan(a + b*x))^(3/2),x)
Output:
int(cos(a + b*x)^3/(d*tan(a + b*x))^(3/2), x)
\[ \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (b x +a \right )}\, \cos \left (b x +a \right )^{3}}{\tan \left (b x +a \right )^{2}}d x \right )}{d^{2}} \] Input:
int(cos(b*x+a)^3/(d*tan(b*x+a))^(3/2),x)
Output:
(sqrt(d)*int((sqrt(tan(a + b*x))*cos(a + b*x)**3)/tan(a + b*x)**2,x))/d**2