Integrand size = 21, antiderivative size = 108 \[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{12 b \sqrt {d \tan (a+b x)}}+\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{6 b}-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b} \] Output:
1/12*d^2*InverseJacobiAM(a-1/4*Pi+b*x,2^(1/2))*sec(b*x+a)*sin(2*b*x+2*a)^( 1/2)/b/(d*tan(b*x+a))^(1/2)+1/6*d*cos(b*x+a)*(d*tan(b*x+a))^(1/2)/b-1/3*d* cos(b*x+a)^3*(d*tan(b*x+a))^(1/2)/b
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {\cos (a+b x) \left (\sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sqrt {\sec ^2(a+b x)}+\cos (2 (a+b x)) \sqrt {\tan (a+b x)}\right ) (d \tan (a+b x))^{3/2}}{6 b \tan ^{\frac {3}{2}}(a+b x)} \] Input:
Integrate[Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]
Output:
-1/6*(Cos[a + b*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Sec[a + b*x]^2] + Cos[2*(a + b*x)]*Sqrt[Tan[a + b*x]])*( d*Tan[a + b*x])^(3/2))/(b*Tan[a + b*x]^(3/2))
Time = 0.56 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3090, 3042, 3092, 3042, 3094, 3042, 3053, 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.
\(\displaystyle \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \tan (a+b x))^{3/2}}{\sec (a+b x)^3}dx\) |
\(\Big \downarrow \) 3090 |
\(\displaystyle \frac {1}{6} d^2 \int \frac {\cos (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} d^2 \int \frac {1}{\sec (a+b x) \sqrt {d \tan (a+b x)}}dx-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3092 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {1}{2} \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}}dx+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {1}{2} \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}}dx+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3094 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {\sqrt {\sin (2 a+2 b x)} \sec (a+b x) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 \sqrt {d \tan (a+b x)}}+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {\sqrt {\sin (2 a+2 b x)} \sec (a+b x) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 \sqrt {d \tan (a+b x)}}+\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{6} d^2 \left (\frac {\cos (a+b x) \sqrt {d \tan (a+b x)}}{b d}+\frac {\sqrt {\sin (2 a+2 b x)} \sec (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b \sqrt {d \tan (a+b x)}}\right )-\frac {d \cos ^3(a+b x) \sqrt {d \tan (a+b x)}}{3 b}\) |
Input:
Int[Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]
Output:
-1/3*(d*Cos[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/b + (d^2*((EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[d*Tan[a + b*x]]) + (Cos[a + b*x]*Sqrt[d*Tan[a + b*x]])/(b*d)))/6
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/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[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)) , x] - Simp[b^2*((n - 1)/(a^2*m)) Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1 ] || (EqQ[m, -1] && EqQ[n, 3/2])) && IntegersQ[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[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
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 = 3.49 (sec) , antiderivative size = 1588, normalized size of antiderivative = 14.70
Input:
int(cos(b*x+a)^3*(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/96/b*(d*tan(b*x+a))^(1/2)*d*(2^(1/2)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x +a)+1)^2)^(1/2)*ln(-(cot(b*x+a)*cos(b*x+a)-2*cot(b*x+a)+2*sin(b*x+a)*(-2*s in(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-2*cos(b*x+a)+csc(b*x+a)-sin(b *x+a)+2)/(-1+cos(b*x+a)))*(-6*cos(b*x+a)^2-6*cos(b*x+a)+3*cot(b*x+a)+3*csc (b*x+a))+(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(-(cot(b*x+a)*c os(b*x+a)-2*cot(b*x+a)+2*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+ 1)^2)^(1/2)-2*cos(b*x+a)+csc(b*x+a)-sin(b*x+a)+2)/(-1+cos(b*x+a)))*(12*cos (b*x+a)^2+12*cos(b*x+a))+2^(1/2)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^ 2)^(1/2)*ln((2*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2 )-cot(b*x+a)*cos(b*x+a)+sin(b*x+a)+2*cos(b*x+a)-csc(b*x+a)+2*cot(b*x+a)-2) /(-1+cos(b*x+a)))*(6*cos(b*x+a)^2+6*cos(b*x+a)-3*cot(b*x+a)-3*csc(b*x+a))+ ln((2*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-cot(b*x +a)*cos(b*x+a)+sin(b*x+a)+2*cos(b*x+a)-csc(b*x+a)+2*cot(b*x+a)-2)/(-1+cos( b*x+a)))*(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*(-12*cos(b*x+a)^2 -12*cos(b*x+a))+2^(1/2)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)* arctan((sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-cos(b *x+a)+1)/(-1+cos(b*x+a)))*(-12*cos(b*x+a)^2-12*cos(b*x+a)+6*cot(b*x+a)+6*c sc(b*x+a))+arctan((sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^ (1/2)-cos(b*x+a)+1)/(-1+cos(b*x+a)))*(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1 )^2)^(1/2)*(24*cos(b*x+a)^2+24*cos(b*x+a))+2^(1/2)*(-2*sin(b*x+a)*cos(b...
\[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \cos \left (b x + a\right )^{3} \,d x } \] 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))*d*cos(b*x + a)^3*tan(b*x + a), x)
Timed out. \[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**3*(d*tan(b*x+a))**(3/2),x)
Output:
Timed out
\[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \cos \left (b x + a\right )^{3} \,d x } \] Input:
integrate(cos(b*x+a)^3*(d*tan(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate((d*tan(b*x + a))^(3/2)*cos(b*x + a)^3, x)
Exception generated. \[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cos(b*x+a)^3*(d*tan(b*x+a))^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo r the root of a polynomial with parameters. This might be wrong.The choice was done
Timed out. \[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \] 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)
\[ \int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \cos \left (b x +a \right )^{3} \tan \left (b x +a \right )d x \right ) d \] 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),x)*d