\(\int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx\) [244]

Optimal result

Integrand size = 19, antiderivative size = 78 \[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b \sqrt {d \tan (a+b x)}}-\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b} \]

[Out]

-1/2*d^2*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(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)-d*cos(b*x+a)*(d*tan(b*x+a))^(1/2)/b

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2690, 2694, 2653, 2720} \[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {d^2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b \sqrt {d \tan (a+b x)}}-\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b} \]

[In]

Int[Cos[a + b*x]*(d*Tan[a + b*x])^(3/2),x]

[Out]

(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]]) - (d*Cos[a +
b*x]*Sqrt[d*Tan[a + b*x]])/b

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[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]

Rule 2690

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] - Dist[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]

Rule 2694

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Co
s[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
]

Rule 2720

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b}+\frac {1}{2} d^2 \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx \\ & = -\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b}+\frac {\left (d^2 \sqrt {\sin (a+b x)}\right ) \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 {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b}+\frac {\left (d^2 \sec (a+b x) \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{2 \sqrt {d \tan (a+b x)}} \\ & = \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)}}{2 b \sqrt {d \tan (a+b x)}}-\frac {d \cos (a+b x) \sqrt {d \tan (a+b x)}}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.74 \[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {d \cos (a+b x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {d \tan (a+b x)}}{b} \]

[In]

Integrate[Cos[a + b*x]*(d*Tan[a + b*x])^(3/2),x]

[Out]

(d*Cos[a + b*x]*(-1 + Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b
*x]])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(95)=190\).

Time = 3.02 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.81

method	result	size
default	\(\frac {\sin \left (b x +a \right ) \left (-\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sin \left (b x +a \right ) \sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {2}}{2 b \left (\cos ^{2}\left (b x +a \right )-1\right )}\)	\(219\)

[In]

int(cos(b*x+a)*(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2/b*sin(b*x+a)*(-(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))*cos(b*x+a)-(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))+
sin(b*x+a)*2^(1/2)*cos(b*x+a))*(d*tan(b*x+a))^(1/2)*d/(cos(b*x+a)^2-1)*2^(1/2)

Fricas [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \cos \left (b x + a\right ) \,d x } \]

[In]

integrate(cos(b*x+a)*(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*tan(b*x + a))*d*cos(b*x + a)*tan(b*x + a), x)

Sympy [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \cos {\left (a + b x \right )}\, dx \]

[In]

integrate(cos(b*x+a)*(d*tan(b*x+a))**(3/2),x)

[Out]

Integral((d*tan(a + b*x))**(3/2)*cos(a + b*x), x)

Maxima [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \cos \left (b x + a\right ) \,d x } \]

[In]

integrate(cos(b*x+a)*(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*tan(b*x + a))^(3/2)*cos(b*x + a), x)

Giac [F(-2)]

Exception generated. \[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cos(b*x+a)*(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]ext_reduce Error: Bad Argument TypeThe choice was done assuming 0=[0,0]
ext_reduce

Mupad [F(-1)]

Timed out. \[ \int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \cos \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \]

[In]

int(cos(a + b*x)*(d*tan(a + b*x))^(3/2),x)

[Out]

int(cos(a + b*x)*(d*tan(a + b*x))^(3/2), x)