\(\int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx\) [59]

Optimal result

Integrand size = 21, antiderivative size = 105 \[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {5 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{12 b} \]

[Out]

-5/6*d*sin(b*x+a)/b/(d*tan(b*x+a))^(1/2)-1/3*d*sin(b*x+a)^3/b/(d*tan(b*x+a))^(1/2)-5/12*csc(b*x+a)*(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))*sin(2*b*x+2*a)^(1/2)*(d*tan(b*x+a))^(1
/2)/b

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2678, 2681, 2653, 2720} \[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}-\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}+\frac {5 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{12 b} \]

[In]

Int[Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-5*d*Sin[a + b*x])/(6*b*Sqrt[d*Tan[a + b*x]]) - (d*Sin[a + b*x]^3)/(3*b*Sqrt[d*Tan[a + b*x]]) + (5*Csc[a + b*
x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(12*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 2678

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-b)*(a*Sin
[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)), x] + Dist[a^2*((m + n - 1)/m), Int[(a*Sin[e + f*x])^(m - 2)*(b*
Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && IntegersQ
[2*m, 2*n]

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[Cos[e + f*x]
^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

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 \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {5}{6} \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {5}{12} \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {\left (5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{12 \sqrt {\sin (a+b x)}} \\ & = -\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {1}{12} \left (5 \csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = -\frac {5 d \sin (a+b x)}{6 b \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {5 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{12 b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.79 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.32 \[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\cos (2 (a+b x)) \sec (a+b x) \left (-5 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sec ^2(a+b x)+(-6+\cos (2 (a+b x))) \sqrt {\sec ^2(a+b x)} \sqrt {\tan (a+b x)}\right ) \sqrt {d \tan (a+b x)}}{6 b \sqrt {\sec ^2(a+b x)} \sqrt {\tan (a+b x)} \left (-1+\tan ^2(a+b x)\right )} \]

[In]

Integrate[Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]],x]

[Out]

-1/6*(Cos[2*(a + b*x)]*Sec[a + b*x]*(-5*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sec
[a + b*x]^2 + (-6 + Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*Sqrt[d*Tan[a + b*x]])/(b*Sqrt[S
ec[a + b*x]^2]*Sqrt[Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.25 (sec) , antiderivative size = 1740, normalized size of antiderivative = 16.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1740\)

[In]

int(sin(b*x+a)^3*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/48/b*csc(b*x+a)*(-6*I*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/
2-1/2*I,1/2*2^(1/2))*cos(b*x+a)+6*I*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+6*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))
^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(b*x+a)-6*I*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(b*x+a)+6*(
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)*EllipticPi((1+cs
c(b*x+a)-cot(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(b*x+a)-32*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(b*x+a)+1+c
ot(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)-8*2^(1/2)*cos(b*x+a)^3*sin(b*x+a)+6*(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)*EllipticPi((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+6*(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)*EllipticPi((1+csc(b*x+a)-cot(b*
x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))-32*(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))+3*(-cos(b*x+a)*sin(b*x+a)/(cos(b
*x+a)+1)^2)^(1/2)*ln(-2*cot(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*csc(b*x+a)*2^(
1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*cot(b*x+a)+2)*cos(b*x+a)-3*(-cos(b*x+a)*sin(b*x+a)/(
cos(b*x+a)+1)^2)^(1/2)*ln(2*cot(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)+2*csc(b*x+a)
*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*cot(b*x+a)+2)*cos(b*x+a)+6*(-cos(b*x+a)*sin(b*x+
a)/(cos(b*x+a)+1)^2)^(1/2)*arctan((-sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x
+a)-1)/(-1+cos(b*x+a)))*cos(b*x+a)-6*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*arctan((sin(b*x+a)*2^(1/2
)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a)))*cos(b*x+a)+28*sin(b*x+a)*2^(1
/2)*cos(b*x+a)+3*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(-2*cot(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x
+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*csc(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*cot(b
*x+a)+2)-3*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(2*cot(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(
-1+cos(b*x+a))^2)^(1/2)+2*csc(b*x+a)*2^(1/2)*(-cot(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))^2)^(1/2)-2*cot(b*x+a)+2
)+6*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*arctan((-sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b
*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a)))-6*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*arctan((sin
(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a))))*(d*tan(b*x+a))
^(1/2)*2^(1/2)

Fricas [F]

\[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right )^{3} \,d x } \]

[In]

integrate(sin(b*x+a)^3*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(-(cos(b*x + a)^2 - 1)*sqrt(d*tan(b*x + a))*sin(b*x + a), x)

Sympy [F(-1)]

Timed out. \[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=\text {Timed out} \]

[In]

integrate(sin(b*x+a)**3*(d*tan(b*x+a))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sin ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right )^{3} \,d x } \]

[In]

integrate(sin(b*x+a)^3*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*tan(b*x + a))*sin(b*x + a)^3, x)

Giac [F(-2)]

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

[In]

integrate(sin(b*x+a)^3*(d*tan(b*x+a))^(1/2),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 TypeDone

Mupad [F(-1)]

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

[In]

int(sin(a + b*x)^3*(d*tan(a + b*x))^(1/2),x)

[Out]

int(sin(a + b*x)^3*(d*tan(a + b*x))^(1/2), x)