Integrand size = 21, antiderivative size = 110 \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac {7 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b} \] Output:
7/3*d^3*sin(b*x+a)^3/b/(d*tan(b*x+a))^(3/2)+7/2*d^2*EllipticE(cos(a+1/4*Pi +b*x),2^(1/2))*sin(b*x+a)/b/sin(2*b*x+2*a)^(1/2)/(d*tan(b*x+a))^(1/2)+2*d* sin(b*x+a)^3*(d*tan(b*x+a))^(1/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.50 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {\left (-28 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sec (a+b x)+2 \cos (a+b x) (13+\cos (2 (a+b x))) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{12 b \sqrt {\sec ^2(a+b x)}} \] Input:
Integrate[Sin[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]
Output:
((-28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x] + 2*C os[a + b*x]*(13 + Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x]) ^(3/2))/(12*b*Sqrt[Sec[a + b*x]^2])
Time = 0.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3074, 3042, 3078, 3042, 3081, 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.
\(\displaystyle \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^3 (d \tan (a+b x))^{3/2}dx\) |
\(\Big \downarrow \) 3074 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \int \frac {\sin ^3(a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \int \frac {\sin (a+b x)^3}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-7 d^2 \left (\frac {\sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )\) |
Input:
Int[Sin[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]
Output:
(2*d*Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/b - 7*d^2*(-1/3*(d*Sin[a + b*x]^ 3)/(b*(d*Tan[a + b*x])^(3/2)) + (EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x] )/(2*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]]))
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_.)*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*(n - 1))), x] - Simp[b^2*((m + n - 1)/(n - 1)) Int[(a*Sin[e + f*x])^m*(b*Ta n[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && In tegersQ[2*m, 2*n] && !(GtQ[m, 1] && !IntegerQ[(m - 1)/2])
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] + Simp[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]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[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])
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. \(287\) vs. \(2(99)=198\).
Time = 1.40 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.62
method | result | size |
default | \(-\frac {\csc \left (b x +a \right ) \left (\left (21 \cos \left (b x +a \right )+21\right ) \sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (-42 \cos \left (b x +a \right )-42\right ) \sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+4 \cos \left (b x +a \right )^{4}-22 \cos \left (b x +a \right )^{2}+42 \cos \left (b x +a \right )-24\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {2}}{24 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\) | \(288\) |
Input:
int(sin(b*x+a)^3*(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/24/b*csc(b*x+a)*((21*cos(b*x+a)+21)*(csc(b*x+a)-cot(b*x+a)+1)^(1/2)*(-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))+(-42*cos(b*x+a)-42)*(csc(b*x +a)-cot(b*x+a)+1)^(1/2)*(-2*csc(b*x+a)+2*cot(b*x+a)+2)^(1/2)*(-csc(b*x+a)+ cot(b*x+a))^(1/2)*EllipticE((csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/2*2^(1/2))+4 *cos(b*x+a)^4-22*cos(b*x+a)^2+42*cos(b*x+a)-24)*(-2*sin(b*x+a)*cos(b*x+a)/ (cos(b*x+a)+1)^2)^(1/2)*(d*tan(b*x+a))^(1/2)*d/(-sin(b*x+a)*cos(b*x+a)/(co s(b*x+a)+1)^2)^(1/2)*2^(1/2)
\[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sin \left (b x + a\right )^{3} \,d x } \] Input:
integrate(sin(b*x+a)^3*(d*tan(b*x+a))^(3/2),x, algorithm="fricas")
Output:
integral(-(d*cos(b*x + a)^2 - d)*sqrt(d*tan(b*x + a))*sin(b*x + a)*tan(b*x + a), x)
Timed out. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(sin(b*x+a)**3*(d*tan(b*x+a))**(3/2),x)
Output:
Timed out
\[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sin \left (b x + a\right )^{3} \,d x } \] Input:
integrate(sin(b*x+a)^3*(d*tan(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate((d*tan(b*x + a))^(3/2)*sin(b*x + a)^3, x)
Exception generated. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sin(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. \[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \] Input:
int(sin(a + b*x)^3*(d*tan(a + b*x))^(3/2),x)
Output:
int(sin(a + b*x)^3*(d*tan(a + b*x))^(3/2), x)
\[ \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \sin \left (b x +a \right )^{3} \tan \left (b x +a \right )d x \right ) d \] Input:
int(sin(b*x+a)^3*(d*tan(b*x+a))^(3/2),x)
Output:
sqrt(d)*int(sqrt(tan(a + b*x))*sin(a + b*x)**3*tan(a + b*x),x)*d