Integrand size = 21, antiderivative size = 144 \[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=-\frac {\sin ^3(a+b x)}{20 b d (d \tan (a+b x))^{3/2}}-\frac {3 \sin ^5(a+b x)}{70 b d (d \tan (a+b x))^{3/2}}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}+\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{40 b d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
-3/40*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi +b*x),2^(1/2))*sin(b*x+a)/b/d^2/sin(2*b*x+2*a)^(1/2)/(d*tan(b*x+a))^(1/2)- 1/20*sin(b*x+a)^3/b/d/(d*tan(b*x+a))^(3/2)-3/70*sin(b*x+a)^5/b/d/(d*tan(b* x+a))^(3/2)+1/7*sin(b*x+a)^7/b/d/(d*tan(b*x+a))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.86 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\frac {\sqrt {d \tan (a+b x)} \left (-\sqrt {\sec ^2(a+b x)} (15 \sin (a+b x)+29 \sin (3 (a+b x))+9 \sin (5 (a+b x))-5 \sin (7 (a+b x)))+112 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sec (a+b x) \tan (a+b x)\right )}{2240 b d^3 \sqrt {\sec ^2(a+b x)}} \]
(Sqrt[d*Tan[a + b*x]]*(-(Sqrt[Sec[a + b*x]^2]*(15*Sin[a + b*x] + 29*Sin[3* (a + b*x)] + 9*Sin[5*(a + b*x)] - 5*Sin[7*(a + b*x)])) + 112*Hypergeometri c2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]))/(2240*b*d ^3*Sqrt[Sec[a + b*x]^2])
Time = 0.75 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3076, 3042, 3078, 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.
| \(\displaystyle \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^7}{(d \tan (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3076 |
| \(\displaystyle \frac {3 \int \frac {\sin ^5(a+b x)}{\sqrt {d \tan (a+b x)}}dx}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\sin (a+b x)^5}{\sqrt {d \tan (a+b x)}}dx}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3078 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \int \frac {\sin ^3(a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \int \frac {\sin (a+b x)^3}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3078 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {1}{2} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {3 \left (\frac {7}{10} \left (\frac {\sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}\right )-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}\right )}{14 d^2}+\frac {\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}\) |
Sin[a + b*x]^7/(7*b*d*(d*Tan[a + b*x])^(3/2)) + (3*(-1/5*(d*Sin[a + b*x]^5 )/(b*(d*Tan[a + b*x])^(3/2)) + (7*(-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]])))/10))/(14*d^2)
3.2.8.3.1 Defintions of rubi rules used
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[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*m)) , x] - Simp[a^2*((n + 1)/(b^2*m)) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
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. \(443\) vs. \(2(151)=302\).
Time = 1.46 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.08
| method | result | size |
| default | \(\frac {\sec \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right ) \left (-1+\cos \left (b x +a \right )\right ) \left (\cos \left (b x +a \right )+1\right ) \left (40 \sqrt {2}\, \left (\cos ^{8}\left (b x +a \right )\right )-108 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )+82 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-21 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+42 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-21 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+7 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-21 \sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {2}}{560 b \sqrt {d \tan \left (b x +a \right )}\, d^{2}}\) | \(444\) |
1/560/b*sec(b*x+a)*csc(b*x+a)^2*(-1+cos(b*x+a))*(cos(b*x+a)+1)*(40*2^(1/2) *cos(b*x+a)^8-108*2^(1/2)*cos(b*x+a)^6+82*2^(1/2)*cos(b*x+a)^4-21*(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)+42*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*( cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2 *2^(1/2))*cos(b*x+a)-21*(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))+42*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+ 1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc(b*x+a)- cot(b*x+a))^(1/2),1/2*2^(1/2))+7*cos(b*x+a)^2*2^(1/2)-21*2^(1/2)*cos(b*x+a ))/(d*tan(b*x+a))^(1/2)/d^2*2^(1/2)
\[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{7}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
integral(-(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2 - 1)*sqrt( d*tan(b*x + a))*sin(b*x + a)/(d^3*tan(b*x + a)^3), x)
Timed out. \[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{7}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{7}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^7}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}} \,d x \]