Integrand size = 23, antiderivative size = 233 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {1155 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}} \] Output:
-1155/8192*arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))* 2^(1/2)/a^(5/2)/d-1/8*sec(d*x+c)^3/d/(a+a*sin(d*x+c))^(5/2)-1155/4096*cos( d*x+c)/a/d/(a+a*sin(d*x+c))^(3/2)-77/512*sec(d*x+c)/a/d/(a+a*sin(d*x+c))^( 3/2)-11/96*sec(d*x+c)^3/a/d/(a+a*sin(d*x+c))^(3/2)+385/1024*sec(d*x+c)/a^2 /d/(a+a*sin(d*x+c))^(1/2)+11/64*sec(d*x+c)^3/a^2/d/(a+a*sin(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.23 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},5,-\frac {1}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^3(c+d x) (a (1+\sin (c+d x)))^{3/2}}{48 a^4 d} \] Input:
Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]
Output:
(Hypergeometric2F1[-3/2, 5, -1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^3*(a* (1 + Sin[c + d*x]))^(3/2))/(48*a^4*d)
Time = 1.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3160, 3042, 3160, 3042, 3166, 3042, 3160, 3042, 3166, 3042, 3129, 3042, 3128, 219}
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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \sin (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a \sin (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {11 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^{3/2}}dx}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {11 \left (\frac {3 \int \frac {\sec ^4(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {3 \int \frac {1}{\cos (c+d x)^4 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \int \frac {\sec ^2(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \int \frac {1}{\cos (c+d x)^2 (\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \int \frac {\sec ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \int \frac {1}{\cos (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \int \frac {1}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \int \frac {1}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \left (-\frac {\int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{2 a d}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {11 \left (\frac {3 \left (\frac {7}{6} a \left (\frac {5 \left (\frac {3}{2} a \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{8 a}-\frac {\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^3(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}\right )}{16 a}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}\) |
Input:
Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]
Output:
-1/8*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^(5/2)) + (11*(-1/6*Sec[c + d*x ]^3/(d*(a + a*Sin[c + d*x])^(3/2)) + (3*(Sec[c + d*x]^3/(3*d*Sqrt[a + a*Si n[c + d*x]]) + (7*a*(-1/4*Sec[c + d*x]/(d*(a + a*Sin[c + d*x])^(3/2)) + (5 *(Sec[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]) + (3*a*(-1/2*ArcTanh[(Sqrt[a]* Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(Sqrt[2]*a^(3/2)*d) - Co s[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))))/2))/(8*a)))/6))/(4*a)))/(16* a)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*(2*m + p + 1))), x] + Simp[(m + p + 1)/(a*(2*m + p + 1)) Int[ (g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] & & IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]], x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)*S qrt[a + b*Sin[e + f*x]])), x] + Simp[a*((2*p + 1)/(2*g^2*(p + 1))) Int[(g *Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e , f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Time = 0.20 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {6930 a^{\frac {11}{2}} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+16170 a^{\frac {11}{2}} \cos \left (d x +c \right )^{4}-14784 a^{\frac {11}{2}} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+3465 \cos \left (d x +c \right )^{4} \sqrt {2}\, a^{4} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-10560 \cos \left (d x +c \right )^{2} a^{\frac {11}{2}}-13860 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \sqrt {2}\, a^{4} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-5632 \sin \left (d x +c \right ) a^{\frac {11}{2}}-27720 \cos \left (d x +c \right )^{2} \sqrt {2}\, a^{4} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-2560 a^{\frac {11}{2}}+27720 \sin \left (d x +c \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{4}+27720 \sqrt {2}\, \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}}{24576 a^{\frac {15}{2}} \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(381\) |
Input:
int(sec(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24576/a^(15/2)*(6930*a^(11/2)*cos(d*x+c)^4*sin(d*x+c)+16170*a^(11/2)*cos (d*x+c)^4-14784*a^(11/2)*cos(d*x+c)^2*sin(d*x+c)+3465*cos(d*x+c)^4*2^(1/2) *a^4*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*(a-a*sin(d*x+c))^ (3/2)-10560*cos(d*x+c)^2*a^(11/2)-13860*sin(d*x+c)*cos(d*x+c)^2*2^(1/2)*a^ 4*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*(a-a*sin(d*x+c))^(3/ 2)-5632*sin(d*x+c)*a^(11/2)-27720*cos(d*x+c)^2*2^(1/2)*a^4*arctanh(1/2*(a- a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*(a-a*sin(d*x+c))^(3/2)-2560*a^(11/2)+ 27720*sin(d*x+c)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2 ))*(a-a*sin(d*x+c))^(3/2)*a^4+27720*2^(1/2)*(a-a*sin(d*x+c))^(3/2)*arctanh (1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^4)/(sin(d*x+c)-1)/(1+sin(d* x+c))^3/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Time = 0.14 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.32 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {3465 \, \sqrt {2} {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (8085 \, \cos \left (d x + c\right )^{4} - 5280 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 672 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) - 1280\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{49152 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")
Output:
1/49152*(3465*sqrt(2)*(3*cos(d*x + c)^5 - 4*cos(d*x + c)^3 + (cos(d*x + c) ^5 - 4*cos(d*x + c)^3)*sin(d*x + c))*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sq rt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(8085*cos( d*x + c)^4 - 5280*cos(d*x + c)^2 + 11*(315*cos(d*x + c)^4 - 672*cos(d*x + c)^2 - 256)*sin(d*x + c) - 1280)*sqrt(a*sin(d*x + c) + a))/(3*a^3*d*cos(d* x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d* x + c)^3)*sin(d*x + c))
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(sec(d*x+c)**4/(a+a*sin(d*x+c))**(5/2),x)
Output:
Integral(sec(c + d*x)**4/(a*(sin(c + d*x) + 1))**(5/2), x)
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a} {\left (\frac {3465 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {256 \, \sqrt {2} {\left (15 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {2 \, {\left (1545 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5153 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5855 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2295 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{49152 \, d} \] Input:
integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
1/49152*sqrt(a)*(3465*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^3 *sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 3465*sqrt(2)*log(-sin(-1/4*pi + 1/ 2*d*x + 1/2*c) + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 256*sqrt(2 )*(15*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3) - 2*(1545*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 5153*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 58 55*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 2295*sqrt(2)*sin(-1/4*pi + 1 /2*d*x + 1/2*c))/((sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^4*a^3*sgn(cos(-1/ 4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int(1/(cos(c + d*x)^4*(a + a*sin(c + d*x))^(5/2)),x)
Output:
int(1/(cos(c + d*x)^4*(a + a*sin(c + d*x))^(5/2)), x)
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{4}}{\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1}d x \right )}{a^{3}} \] Input:
int(sec(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x)
Output:
(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*sec(c + d*x)**4)/(sin(c + d*x)**3 + 3 *sin(c + d*x)**2 + 3*sin(c + d*x) + 1),x))/a**3