Integrand size = 23, antiderivative size = 284 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {3 \left (4 a^2-14 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 (a-b)^{7/2} d}+\frac {3 \left (4 a^2+14 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 (a+b)^{7/2} d}-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}} \]
-3/32*(4*a^2-14*a*b+15*b^2)*arctanh((a+b*sin(d*x+c))^(1/2)/(a-b)^(1/2))/(a -b)^(7/2)/d+3/32*(4*a^2+14*a*b+15*b^2)*arctanh((a+b*sin(d*x+c))^(1/2)/(a+b )^(1/2))/(a+b)^(7/2)/d-3/16*b*(2*a^4-7*a^2*b^2-15*b^4)/(a^2-b^2)^3/d/(a+b* sin(d*x+c))^(1/2)-1/4*sec(d*x+c)^4*(b-a*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d *x+c))^(1/2)+1/16*sec(d*x+c)^2*(b*(a^2+9*b^2)+2*a*(3*a^2-8*b^2)*sin(d*x+c) )/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.38 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.14 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\frac {3}{2} \left (2 a^4-7 a^2 b^2-15 b^4\right ) \left ((a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \sin (c+d x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \sin (c+d x)}{a+b}\right )\right )-4 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (-b+a \sin (c+d x))+3 a \sqrt {a-b} \sqrt {a+b} \left (3 a^2-8 b^2\right ) \left (\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )\right ) \sqrt {a+b \sin (c+d x)}-(a-b) (a+b) \sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 \left (-a^2+b^2\right ) d \sqrt {a+b \sin (c+d x)}} \]
((3*(2*a^4 - 7*a^2*b^2 - 15*b^4)*((a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, ( a + b*Sin[c + d*x])/(a + b)]))/2 - 4*(a - b)^2*(a + b)^2*Sec[c + d*x]^4*(- b + a*Sin[c + d*x]) + 3*a*Sqrt[a - b]*Sqrt[a + b]*(3*a^2 - 8*b^2)*(Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] - Sqrt[a - b]*ArcTanh[S qrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])*Sqrt[a + b*Sin[c + d*x]] - (a - b)*( a + b)*Sec[c + d*x]^2*(b*(a^2 + 9*b^2) + 2*a*(3*a^2 - 8*b^2)*Sin[c + d*x]) )/(16*(a^2 - b^2)^2*(-a^2 + b^2)*d*Sqrt[a + b*Sin[c + d*x]])
Time = 0.64 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.33, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3147, 496, 27, 686, 27, 655, 25, 654, 25, 1480, 220}
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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^5 (a+b \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^5 \int \frac {1}{(a+b \sin (c+d x))^{3/2} \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {b^5 \left (\frac {\int \frac {3 \left (2 a^2-3 b^2\right )+7 a b \sin (c+d x)}{2 (a+b \sin (c+d x))^{3/2} \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{4 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \left (\frac {\int \frac {3 \left (2 a^2-3 b^2\right )+7 a b \sin (c+d x)}{(a+b \sin (c+d x))^{3/2} \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}-\frac {\int -\frac {3 \left (4 a^4-9 b^2 a^2+2 b \left (3 a^2-8 b^2\right ) \sin (c+d x) a+15 b^4\right )}{2 (a+b \sin (c+d x))^{3/2} \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{2 b^2 \left (a^2-b^2\right )}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \int \frac {4 a^4-9 b^2 a^2+2 b \left (3 a^2-8 b^2\right ) \sin (c+d x) a+15 b^4}{(a+b \sin (c+d x))^{3/2} \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \left (-\frac {\int -\frac {a \left (4 a^4-15 b^2 a^2+31 b^4\right )+b \left (2 a^4-7 b^2 a^2-15 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)} \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \left (\frac {\int \frac {a \left (4 a^4-15 b^2 a^2+31 b^4\right )+b \left (2 a^4-7 b^2 a^2-15 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)} \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \left (\frac {2 \int -\frac {b^2 \left (2 a^4-7 b^2 a^2-15 b^4\right ) \sin ^2(c+d x)+2 a \left (a^4-4 b^2 a^2+23 b^4\right )}{b^4 \sin ^4(c+d x)-2 a b^2 \sin ^2(c+d x)+a^2-b^2}d\sqrt {a+b \sin (c+d x)}}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \left (-\frac {2 \int \frac {b^2 \left (2 a^4-7 b^2 a^2-15 b^4\right ) \sin ^2(c+d x)+2 a \left (a^4-4 b^2 a^2+23 b^4\right )}{b^4 \sin ^4(c+d x)-2 a b^2 \sin ^2(c+d x)+a^2-b^2}d\sqrt {a+b \sin (c+d x)}}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \left (\frac {2 \left (\frac {(a+b)^3 \left (4 a^2-14 a b+15 b^2\right ) \int \frac {1}{b^2 \sin ^2(c+d x)-a+b}d\sqrt {a+b \sin (c+d x)}}{2 b}-\frac {(a-b)^3 \left (4 a^2+14 a b+15 b^2\right ) \int \frac {1}{b^2 \sin ^2(c+d x)-a-b}d\sqrt {a+b \sin (c+d x)}}{2 b}\right )}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}+\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {2 a b \left (3 a^2-8 b^2\right ) \sin (c+d x)+b^2 \left (a^2+9 b^2\right )}{2 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}+\frac {3 \left (\frac {2 \left (\frac {(a-b)^3 \left (4 a^2+14 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{2 b \sqrt {a+b}}-\frac {(a+b)^3 \left (4 a^2-14 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{2 b \sqrt {a-b}}\right )}{a^2-b^2}-\frac {2 \left (2 a^4-7 a^2 b^2-15 b^4\right )}{\left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\right )}{4 b^2 \left (a^2-b^2\right )}}{8 b^2 \left (a^2-b^2\right )}-\frac {b^2-a b \sin (c+d x)}{4 b^2 \left (a^2-b^2\right ) \left (b^2-b^2 \sin ^2(c+d x)\right )^2 \sqrt {a+b \sin (c+d x)}}\right )}{d}\) |
(b^5*(-1/4*(b^2 - a*b*Sin[c + d*x])/(b^2*(a^2 - b^2)*Sqrt[a + b*Sin[c + d* x]]*(b^2 - b^2*Sin[c + d*x]^2)^2) + ((b^2*(a^2 + 9*b^2) + 2*a*b*(3*a^2 - 8 *b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]]*(b^2 - b^2 *Sin[c + d*x]^2)) + (3*((2*(-1/2*((a + b)^3*(4*a^2 - 14*a*b + 15*b^2)*ArcT anh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(Sqrt[a - b]*b) + ((a - b)^3*(4 *a^2 + 14*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(2* b*Sqrt[a + b])))/(a^2 - b^2) - (2*(2*a^4 - 7*a^2*b^2 - 15*b^4))/((a^2 - b^ 2)*Sqrt[a + b*Sin[c + d*x]])))/(4*b^2*(a^2 - b^2)))/(8*b^2*(a^2 - b^2))))/ d
3.6.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) ), x] + Simp[1/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(256)=512\).
Time = 0.93 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {\frac {2 b^{5}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \sqrt {a +b \sin \left (d x +c \right )}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {13 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}-\frac {21 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {15 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{8 \left (a -b \right )^{3} \sqrt {-a +b}}-\frac {21 b \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{16 \left (a -b \right )^{3} \sqrt {-a +b}}+\frac {45 b^{2} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{32 \left (a -b \right )^{3} \sqrt {-a +b}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}-\frac {13 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {21 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {15 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{8 \left (a +b \right )^{\frac {7}{2}}}+\frac {21 b \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{16 \left (a +b \right )^{\frac {7}{2}}}+\frac {45 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{32 \left (a +b \right )^{\frac {7}{2}}}}{d}\) | \(602\) |
(2*b^5/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))^(1/2)-3/16*b/(a-b)^3/(b*sin(d*x+c) +b)^2*(a+b*sin(d*x+c))^(3/2)*a+13/32*b^2/(a-b)^3/(b*sin(d*x+c)+b)^2*(a+b*s in(d*x+c))^(3/2)+3/16*b/(a-b)^3/(b*sin(d*x+c)+b)^2*(a+b*sin(d*x+c))^(1/2)* a^2-21/32*b^2/(a-b)^3/(b*sin(d*x+c)+b)^2*(a+b*sin(d*x+c))^(1/2)*a+15/32*b^ 3/(a-b)^3/(b*sin(d*x+c)+b)^2*(a+b*sin(d*x+c))^(1/2)+3/8/(a-b)^3/(-a+b)^(1/ 2)*arctan((a+b*sin(d*x+c))^(1/2)/(-a+b)^(1/2))*a^2-21/16*b/(a-b)^3/(-a+b)^ (1/2)*arctan((a+b*sin(d*x+c))^(1/2)/(-a+b)^(1/2))*a+45/32*b^2/(a-b)^3/(-a+ b)^(1/2)*arctan((a+b*sin(d*x+c))^(1/2)/(-a+b)^(1/2))-3/16*b/(a+b)^3/(b*sin (d*x+c)-b)^2*(a+b*sin(d*x+c))^(3/2)*a-13/32*b^2/(a+b)^3/(b*sin(d*x+c)-b)^2 *(a+b*sin(d*x+c))^(3/2)+3/16*b/(a+b)^3/(b*sin(d*x+c)-b)^2*(a+b*sin(d*x+c)) ^(1/2)*a^2+21/32*b^2/(a+b)^3/(b*sin(d*x+c)-b)^2*(a+b*sin(d*x+c))^(1/2)*a+1 5/32*b^3/(a+b)^3/(b*sin(d*x+c)-b)^2*(a+b*sin(d*x+c))^(1/2)+3/8/(a+b)^(7/2) *arctanh((a+b*sin(d*x+c))^(1/2)/(a+b)^(1/2))*a^2+21/16*b/(a+b)^(7/2)*arcta nh((a+b*sin(d*x+c))^(1/2)/(a+b)^(1/2))*a+45/32*b^2/(a+b)^(7/2)*arctanh((a+ b*sin(d*x+c))^(1/2)/(a+b)^(1/2)))/d
Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (257) = 514\).
Time = 1.05 (sec) , antiderivative size = 3861, normalized size of antiderivative = 13.60 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
[1/256*(3*((4*a^6*b - 2*a^5*b^2 - 17*a^4*b^3 + 8*a^3*b^4 + 38*a^2*b^5 - 46 *a*b^6 + 15*b^7)*cos(d*x + c)^4*sin(d*x + c) + (4*a^7 - 2*a^6*b - 17*a^5*b ^2 + 8*a^4*b^3 + 38*a^3*b^4 - 46*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^4)*sqrt( a + b)*log((b^4*cos(d*x + c)^4 + 128*a^4 + 256*a^3*b + 320*a^2*b^2 + 256*a *b^3 + 72*b^4 - 8*(20*a^2*b^2 + 28*a*b^3 + 9*b^4)*cos(d*x + c)^2 + 8*(16*a ^3 + 24*a^2*b + 20*a*b^2 + 8*b^3 - (10*a*b^2 + 7*b^3)*cos(d*x + c)^2 - (b^ 3*cos(d*x + c)^2 - 24*a^2*b - 28*a*b^2 - 8*b^3)*sin(d*x + c))*sqrt(b*sin(d *x + c) + a)*sqrt(a + b) + 4*(64*a^3*b + 112*a^2*b^2 + 64*a*b^3 + 14*b^4 - (8*a*b^3 + 7*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(cos(d*x + c)^4 - 8*cos(d *x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) - 3*((4*a^6*b + 2*a^ 5*b^2 - 17*a^4*b^3 - 8*a^3*b^4 + 38*a^2*b^5 + 46*a*b^6 + 15*b^7)*cos(d*x + c)^4*sin(d*x + c) + (4*a^7 + 2*a^6*b - 17*a^5*b^2 - 8*a^4*b^3 + 38*a^3*b^ 4 + 46*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^4)*sqrt(a - b)*log((b^4*cos(d*x + c)^4 + 128*a^4 - 256*a^3*b + 320*a^2*b^2 - 256*a*b^3 + 72*b^4 - 8*(20*a^2* b^2 - 28*a*b^3 + 9*b^4)*cos(d*x + c)^2 + 8*(16*a^3 - 24*a^2*b + 20*a*b^2 - 8*b^3 - (10*a*b^2 - 7*b^3)*cos(d*x + c)^2 - (b^3*cos(d*x + c)^2 - 24*a^2* b + 28*a*b^2 - 8*b^3)*sin(d*x + c))*sqrt(b*sin(d*x + c) + a)*sqrt(a - b) + 4*(64*a^3*b - 112*a^2*b^2 + 64*a*b^3 - 14*b^4 - (8*a*b^3 - 7*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c )^2 - 2)*sin(d*x + c) + 8)) - 16*(4*a^6*b - 12*a^4*b^3 + 12*a^2*b^5 - 4...
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for m ore detail
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{5}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]