∫ sec7(c + dx)(a + a sin(c + dx))5∕2 dx [138]

Optimal. Leaf size=159 \[ \frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d} \]

7/48*a*sec(d*x+c)^4*(a+a*sin(d*x+c))^(3/2)/d+1/6*sec(d*x+c)^6*(a+a*sin(d*x+c))^(5/2)/d+35/256*a^(5/2)*arctanh(

1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d-35/128*a^3/d/(a+a*sin(d*x+c))^(1/2)+35/192*a^2*sec(d*x+c

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 2746, 53, 65, 212} \begin {gather*} \frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {35 a^3}{128 d \sqrt {a \sin (c+d x)+a}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{192 d}+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac {7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d} \end {gather*}

(35*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*d) - (35*a^3)/(128*d*Sqrt[a + a*

Sin[c + d*x]]) + (35*a^2*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(192*d) + (7*a*Sec[c + d*x]^4*(a + a*Sin[c +

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1

)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[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]

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S

ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]

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*(p + 1))), x] + Dist[a*((m + p + 1)/(g^2*(p + 1))), Int

[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2,

\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{12} (7 a) \int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{96} \left (35 a^2\right ) \int \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{128} \left (35 a^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{128 d}\\ &=\frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.07, size = 44, normalized size = 0.28 \begin {gather*} -\frac {a^3 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {1}{2} (1+\sin (c+d x))\right )}{8 d \sqrt {a+a \sin (c+d x)}} \end {gather*}

________________________________________________________________________________________


method	result	size

default	\(\frac {2 a^{7} \left (-\frac {-\frac {a^{2} \sqrt {a +a \sin \left (d x +c \right )}\, \left (57 \left (\cos ^{2}\left (d x +c \right )\right )+158 \sin \left (d x +c \right )-190\right )}{48 \left (a \sin \left (d x +c \right )-a \right )^{3}}-\frac {35 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{16 a^{4}}-\frac {1}{16 a^{4} \sqrt {a +a \sin \left (d x +c \right )}}\right )}{d}\)	\(113\)

2*a^7*(-1/16/a^4*(-1/48*a^2*(a+a*sin(d*x+c))^(1/2)*(57*cos(d*x+c)^2+158*sin(d*x+c)-190)/(a*sin(d*x+c)-a)^3-35/

32*2^(1/2)/a^(1/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2)))-1/16/a^4/(a+a*sin(d*x+c))^(1/2))/d

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 185, normalized size = 1.16 \begin {gather*} -\frac {105 \, \sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (105 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{4} - 560 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{5} + 924 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{6} - 384 \, a^{7}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 8 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}}}{1536 \, a d} \end {gather*}

-1/1536*(105*sqrt(2)*a^(7/2)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(a*sin(d

*x + c) + a))) + 4*(105*(a*sin(d*x + c) + a)^3*a^4 - 560*(a*sin(d*x + c) + a)^2*a^5 + 924*(a*sin(d*x + c) + a)

*a^6 - 384*a^7)/((a*sin(d*x + c) + a)^(7/2) - 6*(a*sin(d*x + c) + a)^(5/2)*a + 12*(a*sin(d*x + c) + a)^(3/2)*a

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 208, normalized size = 1.31 \begin {gather*} \frac {105 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{4} + 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (245 \, a^{2} \cos \left (d x + c\right )^{2} - 160 \, a^{2} - 7 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1536 \, {\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2}\right )}} \end {gather*}

1/1536*(105*(sqrt(2)*a^2*cos(d*x + c)^4 + 2*sqrt(2)*a^2*cos(d*x + c)^2*sin(d*x + c) - 2*sqrt(2)*a^2*cos(d*x +

c)^2)*sqrt(a)*log(-(a*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a) + 3*a)/(sin(d*x + c) - 1)) - 4

*(245*a^2*cos(d*x + c)^2 - 160*a^2 - 7*(15*a^2*cos(d*x + c)^2 - 32*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

________________________________________________________________________________________

Giac [A]
time = 5.73, size = 144, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {2} a^{\frac {5}{2}} {\left (\frac {96}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (57 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 136 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 87 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - 105 \, \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 105 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{1536 \, d} \end {gather*}

-1/1536*sqrt(2)*a^(5/2)*(96/cos(-1/4*pi + 1/2*d*x + 1/2*c) + 2*(57*cos(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 136*cos(

-1/4*pi + 1/2*d*x + 1/2*c)^3 + 87*cos(-1/4*pi + 1/2*d*x + 1/2*c))/(cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^3 - 1

05*log(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1) + 105*log(-cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1))*sgn(cos(-1/4*pi + 1

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^7} \,d x \end {gather*}

________________________________________________________________________________________

3.2.38 \(\int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\) [138]