3.2.81 \(\int \frac {\sec ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [181]

Optimal. Leaf size=256 \[ -\frac {3003 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a+a \sin (c+d x)}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}} \]

[Out]

-3003/8192*cos(d*x+c)/d/(a+a*sin(d*x+c))^(3/2)-1001/5120*sec(d*x+c)/d/(a+a*sin(d*x+c))^(3/2)-143/960*sec(d*x+c
)^3/d/(a+a*sin(d*x+c))^(3/2)-1/8*sec(d*x+c)^5/d/(a+a*sin(d*x+c))^(3/2)-3003/16384*arctanh(1/2*cos(d*x+c)*a^(1/
2)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))*2^(1/2)/a^(3/2)/d+1001/2048*sec(d*x+c)/a/d/(a+a*sin(d*x+c))^(1/2)+143/640*s
ec(d*x+c)^3/a/d/(a+a*sin(d*x+c))^(1/2)+13/80*sec(d*x+c)^5/a/d/(a+a*sin(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2760, 2766, 2729, 2728, 212} \begin {gather*} -\frac {3003 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a \sin (c+d x)+a}}-\frac {143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a \sin (c+d x)+a}}-\frac {1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-3003*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8192*Sqrt[2]*a^(3/2)*d) - (3003*Co
s[c + d*x])/(8192*d*(a + a*Sin[c + d*x])^(3/2)) - (1001*Sec[c + d*x])/(5120*d*(a + a*Sin[c + d*x])^(3/2)) - (1
43*Sec[c + d*x]^3)/(960*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^5/(8*d*(a + a*Sin[c + d*x])^(3/2)) + (100
1*Sec[c + d*x])/(2048*a*d*Sqrt[a + a*Sin[c + d*x]]) + (143*Sec[c + d*x]^3)/(640*a*d*Sqrt[a + a*Sin[c + d*x]])
+ (13*Sec[c + d*x]^5)/(80*a*d*Sqrt[a + a*Sin[c + d*x]])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

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] + Dist[(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]

Rule 2760

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1))), x] + Dist[(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]

Rule 2766

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)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[a*((2*p + 1)/(2*g^2*(p + 1))), In
t[(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]

Rubi steps

\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{16 a}\\ &=-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {143}{160} \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {429 \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{640 a}\\ &=-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {1001 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{1280}\\ &=-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {1001 \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{2048 a}\\ &=-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a+a \sin (c+d x)}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {3003 \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx}{4096}\\ &=-\frac {3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a+a \sin (c+d x)}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {3003 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{16384 a}\\ &=-\frac {3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a+a \sin (c+d x)}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}-\frac {3003 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8192 a d}\\ &=-\frac {3003 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac {1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac {143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {1001 \sec (c+d x)}{2048 a d \sqrt {a+a \sin (c+d x)}}+\frac {143 \sec ^3(c+d x)}{640 a d \sqrt {a+a \sin (c+d x)}}+\frac {13 \sec ^5(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.91, size = 444, normalized size = 1.73 \begin {gather*} \frac {-8860+\frac {3840 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {1920}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {9920 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {4960}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {17720 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+32490 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-16245 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2+(45045+45045 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+\frac {1536 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {6400 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {28800 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}}{122880 d (a (1+\sin (c+d x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-8860 + (3840*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - 1920/(Cos[(c + d*x)/2] + Sin[(c + d
*x)/2])^4 + (9920*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 4960/(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^2 + (17720*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 32490*Sin[(c + d*x)/2]*(Cos[(c
+ d*x)/2] + Sin[(c + d*x)/2]) - 16245*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (45045 + 45045*I)*(-1)^(3/4)*
ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (1536*(Cos[(
c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (6400*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (28800*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(
c + d*x)/2] - Sin[(c + d*x)/2]))/(122880*d*(a*(1 + Sin[c + d*x]))^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.69, size = 367, normalized size = 1.43

method	result	size
default	\(-\frac {-120120 a^{\frac {13}{2}} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (-54912 a^{\frac {13}{2}}-180180 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (-39936 a^{\frac {13}{2}}+360360 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \sin \left (d x +c \right )+90090 a^{\frac {13}{2}} \left (\cos ^{6}\left (d x +c \right )\right )+9009 \left (-8 a^{\frac {13}{2}}+5 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (-18304 a^{\frac {13}{2}}-360360 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-9216 a^{\frac {13}{2}}+360360 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}}{245760 a^{\frac {15}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \left (1+\sin \left (d x +c \right )\right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(367\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/245760/a^(15/2)*(-120120*a^(13/2)*sin(d*x+c)*cos(d*x+c)^4+(-54912*a^(13/2)-180180*(a-a*sin(d*x+c))^(5/2)*2^
(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^4)*cos(d*x+c)^2*sin(d*x+c)+(-39936*a^(13/2)+360360
*(a-a*sin(d*x+c))^(5/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^4)*sin(d*x+c)+90090*a^(1
3/2)*cos(d*x+c)^6+9009*(-8*a^(13/2)+5*(a-a*sin(d*x+c))^(5/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2
)/a^(1/2))*a^4)*cos(d*x+c)^4+(-18304*a^(13/2)-360360*(a-a*sin(d*x+c))^(5/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c
))^(1/2)*2^(1/2)/a^(1/2))*a^4)*cos(d*x+c)^2-9216*a^(13/2)+360360*(a-a*sin(d*x+c))^(5/2)*2^(1/2)*arctanh(1/2*(a
-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^4)/(sin(d*x+c)-1)^2/(1+sin(d*x+c))^3/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)
/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 290, normalized size = 1.13 \begin {gather*} \frac {45045 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{7} - 2 \, \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{5}\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 (45045 \, \cos \left (d x + c\right )^{6} - 36036 \, \cos \left (d x + c\right )^{4} - 9152 \, \cos \left (d x + c\right )^{2} - 156 \, {\left (385 \, \cos \left (d x + c\right )^{4} + 176 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) - 4608\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{491520 \, {\left (a^{2} d \cos \left (d x + c\right )^{7} - 2 \, a^{2} d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/491520*(45045*sqrt(2)*(cos(d*x + c)^7 - 2*cos(d*x + c)^5*sin(d*x + c) - 2*cos(d*x + c)^5)*sqrt(a)*log(-(a*co
s(d*x + c)^2 - 2*sqrt(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*(45045*cos(d*x + c)^6 - 36036*cos(d*x + c)^4 - 9152*cos(d*x + c)^2 - 156*(385*cos(d*x + c)^4 + 176*
cos(d*x + c)^2 + 128)*sin(d*x + c) - 4608)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^7 - 2*a^2*d*cos(d*x +
c)^5*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6/(a+a*sin(d*x+c))**(3/2),x)

[Out]

Integral(sec(c + d*x)**6/(a*(sin(c + d*x) + 1))**(3/2), x)

________________________________________________________________________________________

Giac [A]
time = 4.03, size = 271, normalized size = 1.06 \begin {gather*} \frac {\sqrt {a} {\left (\frac {45045 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {45045 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {10 \, {\left (3249 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 10633 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 11767 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4431 \, \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^{2} \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 (225 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{2} \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 )^{5}}\right )}}{491520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/491520*sqrt(a)*(45045*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c
))) - 45045*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 10*(3
249*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 10633*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 11767*sqrt(2)*
sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 4431*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^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 256*sqrt(2)*(225*sin(-1/4*pi + 1/2*d*x + 1/2*c)^4 + 25*
sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 3)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^
5))/d

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^6*(a + a*sin(c + d*x))^(3/2)),x)

[Out]

int(1/(cos(c + d*x)^6*(a + a*sin(c + d*x))^(3/2)), x)

________________________________________________________________________________________