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

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

[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]])

________________________________________________________________________________________

Rubi [A]  time = 0.429009, antiderivative size = 256, normalized size of antiderivative = 1., 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 = {2681, 2687, 2650, 2649, 206} \[ -\frac{3003 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}} \]

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 2681

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] + 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 2687

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)), 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]

Rule 2650

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 2649

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 \operatorname{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]  time = 1.40622, size = 444, normalized size = 1.73 \[ \frac{\frac{28800 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{6400 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{1536 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^5}-16245 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2+32490 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{17720 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-\frac{4960}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{9920 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}-\frac{1920}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}+\frac{3840 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}+(45045+45045 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )-8860}{122880 d (a (\sin (c+d x)+1))^{3/2}} \]

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))

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Maple [A]  time = 0.178, size = 367, normalized size = 1.4 \begin{align*} -{\frac{1}{245760\, \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d} \left ( -120120\,{a}^{13/2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -54912\,{a}^{13/2}-180180\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) + \left ( -39936\,{a}^{13/2}+360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \sin \left ( dx+c \right ) +90090\,{a}^{13/2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}+9009\, \left ( -8\,{a}^{13/2}+5\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -18304\,{a}^{13/2}-360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-9216\,{a}^{13/2}+360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ){a}^{-{\frac{15}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

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)

[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

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

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

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Fricas [A]  time = 2.89251, size = 805, normalized size = 3.14 \begin{align*} \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{align*}

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)

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

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

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]

sage2