3.410 \(\int \frac{\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=249 \[ \frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^3}-\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^3}+\frac{\left (3 a^2-6 a b+35 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d (a-b)^3}+\frac{3 a-11 b}{16 d (a-b)^2 (1-\sin (c+d x))}-\frac{3 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a-b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2} \]

[Out]

(b^(5/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^3*d) + ((3*a^2 - 6*a*b + 35*b^
2)*ArcTanh[Sin[c + d*x]])/(8*(a - b)^3*d) - (b^(5/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt
[a] - Sqrt[b])^3*d) + 1/(16*(a - b)*d*(1 - Sin[c + d*x])^2) + (3*a - 11*b)/(16*(a - b)^2*d*(1 - Sin[c + d*x]))
- 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (3*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))

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Rubi [A]  time = 0.296795, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3223, 1171, 207, 1167, 205, 208} \[ \frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^3}-\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^3}+\frac{\left (3 a^2-6 a b+35 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d (a-b)^3}+\frac{3 a-11 b}{16 d (a-b)^2 (1-\sin (c+d x))}-\frac{3 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a-b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

(b^(5/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^3*d) + ((3*a^2 - 6*a*b + 35*b^
2)*ArcTanh[Sin[c + d*x]])/(8*(a - b)^3*d) - (b^(5/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt
[a] - Sqrt[b])^3*d) + 1/(16*(a - b)*d*(1 - Sin[c + d*x])^2) + (3*a - 11*b)/(16*(a - b)^2*d*(1 - Sin[c + d*x]))
- 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (3*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 207

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

Rule 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3 \left (a-b x^4\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{8 (a-b) (-1+x)^3}+\frac{3 a-11 b}{16 (a-b)^2 (-1+x)^2}+\frac{1}{8 (a-b) (1+x)^3}+\frac{3 a-11 b}{16 (a-b)^2 (1+x)^2}+\frac{-3 a^2+6 a b-35 b^2}{8 (a-b)^3 \left (-1+x^2\right )}+\frac{b^2 \left (-3 a-b-(a+3 b) x^2\right )}{(a-b)^3 \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac{3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac{1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac{3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-3 a-b+(-a-3 b) x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{(a-b)^3 d}-\frac{\left (3 a^2-6 a b+35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sin (c+d x)\right )}{8 (a-b)^3 d}\\ &=\frac{\left (3 a^2-6 a b+35 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 (a-b)^3 d}+\frac{1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac{3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac{1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac{3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} \left (\sqrt{a}-\sqrt{b}\right )^3 d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^3 d}\\ &=\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt{a}+\sqrt{b}\right )^3 d}+\frac{\left (3 a^2-6 a b+35 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 (a-b)^3 d}-\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt{a}-\sqrt{b}\right )^3 d}+\frac{1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac{3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac{1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac{3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}\\ \end{align*}

Mathematica [C]  time = 5.65029, size = 317, normalized size = 1.27 \[ \frac{\frac{4 b^{5/4} \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt{a}-\sqrt{b}\right )^3}+\frac{4 i b^{5/4} \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt{a}+\sqrt{b}\right )^3}-\frac{4 i b^{5/4} \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt{a}+\sqrt{b}\right )^3}-\frac{4 b^{5/4} \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt{a}-\sqrt{b}\right )^3}+\frac{2 \left (3 a^2-6 a b+35 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{(a-b)^3}+\frac{11 b-3 a}{(a-b)^2 (\sin (c+d x)-1)}+\frac{11 b-3 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a-b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}}{16 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

((2*(3*a^2 - 6*a*b + 35*b^2)*ArcTanh[Sin[c + d*x]])/(a - b)^3 + (4*b^(5/4)*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]]
)/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + ((4*I)*b^(5/4)*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] +
Sqrt[b])^3) - ((4*I)*b^(5/4)*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^3) - (4*b^(5/
4)*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + 1/((a - b)*(-1 + Sin[c + d*x])^2) +
(-3*a + 11*b)/((a - b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (-3*a + 11*b)/((a - b)^2*(1
+ Sin[c + d*x])))/(16*d)

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Maple [B]  time = 0.119, size = 660, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x)

[Out]

1/2/d/(8*a-8*b)/(sin(d*x+c)-1)^2-3/16/d/(a-b)^2/(sin(d*x+c)-1)*a+11/16/d/(a-b)^2/(sin(d*x+c)-1)*b-3/16/d/(a-b)
^3*ln(sin(d*x+c)-1)*a^2+3/8/d/(a-b)^3*ln(sin(d*x+c)-1)*a*b-35/16/d/(a-b)^3*ln(sin(d*x+c)-1)*b^2-1/2/d/(8*a-8*b
)/(1+sin(d*x+c))^2-3/16/d/(a-b)^2/(1+sin(d*x+c))*a+11/16/d/(a-b)^2/(1+sin(d*x+c))*b+3/16/d/(a-b)^3*ln(1+sin(d*
x+c))*a^2-3/8/d/(a-b)^3*ln(1+sin(d*x+c))*a*b+35/16/d/(a-b)^3*ln(1+sin(d*x+c))*b^2-3/2/d*b^2/(a-b)^3*(a/b)^(1/4
)*arctan(sin(d*x+c)/(a/b)^(1/4))-1/2/d*b^3/(a-b)^3*(a/b)^(1/4)/a*arctan(sin(d*x+c)/(a/b)^(1/4))-3/4/d*b^2/(a-b
)^3*(a/b)^(1/4)*ln((sin(d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1/4)))-1/4/d*b^3/(a-b)^3*(a/b)^(1/4)/a*ln((sin(
d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1/4)))+1/2/d*b/(a-b)^3/(a/b)^(1/4)*a*arctan(sin(d*x+c)/(a/b)^(1/4))+3/2
/d*b^2/(a-b)^3/(a/b)^(1/4)*arctan(sin(d*x+c)/(a/b)^(1/4))-1/4/d*b/(a-b)^3/(a/b)^(1/4)*a*ln((sin(d*x+c)+(a/b)^(
1/4))/(sin(d*x+c)-(a/b)^(1/4)))-3/4/d*b^2/(a-b)^3/(a/b)^(1/4)*ln((sin(d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1
/4)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 17.9734, size = 8404, normalized size = 33.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/16*(4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 -
20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*
a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 92
4*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a
^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^
2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5
- 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^1
0 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^
8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b^4 + 60
*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^
3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 +
30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6
- 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15
*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a
^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqr
t((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*
a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9
+ 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*
b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 -
3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^
5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^1
2*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12
*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 +
20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^
5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2
- 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*
b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b
^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*
b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8
+ 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^
5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7
- 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4
+ 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*
a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 3
0*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 -
792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b
^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20
*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^
2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*
a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6
*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sq
rt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*
d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*
b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a
^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 -
6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*(
(a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5
+ 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 -
220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b
^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^
2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt
((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a
^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 +
66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b
^5 + a*b^6)*d^2))) - (3*a^2 - 6*a*b + 35*b^2)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + (3*a^2 - 6*a*b + 35*b^2)*
cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*((3*a^2 - 14*a*b + 11*b^2)*cos(d*x + c)^2 + 2*a^2 - 4*a*b + 2*b^2)*s
in(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(d*x + c)^4)

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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)**5/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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Giac [B]  time = 6.37874, size = 851, normalized size = 3.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/16*(8*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4)
+ 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 8*((
-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*
x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - 4*((-a*b^3)^(3
/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(
-a/b))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 4*((-a*b^3)^(3/4)*(a + 3*b) -
(-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(sqrt(2)
*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c
) + 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c) - 1))/(a^3 - 3*a^2*b +
3*a*b^2 - b^3) + 2*(3*a*sin(d*x + c)^3 - 11*b*sin(d*x + c)^3 - 5*a*sin(d*x + c) + 13*b*sin(d*x + c))/((a^2 -
2*a*b + b^2)*(sin(d*x + c)^2 - 1)^2))/d