3.4.22 \(\int \frac {\sec ^3(x)}{(a+b \sin ^2(x))^2} \, dx\) [322]

Optimal. Leaf size=109 \[ \frac {b^{3/2} (5 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^3}+\frac {(a+5 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^3}-\frac {(a-b) b \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )} \]

[Out]

1/2*b^(3/2)*(5*a+b)*arctan(sin(x)*b^(1/2)/a^(1/2))/a^(3/2)/(a+b)^3+1/2*(a+5*b)*arctanh(sin(x))/(a+b)^3-1/2*(a-
b)*b*sin(x)/a/(a+b)^2/(a+b*sin(x)^2)+1/2*sec(x)*tan(x)/(a+b)/(a+b*sin(x)^2)

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Rubi [A]
time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3269, 425, 541, 536, 212, 211} \begin {gather*} \frac {b^{3/2} (5 a+b) \text {ArcTan}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^3}-\frac {b (a-b) \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {(a+5 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^3}+\frac {\tan (x) \sec (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^3/(a + b*Sin[x]^2)^2,x]

[Out]

(b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^3) + ((a + 5*b)*ArcTanh[Sin[x]])/(2*(a
+ b)^3) - ((a - b)*b*Sin[x])/(2*a*(a + b)^2*(a + b*Sin[x]^2)) + (Sec[x]*Tan[x])/(2*(a + b)*(a + b*Sin[x]^2))

Rule 211

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

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3269

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\sec ^3(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )}+\frac {\text {Subst}\left (\int \frac {a+2 b+3 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sin (x)\right )}{2 (a+b)}\\ &=-\frac {(a-b) b \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )}-\frac {\text {Subst}\left (\int \frac {-2 \left (a^2+4 a b+b^2\right )-2 (a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{4 a (a+b)^2}\\ &=-\frac {(a-b) b \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )}+\frac {\left (b^2 (5 a+b)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a (a+b)^3}+\frac {(a+5 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{2 (a+b)^3}\\ &=\frac {b^{3/2} (5 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^3}+\frac {(a+5 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^3}-\frac {(a-b) b \sin (x)}{2 a (a+b)^2 \left (a+b \sin ^2(x)\right )}+\frac {\sec (x) \tan (x)}{2 (a+b) \left (a+b \sin ^2(x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 183, normalized size = 1.68 \begin {gather*} \frac {-\frac {b^{3/2} (5 a+b) \tan ^{-1}\left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{a^{3/2}}+\frac {b^{3/2} (5 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{a^{3/2}}-2 (a+5 b) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+2 (a+5 b) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {4 b^2 (a+b) \sin (x)}{a (2 a+b-b \cos (2 x))}}{4 (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b^(3/2)*(5*a + b)*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]])/a^(3/2)) + (b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/
Sqrt[a]])/a^(3/2) - 2*(a + 5*b)*Log[Cos[x/2] - Sin[x/2]] + 2*(a + 5*b)*Log[Cos[x/2] + Sin[x/2]] + (a + b)/(Cos
[x/2] - Sin[x/2])^2 - (a + b)/(Cos[x/2] + Sin[x/2])^2 + (4*b^2*(a + b)*Sin[x])/(a*(2*a + b - b*Cos[2*x])))/(4*
(a + b)^3)

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Maple [A]
time = 0.41, size = 119, normalized size = 1.09

method	result	size
default	\(\frac {b^{2} \left (\frac {\left (a +b \right ) \sin \left (x \right )}{2 a \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}+\frac {\left (5 a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (x \right )-1\right )}+\frac {\left (-a -5 b \right ) \ln \left (\sin \left (x \right )-1\right )}{4 \left (a +b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (1+\sin \left (x \right )\right )}+\frac {\left (a +5 b \right ) \ln \left (1+\sin \left (x \right )\right )}{4 \left (a +b \right )^{3}}\)	\(119\)
risch	\(-\frac {i \left (a b \,{\mathrm e}^{7 i x}-b^{2} {\mathrm e}^{7 i x}-4 a^{2} {\mathrm e}^{5 i x}-3 a b \,{\mathrm e}^{5 i x}-b^{2} {\mathrm e}^{5 i x}+4 \,{\mathrm e}^{3 i x} a^{2}+3 b \,{\mathrm e}^{3 i x} a +b^{2} {\mathrm e}^{3 i x}-{\mathrm e}^{i x} a b +{\mathrm e}^{i x} b^{2}\right )}{\left (a +b \right )^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} a \left (b \,{\mathrm e}^{4 i x}-4 a \,{\mathrm e}^{2 i x}-2 b \,{\mathrm e}^{2 i x}+b \right )}+\frac {\ln \left ({\mathrm e}^{i x}+i\right ) a}{2 a^{3}+6 a^{2} b +6 a \,b^{2}+2 b^{3}}+\frac {5 \ln \left ({\mathrm e}^{i x}+i\right ) b}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{i x}-i\right ) a}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {5 \ln \left ({\mathrm e}^{i x}-i\right ) b}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{3}}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a^{2} \left (a +b \right )^{3}}-\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{3}}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a^{2} \left (a +b \right )^{3}}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a+b)^3*b^2*(1/2*(a+b)/a*sin(x)/(a+b*sin(x)^2)+1/2*(5*a+b)/a/(a*b)^(1/2)*arctan(b*sin(x)/(a*b)^(1/2)))-1/4/(
a+b)^2/(sin(x)-1)+1/4/(a+b)^3*(-a-5*b)*ln(sin(x)-1)-1/4/(a+b)^2/(1+sin(x))+1/4*(a+5*b)/(a+b)^3*ln(1+sin(x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (93) = 186\).
time = 0.48, size = 220, normalized size = 2.02 \begin {gather*} \frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) - 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} - \frac {{\left (a b - b^{2}\right )} \sin \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sin \left (x\right )}{2 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \sin \left (x\right )^{4} - a^{4} - 2 \, a^{3} b - a^{2} b^{2} + {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} \sin \left (x\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a + 5*b)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(a + 5*b)*log(sin(x) - 1)/(a^3 + 3*a^2*b +
3*a*b^2 + b^3) + 1/2*(5*a*b^2 + b^3)*arctan(b*sin(x)/sqrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b
)) - 1/2*((a*b - b^2)*sin(x)^3 + (a^2 + b^2)*sin(x))/((a^3*b + 2*a^2*b^2 + a*b^3)*sin(x)^4 - a^4 - 2*a^3*b - a
^2*b^2 + (a^4 + a^3*b - a^2*b^2 - a*b^3)*sin(x)^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (93) = 186\).
time = 0.49, size = 560, normalized size = 5.14 \begin {gather*} \left [\frac {{\left ({\left (5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} - {\left (5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) - {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )}}, \frac {2 \, {\left ({\left (5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} - {\left (5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \left (x\right )\right ) + {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) - {\left ({\left (a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{4} - {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((5*a*b^2 + b^3)*cos(x)^4 - (5*a^2*b + 6*a*b^2 + b^3)*cos(x)^2)*sqrt(-b/a)*log(-(b*cos(x)^2 - 2*a*sqrt(-
b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)
*log(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(-sin(x) + 1) - 2*(a^3
+ 2*a^2*b + a*b^2 - (a^2*b - b^3)*cos(x)^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4 - (a^5
+ 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2), 1/4*(2*((5*a*b^2 + b^3)*cos(x)^4 - (5*a^2*b + 6*a*b^2 +
b^3)*cos(x)^2)*sqrt(b/a)*arctan(sqrt(b/a)*sin(x)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*co
s(x)^2)*log(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(-sin(x) + 1) -
2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - b^3)*cos(x)^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4
- (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**3/(a+b*sin(x)**2)**2,x)

[Out]

Integral(sec(x)**3/(a + b*sin(x)**2)**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (93) = 186\).
time = 0.48, size = 194, normalized size = 1.78 \begin {gather*} \frac {{\left (a + 5 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (a + 5 \, b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} - \frac {a b \sin \left (x\right )^{3} - b^{2} \sin \left (x\right )^{3} + a^{2} \sin \left (x\right ) + b^{2} \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{4} + a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} - a\right )} {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a + 5*b)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(a + 5*b)*log(-sin(x) + 1)/(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3) + 1/2*(5*a*b^2 + b^3)*arctan(b*sin(x)/sqrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*
b)) - 1/2*(a*b*sin(x)^3 - b^2*sin(x)^3 + a^2*sin(x) + b^2*sin(x))/((b*sin(x)^4 + a*sin(x)^2 - b*sin(x)^2 - a)*
(a^3 + 2*a^2*b + a*b^2))

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Mupad [B]
time = 15.06, size = 2009, normalized size = 18.43 \begin {gather*} \frac {\ln \left (\sin \left (x\right )+1\right )\,\left (a+5\,b\right )}{4\,{\left (a+b\right )}^3}-\ln \left (\sin \left (x\right )-1\right )\,\left (\frac {b}{{\left (a+b\right )}^3}+\frac {1}{4\,{\left (a+b\right )}^2}\right )-\frac {\frac {\sin \left (x\right )\,\left (a^2+b^2\right )}{2\,a\,\left (a^2+2\,a\,b+b^2\right )}+\frac {b\,{\sin \left (x\right )}^3\,\left (a-b\right )}{2\,a\,\left (a^2+2\,a\,b+b^2\right )}}{b\,{\sin \left (x\right )}^4+\left (a-b\right )\,{\sin \left (x\right )}^2-a}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sin \left (x\right )\,\left (a^4\,b^3+10\,a^3\,b^4+50\,a^2\,b^5+10\,a\,b^6+b^7\right )}{2\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}+\frac {\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (\frac {2\,a^9\,b^2+20\,a^8\,b^3+80\,a^7\,b^4+172\,a^6\,b^5+220\,a^5\,b^6+172\,a^4\,b^7+80\,a^3\,b^8+20\,a^2\,b^9+2\,a\,b^{10}}{a^8+6\,a^7\,b+15\,a^6\,b^2+20\,a^5\,b^3+15\,a^4\,b^4+6\,a^3\,b^5+a^2\,b^6}-\frac {\sin \left (x\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (-16\,a^9\,b^2-80\,a^8\,b^3-144\,a^7\,b^4-80\,a^6\,b^5+80\,a^5\,b^6+144\,a^4\,b^7+80\,a^3\,b^8+16\,a^2\,b^9\right )}{8\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}\right )}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,1{}\mathrm {i}}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}+\frac {\left (\frac {\sin \left (x\right )\,\left (a^4\,b^3+10\,a^3\,b^4+50\,a^2\,b^5+10\,a\,b^6+b^7\right )}{2\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}-\frac {\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (\frac {2\,a^9\,b^2+20\,a^8\,b^3+80\,a^7\,b^4+172\,a^6\,b^5+220\,a^5\,b^6+172\,a^4\,b^7+80\,a^3\,b^8+20\,a^2\,b^9+2\,a\,b^{10}}{a^8+6\,a^7\,b+15\,a^6\,b^2+20\,a^5\,b^3+15\,a^4\,b^4+6\,a^3\,b^5+a^2\,b^6}+\frac {\sin \left (x\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (-16\,a^9\,b^2-80\,a^8\,b^3-144\,a^7\,b^4-80\,a^6\,b^5+80\,a^5\,b^6+144\,a^4\,b^7+80\,a^3\,b^8+16\,a^2\,b^9\right )}{8\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}\right )}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,1{}\mathrm {i}}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}}{\frac {-\frac {5\,a^3\,b^4}{4}-\frac {21\,a^2\,b^5}{4}+\frac {21\,a\,b^6}{4}+\frac {5\,b^7}{4}}{a^8+6\,a^7\,b+15\,a^6\,b^2+20\,a^5\,b^3+15\,a^4\,b^4+6\,a^3\,b^5+a^2\,b^6}+\frac {\left (\frac {\sin \left (x\right )\,\left (a^4\,b^3+10\,a^3\,b^4+50\,a^2\,b^5+10\,a\,b^6+b^7\right )}{2\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}+\frac {\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (\frac {2\,a^9\,b^2+20\,a^8\,b^3+80\,a^7\,b^4+172\,a^6\,b^5+220\,a^5\,b^6+172\,a^4\,b^7+80\,a^3\,b^8+20\,a^2\,b^9+2\,a\,b^{10}}{a^8+6\,a^7\,b+15\,a^6\,b^2+20\,a^5\,b^3+15\,a^4\,b^4+6\,a^3\,b^5+a^2\,b^6}-\frac {\sin \left (x\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (-16\,a^9\,b^2-80\,a^8\,b^3-144\,a^7\,b^4-80\,a^6\,b^5+80\,a^5\,b^6+144\,a^4\,b^7+80\,a^3\,b^8+16\,a^2\,b^9\right )}{8\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}\right )}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}-\frac {\left (\frac {\sin \left (x\right )\,\left (a^4\,b^3+10\,a^3\,b^4+50\,a^2\,b^5+10\,a\,b^6+b^7\right )}{2\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}-\frac {\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (\frac {2\,a^9\,b^2+20\,a^8\,b^3+80\,a^7\,b^4+172\,a^6\,b^5+220\,a^5\,b^6+172\,a^4\,b^7+80\,a^3\,b^8+20\,a^2\,b^9+2\,a\,b^{10}}{a^8+6\,a^7\,b+15\,a^6\,b^2+20\,a^5\,b^3+15\,a^4\,b^4+6\,a^3\,b^5+a^2\,b^6}+\frac {\sin \left (x\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,\left (-16\,a^9\,b^2-80\,a^8\,b^3-144\,a^7\,b^4-80\,a^6\,b^5+80\,a^5\,b^6+144\,a^4\,b^7+80\,a^3\,b^8+16\,a^2\,b^9\right )}{8\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )\,\left (a^6+4\,a^5\,b+6\,a^4\,b^2+4\,a^3\,b^3+a^2\,b^4\right )}\right )}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}}{4\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )}}\right )\,\left (5\,a+b\right )\,\sqrt {-a^3\,b^3}\,1{}\mathrm {i}}{2\,\left (a^6+3\,a^5\,b+3\,a^4\,b^2+a^3\,b^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^3*(a + b*sin(x)^2)^2),x)

[Out]

(log(sin(x) + 1)*(a + 5*b))/(4*(a + b)^3) - log(sin(x) - 1)*(b/(a + b)^3 + 1/(4*(a + b)^2)) - ((sin(x)*(a^2 +
b^2))/(2*a*(2*a*b + a^2 + b^2)) + (b*sin(x)^3*(a - b))/(2*a*(2*a*b + a^2 + b^2)))/(b*sin(x)^4 - a + sin(x)^2*(
a - b)) - (atan(((((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 + a^4*b^3))/(2*(4*a^5*b + a^6 + a^2*b^4 +
4*a^3*b^3 + 6*a^4*b^2)) + ((5*a + b)*(-a^3*b^3)^(1/2)*((2*a*b^10 + 20*a^2*b^9 + 80*a^3*b^8 + 172*a^4*b^7 + 22
0*a^5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8*b^3 + 2*a^9*b^2)/(6*a^7*b + a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b
^4 + 20*a^5*b^3 + 15*a^6*b^2) - (sin(x)*(5*a + b)*(-a^3*b^3)^(1/2)*(16*a^2*b^9 + 80*a^3*b^8 + 144*a^4*b^7 + 80
*a^5*b^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^3 - 16*a^9*b^2))/(8*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)*(4*a^
5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2))))/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)))*(5*a + b)*(-a^3*b^3
)^(1/2)*1i)/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)) + (((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 +
a^4*b^3))/(2*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2)) - ((5*a + b)*(-a^3*b^3)^(1/2)*((2*a*b^10 + 20*
a^2*b^9 + 80*a^3*b^8 + 172*a^4*b^7 + 220*a^5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8*b^3 + 2*a^9*b^2)/(6*a^7*b
+ a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4 + 20*a^5*b^3 + 15*a^6*b^2) + (sin(x)*(5*a + b)*(-a^3*b^3)^(1/2)*(16*
a^2*b^9 + 80*a^3*b^8 + 144*a^4*b^7 + 80*a^5*b^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^3 - 16*a^9*b^2))/(8*(3*a
^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2))))/(4*(3*a^5*b + a^6 + a^3
*b^3 + 3*a^4*b^2)))*(5*a + b)*(-a^3*b^3)^(1/2)*1i)/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)))/(((21*a*b^6)/4 +
(5*b^7)/4 - (21*a^2*b^5)/4 - (5*a^3*b^4)/4)/(6*a^7*b + a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4 + 20*a^5*b^3 +
15*a^6*b^2) + (((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 + a^4*b^3))/(2*(4*a^5*b + a^6 + a^2*b^4 + 4*
a^3*b^3 + 6*a^4*b^2)) + ((5*a + b)*(-a^3*b^3)^(1/2)*((2*a*b^10 + 20*a^2*b^9 + 80*a^3*b^8 + 172*a^4*b^7 + 220*a
^5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8*b^3 + 2*a^9*b^2)/(6*a^7*b + a^8 + a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4
+ 20*a^5*b^3 + 15*a^6*b^2) - (sin(x)*(5*a + b)*(-a^3*b^3)^(1/2)*(16*a^2*b^9 + 80*a^3*b^8 + 144*a^4*b^7 + 80*a^
5*b^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^3 - 16*a^9*b^2))/(8*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)*(4*a^5*b
+ a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2))))/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)))*(5*a + b)*(-a^3*b^3)^(
1/2))/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2)) - (((sin(x)*(10*a*b^6 + b^7 + 50*a^2*b^5 + 10*a^3*b^4 + a^4*b^
3))/(2*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2)) - ((5*a + b)*(-a^3*b^3)^(1/2)*((2*a*b^10 + 20*a^2*b^
9 + 80*a^3*b^8 + 172*a^4*b^7 + 220*a^5*b^6 + 172*a^6*b^5 + 80*a^7*b^4 + 20*a^8*b^3 + 2*a^9*b^2)/(6*a^7*b + a^8
+ a^2*b^6 + 6*a^3*b^5 + 15*a^4*b^4 + 20*a^5*b^3 + 15*a^6*b^2) + (sin(x)*(5*a + b)*(-a^3*b^3)^(1/2)*(16*a^2*b^
9 + 80*a^3*b^8 + 144*a^4*b^7 + 80*a^5*b^6 - 80*a^6*b^5 - 144*a^7*b^4 - 80*a^8*b^3 - 16*a^9*b^2))/(8*(3*a^5*b +
a^6 + a^3*b^3 + 3*a^4*b^2)*(4*a^5*b + a^6 + a^2*b^4 + 4*a^3*b^3 + 6*a^4*b^2))))/(4*(3*a^5*b + a^6 + a^3*b^3 +
3*a^4*b^2)))*(5*a + b)*(-a^3*b^3)^(1/2))/(4*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2))))*(5*a + b)*(-a^3*b^3)^(1/
2)*1i)/(2*(3*a^5*b + a^6 + a^3*b^3 + 3*a^4*b^2))

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