3.1468 \(\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=229 \[ \frac{2 b^3 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{5/2}}+\frac{2 b^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{5/2}}+\frac{b^4 \cos (c+d x)}{a d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)} \]

[Out]

(2*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (2*b^3*(3*a^2 - b^2)*ArcTan[(
b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c +
d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])
/(a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 0.310499, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2897, 3770, 2648, 2664, 12, 2660, 618, 204} \[ \frac{2 b^3 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{5/2}}+\frac{2 b^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{5/2}}+\frac{b^4 \cos (c+d x)}{a d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (2*b^3*(3*a^2 - b^2)*ArcTan[(
b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c +
d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])
/(a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))

Rule 2897

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2648

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

Rule 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

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

Mathematica [A]  time = 2.21898, size = 203, normalized size = 0.89 \[ \frac{-\frac{2 \left (b^5-4 a^2 b^3\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{5/2}}+\frac{\frac{a b^4 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^2}+\sin \left (\frac{1}{2} (c+d x)\right ) \left (\frac{1}{(a+b)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{1}{(a-b)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(-4*a^2*b^3 + b^5)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)) + Sin[(c + d
*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2
]))) + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]] + (a*b^4*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[
c + d*x])))/a^2)/d

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Maple [A]  time = 0.15, size = 304, normalized size = 1.3 \begin{align*} -{\frac{1}{d \left ( a+b \right ) ^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+2\,{\frac{{b}^{5}\tan \left ( 1/2\,dx+c/2 \right ) }{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+2\,{\frac{{b}^{4}}{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}a \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+8\,{\frac{{b}^{3}}{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-2\,{\frac{{b}^{5}}{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}{a}^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }+{\frac{1}{d \left ( a-b \right ) ^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)*sec(d*x+c)^2/(a+b*sin(d*x+c))^2,x)

[Out]

-1/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)+2/d*b^5/(a-b)^2/(a+b)^2/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b
+a)*tan(1/2*d*x+1/2*c)+2/d*b^4/(a-b)^2/(a+b)^2/a/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)+8/d*b^3/(a-
b)^2/(a+b)^2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-2/d*b^5/(a-b)^2/(a+b)^2/
a^2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+1/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1
)+1/d/a^2*ln(tan(1/2*d*x+1/2*c))

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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(csc(d*x+c)*sec(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.80245, size = 2133, normalized size = 9.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*a^7 - 4*a^5*b^2 + 2*a^3*b^4 + 2*(2*a^5*b^2 - a^3*b^4 - a*b^6)*cos(d*x + c)^2 + ((4*a^2*b^4 - b^6)*cos(
d*x + c)*sin(d*x + c) + (4*a^3*b^3 - a*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2
- 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos
(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - ((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)*sin(d*x +
c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + ((a^6*b - 3*a^4*b^3 +
3*a^2*b^5 - b^7)*cos(d*x + c)*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(-1/2*cos(
d*x + c) + 1/2) - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*sin(d*x + c))/((a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*c
os(d*x + c)*sin(d*x + c) + (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)), 1/2*(2*a^7 - 4*a^5*b^2 + 2
*a^3*b^4 + 2*(2*a^5*b^2 - a^3*b^4 - a*b^6)*cos(d*x + c)^2 - 2*((4*a^2*b^4 - b^6)*cos(d*x + c)*sin(d*x + c) + (
4*a^3*b^3 - a*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))
- ((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos
(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + ((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)*sin(d*x + c) + (a
^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(a^6*b - 2*a^4*b^3 + a^2*b^
5)*sin(d*x + c))/((a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)*sin(d*x + c) + (a^9 - 3*a^7*b^2 + 3
*a^5*b^4 - a^3*b^6)*d*cos(d*x + c))]

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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(csc(d*x+c)*sec(d*x+c)**2/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.2666, size = 424, normalized size = 1.85 \begin{align*} \frac{\frac{2 \,{\left (4 \, a^{2} b^{3} - b^{5}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt{a^{2} - b^{2}}} + \frac{2 \,{\left (2 \, a^{4} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{5} - a^{3} b^{2} - a b^{4}\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}} + \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(4*a^2*b^3 - b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 -
b^2)))/((a^6 - 2*a^4*b^2 + a^2*b^4)*sqrt(a^2 - b^2)) + 2*(2*a^4*b*tan(1/2*d*x + 1/2*c)^3 + b^5*tan(1/2*d*x +
1/2*c)^3 - a^5*tan(1/2*d*x + 1/2*c)^2 + 3*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + a*b^4*tan(1/2*d*x + 1/2*c)^2 - 2*a^
2*b^3*tan(1/2*d*x + 1/2*c) - b^5*tan(1/2*d*x + 1/2*c) - a^5 - a^3*b^2 - a*b^4)/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a
*tan(1/2*d*x + 1/2*c)^4 + 2*b*tan(1/2*d*x + 1/2*c)^3 - 2*b*tan(1/2*d*x + 1/2*c) - a)) + log(abs(tan(1/2*d*x +
1/2*c)))/a^2)/d