3.114 \(\int \frac {c+d x}{(a+a \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=148 \[ -\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f}-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{6 a^2 f}-\frac {d \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{6 a^2 f^2}+\frac {2 d \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{3 a^2 f^2} \]
[Out]
-1/3*(d*x+c)*cot(1/2*e+1/4*Pi+1/2*f*x)/a^2/f-1/6*d*csc(1/2*e+1/4*Pi+1/2*f*x)^2/a^2/f^2-1/6*(d*x+c)*cot(1/2*e+1
/4*Pi+1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)^2/a^2/f+2/3*d*ln(sin(1/2*e+1/4*Pi+1/2*f*x))/a^2/f^2
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Rubi [A] time = 0.09, antiderivative size = 148, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{3318, 4185, 4184, 3475} \[ -\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f}-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{6 a^2 f}-\frac {d \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{6 a^2 f^2}+\frac {2 d \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{3 a^2 f^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)/(a + a*Sin[e + f*x])^2,x]
[Out]
-((c + d*x)*Cot[e/2 + Pi/4 + (f*x)/2])/(3*a^2*f) - (d*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f^2) - ((c + d*x)*Co
t[e/2 + Pi/4 + (f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f) + (2*d*Log[Sin[e/2 + Pi/4 + (f*x)/2]])/(3*a^2*f
^2)
Rule 3318
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 4185
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int (c+d x) \csc ^4\left (\frac {1}{2} \left (e+\frac {\pi }{2}\right )+\frac {f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac {d \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}\\ &=-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {d \int \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {(c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 225, normalized size = 1.52 \[ -\frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {3}{2} (e+f x)\right ) \left (2 c f+2 d \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-d e+d f x\right )+2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (-3 c f+d \cos (e+f x) \left (-2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+e+f x\right )-4 d \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+2 d e-d f x+d\right )+d \cos \left (\frac {1}{2} (e+f x)\right ) \left (-6 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+3 e+3 f x+2\right )\right )}{6 a^2 f^2 (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)/(a + a*Sin[e + f*x])^2,x]
[Out]
-1/6*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*Cos[(e + f*x)/2]*(2 + 3*e + 3*f*x - 6*Log[Cos[(e + f*x)/2] + Si
n[(e + f*x)/2]]) + Cos[(3*(e + f*x))/2]*(-(d*e) + 2*c*f + d*f*x + 2*d*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]
) + 2*(d + 2*d*e - 3*c*f - d*f*x + d*Cos[e + f*x]*(e + f*x - 2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 4*d
*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[(e + f*x)/2]))/(a^2*f^2*(1 + Sin[e + f*x])^2)
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fricas [A] time = 0.69, size = 204, normalized size = 1.38 \[ \frac {d f x + {\left (d f x + c f\right )} \cos \left (f x + e\right )^{2} + c f + {\left (2 \, d f x + 2 \, c f + d\right )} \cos \left (f x + e\right ) + {\left (d \cos \left (f x + e\right )^{2} - d \cos \left (f x + e\right ) - {\left (d \cos \left (f x + e\right ) + 2 \, d\right )} \sin \left (f x + e\right ) - 2 \, d\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (d f x + c f - {\left (d f x + c f\right )} \cos \left (f x + e\right ) - d\right )} \sin \left (f x + e\right ) + d}{3 \, {\left (a^{2} f^{2} \cos \left (f x + e\right )^{2} - a^{2} f^{2} \cos \left (f x + e\right ) - 2 \, a^{2} f^{2} - {\left (a^{2} f^{2} \cos \left (f x + e\right ) + 2 \, a^{2} f^{2}\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
1/3*(d*f*x + (d*f*x + c*f)*cos(f*x + e)^2 + c*f + (2*d*f*x + 2*c*f + d)*cos(f*x + e) + (d*cos(f*x + e)^2 - d*c
os(f*x + e) - (d*cos(f*x + e) + 2*d)*sin(f*x + e) - 2*d)*log(sin(f*x + e) + 1) - (d*f*x + c*f - (d*f*x + c*f)*
cos(f*x + e) - d)*sin(f*x + e) + d)/(a^2*f^2*cos(f*x + e)^2 - a^2*f^2*cos(f*x + e) - 2*a^2*f^2 - (a^2*f^2*cos(
f*x + e) + 2*a^2*f^2)*sin(f*x + e))
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giac [B] time = 41.37, size = 3094, normalized size = 20.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
-1/3*(2*d*f*x*tan(1/2*f*x)^3*tan(1/2*e)^3 + 2*c*f*tan(1/2*f*x)^3*tan(1/2*e)^3 - d*log(2*(tan(1/2*f*x)^4*tan(1/
2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1
/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) +
2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^3*tan(1/2*e)^3 - 6*d*f*x*tan(1/2*f*x)^2*tan(1/2*e)^2 + 3*d*
log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x
)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(
1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^3*tan(1/2*e)^2 + 3*d*log(2*(tan
(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*ta
n(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 +
2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^2*tan(1/2*e)^3 + d*tan(1/2*f*x)^3*tan(1/2
*e)^3 + 2*d*f*x*tan(1/2*f*x)^3 + 6*d*f*x*tan(1/2*f*x)^2*tan(1/2*e) - 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 -
2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 +
2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2
*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^3*tan(1/2*e) + 6*d*f*x*tan(1/2*f*x)*tan(1/2*e)^2 - 6*c*f*tan(1/2*f*x
)^2*tan(1/2*e)^2 - 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan
(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 +
2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^2*tan(1/
2*e)^2 - d*tan(1/2*f*x)^3*tan(1/2*e)^2 + 2*d*f*x*tan(1/2*e)^3 - 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan
(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*ta
n(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) +
1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)*tan(1/2*e)^3 - d*tan(1/2*f*x)^2*tan(1/2*e)^3 + 2*c*f*tan(1/2*f*x)^3 + d*l
og(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)
^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1
/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^3 + 6*d*f*x*tan(1/2*f*x)*tan(1/2
*e) + 6*c*f*tan(1/2*f*x)^2*tan(1/2*e) - 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) -
2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2
*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*
tan(1/2*f*x)^2*tan(1/2*e) + d*tan(1/2*f*x)^3*tan(1/2*e) + 6*c*f*tan(1/2*f*x)*tan(1/2*e)^2 - 3*d*log(2*(tan(1/2
*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/
2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*t
an(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)*tan(1/2*e)^2 - d*tan(1/2*f*x)^2*tan(1/2*e)^2
+ 2*c*f*tan(1/2*e)^3 + d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*t
an(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2
+ 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*e)^3 + d*ta
n(1/2*f*x)*tan(1/2*e)^3 + 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x
)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*
e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x)^2
- d*tan(1/2*f*x)^3 + 6*c*f*tan(1/2*f*x)*tan(1/2*e) + 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^
4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)
^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1
/2*e)^2 + 1))*tan(1/2*f*x)*tan(1/2*e) - d*tan(1/2*f*x)^2*tan(1/2*e) + 3*d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 -
2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2
+ 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/
2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*e)^2 - d*tan(1/2*f*x)*tan(1/2*e)^2 - d*tan(1/2*e)^3 - 2*d*f*x + 3*d*log(
2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4
+ 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*
e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*f*x) - d*tan(1/2*f*x)^2 + 3*d*log(2*(tan
(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2 + tan(1/2*f*x)^4 + 2*ta
n(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2*f*x)^2 + tan(1/2*e)^2 +
2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1))*tan(1/2*e) + d*tan(1/2*f*x)*tan(1/2*e) - d*tan(1/2*e)^
2 - 2*c*f + d*log(2*(tan(1/2*f*x)^4*tan(1/2*e)^2 - 2*tan(1/2*f*x)^4*tan(1/2*e) - 2*tan(1/2*f*x)^3*tan(1/2*e)^2
+ tan(1/2*f*x)^4 + 2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*tan(1/2*f*x)^3 - 2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*tan(1/2
*f*x)^2 + tan(1/2*e)^2 + 2*tan(1/2*f*x) + 2*tan(1/2*e) + 1)/(tan(1/2*e)^2 + 1)) - d*tan(1/2*f*x) - d*tan(1/2*e
) - d)/(a^2*f^2*tan(1/2*f*x)^3*tan(1/2*e)^3 - 3*a^2*f^2*tan(1/2*f*x)^3*tan(1/2*e)^2 - 3*a^2*f^2*tan(1/2*f*x)^2
*tan(1/2*e)^3 + 3*a^2*f^2*tan(1/2*f*x)^3*tan(1/2*e) + 3*a^2*f^2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 3*a^2*f^2*tan(1/
2*f*x)*tan(1/2*e)^3 - a^2*f^2*tan(1/2*f*x)^3 + 3*a^2*f^2*tan(1/2*f*x)^2*tan(1/2*e) + 3*a^2*f^2*tan(1/2*f*x)*ta
n(1/2*e)^2 - a^2*f^2*tan(1/2*e)^3 - 3*a^2*f^2*tan(1/2*f*x)^2 - 3*a^2*f^2*tan(1/2*f*x)*tan(1/2*e) - 3*a^2*f^2*t
an(1/2*e)^2 - 3*a^2*f^2*tan(1/2*f*x) - 3*a^2*f^2*tan(1/2*e) - a^2*f^2)
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maple [B] time = 0.39, size = 233, normalized size = 1.57 \[ \frac {2 c}{a^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {4 c}{3 a^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 c}{a^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2 x d}{3 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} f}+\frac {2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} f^{2}}+\frac {2 d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} f^{2}}+\frac {2 x d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} f}+\frac {2 d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{3 a^{2} f^{2}}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)/(a+a*sin(f*x+e))^2,x)
[Out]
2/a^2*c/f/(tan(1/2*f*x+1/2*e)+1)^2-4/3/a^2*c/f/(tan(1/2*f*x+1/2*e)+1)^3-2/a^2*c/f/(tan(1/2*f*x+1/2*e)+1)-2/3/a
^2/(tan(1/2*f*x+1/2*e)+1)^3/f*x*d+2/3/a^2/(tan(1/2*f*x+1/2*e)+1)^3*d/f^2*tan(1/2*f*x+1/2*e)+2/3/a^2/(tan(1/2*f
*x+1/2*e)+1)^3*d/f^2*tan(1/2*f*x+1/2*e)^2+2/3/a^2/(tan(1/2*f*x+1/2*e)+1)^3/f*x*d*tan(1/2*f*x+1/2*e)^3+2/3/a^2*
d/f^2*ln(tan(1/2*f*x+1/2*e)+1)-1/3/a^2*d/f^2*ln(1+tan(1/2*f*x+1/2*e)^2)
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maxima [B] time = 1.79, size = 910, normalized size = 6.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
1/3*(2*d*e*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2*f + 3*a^2*f*si
n(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*f*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*f*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3) + (2*(f*x + 3*(f*x + e)*sin(f*x + e) + e + cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 2*(9
*(f*x + e)*cos(f*x + e) - 6*sin(f*x + e) - 1)*cos(2*f*x + 2*e) - 6*cos(2*f*x + 2*e)^2 - 6*cos(f*x + e)^2 - (6*
(cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - cos(3*f*x + 3*e)^2 + 6*(3*sin(f*x + e) + 1)*cos(2*f*x + 2
*e) - 9*cos(2*f*x + 2*e)^2 - 9*cos(f*x + e)^2 - 2*(3*cos(2*f*x + 2*e) - 3*sin(f*x + e) - 1)*sin(3*f*x + 3*e) -
sin(3*f*x + 3*e)^2 - 18*cos(f*x + e)*sin(2*f*x + 2*e) - 9*sin(2*f*x + 2*e)^2 - 9*sin(f*x + e)^2 - 6*sin(f*x +
e) - 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) - 2*(3*(f*x + e)*cos(f*x + e) + cos(2*f*x +
2*e) - sin(f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*sin(f*x + e) + e + 2*cos(f*x + e))*sin(2*f*x + 2
*e) - 6*sin(2*f*x + 2*e)^2 - 6*sin(f*x + e)^2 - 2*sin(f*x + e))*d/(a^2*f*cos(3*f*x + 3*e)^2 + 9*a^2*f*cos(2*f*
x + 2*e)^2 + 9*a^2*f*cos(f*x + e)^2 + a^2*f*sin(3*f*x + 3*e)^2 + 18*a^2*f*cos(f*x + e)*sin(2*f*x + 2*e) + 9*a^
2*f*sin(2*f*x + 2*e)^2 + 9*a^2*f*sin(f*x + e)^2 + 6*a^2*f*sin(f*x + e) + a^2*f - 6*(a^2*f*cos(f*x + e) + a^2*f
*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(3*a^2*f*sin(f*x + e) + a^2*f)*cos(2*f*x + 2*e) + 2*(3*a^2*f*cos(2*f*x
+ 2*e) - 3*a^2*f*sin(f*x + e) - a^2*f)*sin(3*f*x + 3*e)) - 2*c*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x
+ e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3))/f
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mupad [B] time = 4.81, size = 183, normalized size = 1.24 \[ \frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{3\,a^2\,f^2}-\frac {\left (c\,f+d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1+{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}\right )}-\frac {d\,x\,2{}\mathrm {i}}{3\,a^2\,f}-\frac {d\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f\,\left (3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,3{}\mathrm {i}-{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)/(a + a*sin(e + f*x))^2,x)
[Out]
(2*d*log(exp(e*1i)*exp(f*x*1i) + 1i))/(3*a^2*f^2) - ((c*f - d*1i + d*f*x)*2i)/(3*a^2*f^2*(exp(e*1i + f*x*1i)*2
i + exp(e*2i + f*x*2i) - 1)) - (d*x*2i)/(3*a^2*f) - (d*2i)/(3*a^2*f^2*(exp(e*1i + f*x*1i) + 1i)) + (exp(e*1i +
f*x*1i)*(c + d*x)*4i)/(3*a^2*f*(3*exp(e*1i + f*x*1i) - exp(e*2i + f*x*2i)*3i - exp(e*3i + f*x*3i) + 1i))
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sympy [A] time = 2.31, size = 1336, normalized size = 9.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)/(a+a*sin(f*x+e))**2,x)
[Out]
Piecewise((-6*c*f*tan(e/2 + f*x/2)**2/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a
**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - 6*c*f*tan(e/2 + f*x/2)/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f*
*2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - 4*c*f/(3*a**2*f**2*tan(e/2 + f*x/2)**3
+ 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) + 2*d*f*x*tan(e/2 + f*x/2)**3/
(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**
2) - 2*d*f*x/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2)
+ 3*a**2*f**2) + 2*d*log(tan(e/2 + f*x/2) + 1)*tan(e/2 + f*x/2)**3/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*
f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) + 6*d*log(tan(e/2 + f*x/2) + 1)*tan(e/2
+ f*x/2)**2/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2)
+ 3*a**2*f**2) + 6*d*log(tan(e/2 + f*x/2) + 1)*tan(e/2 + f*x/2)/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**
2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) + 2*d*log(tan(e/2 + f*x/2) + 1)/(3*a**2*f*
*2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - d*log
(tan(e/2 + f*x/2)**2 + 1)*tan(e/2 + f*x/2)**3/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)*
*2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - 3*d*log(tan(e/2 + f*x/2)**2 + 1)*tan(e/2 + f*x/2)**2/(3*a**
2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - 3
*d*log(tan(e/2 + f*x/2)**2 + 1)*tan(e/2 + f*x/2)/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/
2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) - d*log(tan(e/2 + f*x/2)**2 + 1)/(3*a**2*f**2*tan(e/2 + f*
x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2*f**2) + 2*d*tan(e/2 + f*x/2)
**2/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2*tan(e/2 + f*x/2) + 3*a**2
*f**2) + 2*d*tan(e/2 + f*x/2)/(3*a**2*f**2*tan(e/2 + f*x/2)**3 + 9*a**2*f**2*tan(e/2 + f*x/2)**2 + 9*a**2*f**2
*tan(e/2 + f*x/2) + 3*a**2*f**2), Ne(f, 0)), ((c*x + d*x**2/2)/(a*sin(e) + a)**2, True))
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