3.108 \(\int \csc ^3(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\)
Optimal. Leaf size=167 \[ \frac {b \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{f}-\frac {\sqrt {a} (a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{2 f}+\frac {\sqrt {b} (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{2 f}-\frac {\cot (e+f x) \csc (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{2 f} \]
[Out]
-1/2*cot(f*x+e)*csc(f*x+e)*(a-b+b*sec(f*x+e)^2)^(3/2)/f-1/2*(a+3*b)*arctanh(sec(f*x+e)*a^(1/2)/(a-b+b*sec(f*x+
e)^2)^(1/2))*a^(1/2)/f+1/2*(3*a+b)*arctanh(sec(f*x+e)*b^(1/2)/(a-b+b*sec(f*x+e)^2)^(1/2))*b^(1/2)/f+b*sec(f*x+
e)*(a-b+b*sec(f*x+e)^2)^(1/2)/f
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Rubi [A] time = 0.21, antiderivative size = 167, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used =
{3664, 467, 528, 523, 217, 206, 377, 207} \[ \frac {b \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{f}-\frac {\sqrt {a} (a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{2 f}+\frac {\sqrt {b} (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{2 f}-\frac {\cot (e+f x) \csc (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{2 f} \]
Antiderivative was successfully verified.
[In]
Int[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
-(Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (Sqrt[b]*(3*a + b)
*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e
+ f*x]^2])/f - (Cot[e + f*x]*Csc[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(2*f)
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])
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 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 467
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 523
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 528
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]
Rule 3664
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \csc ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a-b+b x^2\right )^{3/2}}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \csc (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{2 f}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a-b+b x^2} \left (a-b+4 b x^2\right )}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{2 f}\\ &=\frac {b \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{f}-\frac {\cot (e+f x) \csc (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{2 f}+\frac {\operatorname {Subst}\left (\int \frac {2 \left (a^2-b^2\right )+2 b (3 a+b) x^2}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=\frac {b \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{f}-\frac {\cot (e+f x) \csc (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{2 f}+\frac {(b (3 a+b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 f}+\frac {(a (a+3 b)) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 f}\\ &=\frac {b \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{f}-\frac {\cot (e+f x) \csc (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{2 f}+\frac {(b (3 a+b)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {(a (a+3 b)) \operatorname {Subst}\left (\int \frac {1}{-1+a x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}\\ &=-\frac {\sqrt {a} (a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {\sqrt {b} (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {b \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{f}-\frac {\cot (e+f x) \csc (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{2 f}\\ \end {align*}
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Mathematica [B] time = 6.77, size = 1022, normalized size = 6.12 \[ \frac {\sqrt {\frac {\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac {1}{2} b \sec (e+f x)-\frac {1}{2} a \cot (e+f x) \csc (e+f x)\right )}{f}+\frac {\frac {\left (a^2-b^2\right ) \left (2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )}{\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )+\sqrt {b} \left (\tanh ^{-1}\left (\frac {-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a}{\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )+\tanh ^{-1}\left (\frac {2 b+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )}{\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )\right )\right ) (\cos (e+f x)+1) \sqrt {\frac {\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}{\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )^2}}}{4 \sqrt {a} \sqrt {b} \sqrt {a+b+(a-b) \cos (2 (e+f x))} \sqrt {\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}-\frac {\left (a^2+6 b a+b^2\right ) \left (2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )}{\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )-\sqrt {b} \left (\tanh ^{-1}\left (\frac {-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a}{\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )+\tanh ^{-1}\left (\frac {2 b+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )}{\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}\right )\right )\right ) (\cos (e+f x)+1) \sqrt {\frac {\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}{\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )^2}}}{4 \sqrt {a} \sqrt {b} \sqrt {a+b+(a-b) \cos (2 (e+f x))} \sqrt {\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2}}}{2 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1/2*(a*Cot[e + f*x]*Csc[e +
f*x]) + (b*Sec[e + f*x])/2))/f + (-1/4*((a^2 + 6*a*b + b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^
2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2
+ 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(
2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(
1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1
+ Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1
+ Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*
Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]) + ((a^2 - b^2)
*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^
2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/
2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e
+ f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e +
f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e
+ f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sq
rt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/
2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]))/(2*f)
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fricas [A] time = 1.97, size = 994, normalized size = 5.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*s
qrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + ((3*a + b)*cos(
f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x +
e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*c
os(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), -1/4*(2*((3*a + b)*cos(f*x + e)^3 - (
3*a + b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)
/b) - ((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*s
qrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*((a + b)*cos(
f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), 1/4*(2
*((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b
)/cos(f*x + e)^2)*cos(f*x + e)/a) + ((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*
cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^
2) + 2*((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*c
os(f*x + e)), 1/2*(((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*
cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - ((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt
(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((a + b)*cos(f*x + e)
^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e))]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nost
ep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/
t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign:
(2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check
sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to
check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unab
le to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/
2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_n
ostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*
pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2
)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_no
step/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*p
i/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign
: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to chec
k sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable t
o check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Un
able to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_noste
p/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t
_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-
2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep
/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_
nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2
*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check si
gn: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to ch
eck sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable
to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)
Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nos
tep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi
/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>
(-2*pi/t_nostep/2)Warning, integration of abs or sign assumes constant sign by intervals (correct if the argum
ent is real):Check [abs(t_nostep^2-1)]Discontinuities at zeroes of t_nostep^2-1 were not checkedUnable to chec
k sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable t
o check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Un
able to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_noste
p/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t
_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-
2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep
/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_
nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2
*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check si
gn: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to ch
eck sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable
to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)
Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nos
tep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi
/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>
(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nost
ep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/
t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2
*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to chec
k sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2
*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to chec
k sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign:
(2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check
sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Warning, integration of abs or sign assumes constant sign by interva
ls (correct if the argument is real):Check [abs(t_nostep^2-1)]Evaluation time: 2.5Unable to divide, perhaps du
e to rounding error%%%{%%{[-16384,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,9]%%%}+%%%{%%{[%%%{81920,[1]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[6,8]%%%}+%%%{%%{[%%%{-163840,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,7]%%%}+%%%{%%{[%%%{163840,[3
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,6]%%%}+%%%{%%{[%%%{-81920,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,5]%%%}+%%%
{%%{[%%%{16384,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,4]%%%}+%%%{%%%{-98304,[1]%%%},[5,9]%%%}+%%%{%%%{491520,[2
]%%%},[5,8]%%%}+%%%{%%%{-983040,[3]%%%},[5,7]%%%}+%%%{%%%{983040,[4]%%%},[5,6]%%%}+%%%{%%%{-491520,[5]%%%},[5,
5]%%%}+%%%{%%%{98304,[6]%%%},[5,4]%%%}+%%%{%%{[196608,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,10]%%%}+%%%{%%{[%%%{-12288
00,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,9]%%%}+%%%{%%{[%%%{3194880,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,8]%%
%}+%%%{%%{[%%%{-4423680,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,7]%%%}+%%%{%%{[%%%{3440640,[4]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[4,6]%%%}+%%%{%%{[%%%{-1425408,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,5]%%%}+%%%{%%{[%%%{245760,[
6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,4]%%%}+%%%{%%%{786432,[1]%%%},[3,10]%%%}+%%%{%%%{-4259840,[2]%%%},[3,9]%%
%}+%%%{%%%{9502720,[3]%%%},[3,8]%%%}+%%%{%%%{-11141120,[4]%%%},[3,7]%%%}+%%%{%%%{7208960,[5]%%%},[3,6]%%%}+%%%
{%%%{-2424832,[6]%%%},[3,5]%%%}+%%%{%%%{327680,[7]%%%},[3,4]%%%}+%%%{%%{[-786432,0]:[1,0,%%%{-1,[1]%%%}]%%},[2
,11]%%%}+%%%{%%{[%%%{5111808,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,10]%%%}+%%%{%%{[%%%{-14008320,[2]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[2,9]%%%}+%%%{%%{[%%%{20889600,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,8]%%%}+%%%{%%{[%%%{
-18186240,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,7]%%%}+%%%{%%{[%%%{9142272,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[2,6]%%%}+%%%{%%{[%%%{-2408448,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,5]%%%}+%%%{%%{[%%%{245760,[7]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[2,4]%%%}+%%%{%%%{-1572864,[1]%%%},[1,11]%%%}+%%%{%%%{8650752,[2]%%%},[1,10]%%%}+%%%{%%%{-
19759104,[3]%%%},[1,9]%%%}+%%%{%%%{24084480,[4]%%%},[1,8]%%%}+%%%{%%%{-16711680,[5]%%%},[1,7]%%%}+%%%{%%%{6488
064,[6]%%%},[1,6]%%%}+%%%{%%%{-1277952,[7]%%%},[1,5]%%%}+%%%{%%%{98304,[8]%%%},[1,4]%%%}+%%%{%%{[1048576,0]:[1
,0,%%%{-1,[1]%%%}]%%},[0,12]%%%}+%%%{%%{[%%%{-6029312,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,11]%%%}+%%%{%%{[%%
%{14614528,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,10]%%%}+%%%{%%{[%%%{-19349504,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[0,9]%%%}+%%%{%%{[%%%{15155200,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,8]%%%}+%%%{%%{[%%%{-7110656,[5]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[0,7]%%%}+%%%{%%{[%%%{1933312,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,6]%%%}+%%%{%%{[%
%%{-278528,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,5]%%%}+%%%{%%{[%%%{16384,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[
0,4]%%%} / %%%{%%{poly1[%%%{-1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0]%%%}+%%%{%%%{-6,[2]%%%},[5,0]%%%}+%%%{%
%{[%%%{12,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1]%%%}+%%%{%%{poly1[%%%{-15,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[4,0]%%%}+%%%{%%%{48,[2]%%%},[3,1]%%%}+%%%{%%%{-20,[3]%%%},[3,0]%%%}+%%%{%%{[%%%{-48,[1]%%%},0]:[1,0,%%%{-1,[
1]%%%}]%%},[2,2]%%%}+%%%{%%{[%%%{72,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1]%%%}+%%%{%%{poly1[%%%{-15,[3]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0]%%%}+%%%{%%%{-96,[2]%%%},[1,2]%%%}+%%%{%%%{48,[3]%%%},[1,1]%%%}+%%%{%%%{-6,[4]
%%%},[1,0]%%%}+%%%{%%{[%%%{64,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,3]%%%}+%%%{%%{poly1[%%%{-48,[2]%%%},0]:[1,
0,%%%{-1,[1]%%%}]%%},[0,2]%%%}+%%%{%%{[%%%{12,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,1]%%%}+%%%{%%{poly1[%%%{-1
,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [B] time = 0.99, size = 2904, normalized size = 17.39 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x)
[Out]
-1/8/f*(-1+cos(f*x+e))^2*(6*cos(f*x+e)^3*a^(7/2)*arctanh(1/8*(-1+cos(f*x+e))*(4^(1/2)*cos(f*x+e)-4^(1/2)-2*cos
(f*x+e)-2)/sin(f*x+e)^2/((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*4^(1/2))*b+10*cos(f
*x+e)^3*ln(-4*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+(
(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^
(1/2))*b^(7/2)*a-10*cos(f*x+e)^3*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/
(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(
f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(7/2)*a+2*cos(f*x+e)^3*a^(5/2)*arctanh(1/8*(-1+cos(f*x+e))*(4^(1/2)*cos(f*x+
e)-4^(1/2)-2*cos(f*x+e)-2)/sin(f*x+e)^2/((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*4^(
1/2))*b^2-18*cos(f*x+e)^3*ln(-4*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(
f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+
b)/sin(f*x+e)^2/a^(1/2))*b^(5/2)*a^2+18*cos(f*x+e)^3*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^
2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)
-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(5/2)*a^2-6*cos(f*x+e)^2*a^(7/2)*arctanh(1/8*(-1+cos(f*x
+e))*(4^(1/2)*cos(f*x+e)-4^(1/2)-2*cos(f*x+e)-2)/sin(f*x+e)^2/((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e)
)^2)^(1/2)*b^(1/2)*4^(1/2))*b+2*cos(f*x+e)^2*a^(7/2)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2
)*b^(1/2)+2*cos(f*x+e)^2*a^(5/2)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*b^(3/2)-10*cos(f*x
+e)^2*ln(-4*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a
*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1
/2))*b^(7/2)*a+10*cos(f*x+e)^2*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1
+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*
x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(7/2)*a-3*cos(f*x+e)^3*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e
)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/
2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(3/2)*a^3-3*cos(f*x+e)^3*ln(-4*(a^(1/2)*cos(f*x+e)*((a
*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(
1/2)*a^(1/2)+a*cos(f*x+e)-b*cos(f*x+e)+b)/(-1+cos(f*x+e)))*b^(3/2)*a^3-2*cos(f*x+e)^2*a^(5/2)*arctanh(1/8*(-1+
cos(f*x+e))*(4^(1/2)*cos(f*x+e)-4^(1/2)-2*cos(f*x+e)-2)/sin(f*x+e)^2/((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos
(f*x+e))^2)^(1/2)*b^(1/2)*4^(1/2))*b^2+18*cos(f*x+e)^2*ln(-4*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e
)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/
2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(5/2)*a^2-18*cos(f*x+e)^2*ln(-2*(-1+cos(f*x+e))*(a^(1/
2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1
+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(5/2)*a^2-cos(f*x+e)^3*ln(-
2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e
)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/sin(f*x+e)^2/a^(1/2))*b^(1/
2)*a^4-cos(f*x+e)^3*ln(-4*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)+((a*c
os(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)+a*cos(f*x+e)-b*cos(f*x+e)+b)/(-1+cos(f*x+e)))*b^
(1/2)*a^4+3*cos(f*x+e)^2*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f
*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b
)/sin(f*x+e)^2/a^(1/2))*b^(3/2)*a^3+3*cos(f*x+e)^2*ln(-4*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b
)/(1+cos(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)+a*cos(f*x+e)-b*co
s(f*x+e)+b)/(-1+cos(f*x+e)))*b^(3/2)*a^3-2*a^(5/2)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*
b^(3/2)+cos(f*x+e)^2*ln(-2*(-1+cos(f*x+e))*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e
))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+b*cos(f*x+e)+b)/si
n(f*x+e)^2/a^(1/2))*b^(1/2)*a^4+cos(f*x+e)^2*ln(-4*(a^(1/2)*cos(f*x+e)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+c
os(f*x+e))^2)^(1/2)+((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)+a*cos(f*x+e)-b*cos(f*x+
e)+b)/(-1+cos(f*x+e)))*b^(1/2)*a^4)*cos(f*x+e)*4^(1/2)*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/cos(f*x+e)^2)^(3/2)/
((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)/(1+cos(f*x+e))^2)^(3/2)/sin(f*x+e)^6/a^(5/2)/b^(1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*tan(f*x + e)^2 + a)^(3/2)*csc(f*x + e)^3, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*tan(e + f*x)^2)^(3/2)/sin(e + f*x)^3,x)
[Out]
int((a + b*tan(e + f*x)^2)^(3/2)/sin(e + f*x)^3, x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)**3*(a+b*tan(f*x+e)**2)**(3/2),x)
[Out]
Timed out
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