3.5.9 \(\int \frac {\sec ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [409]
Optimal. Leaf size=175 \[ \frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^2 d}+\frac {(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 (a-b)^2 d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {1}{4 (a-b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \]
[Out]
1/2*(a-5*b)*arctanh(sin(d*x+c))/(a-b)^2/d+1/4/(a-b)/d/(1-sin(d*x+c))-1/4/(a-b)/d/(1+sin(d*x+c))+1/2*b^(3/4)*ar
ctanh(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)-b^(1/2))^2+1/2*b^(3/4)*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))
/a^(3/4)/d/(a^(1/2)+b^(1/2))^2
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Rubi [A]
time = 0.14, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3302, 1185,
213, 1181, 211, 214} \begin {gather*} \frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^2}+\frac {b^{3/4} \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 )^2}+\frac {1}{4 d (a-b) (1-\sin (c+d x))}-\frac {1}{4 d (a-b) (\sin (c+d x)+1)}+\frac {(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 d (a-b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sec[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]
[Out]
(b^(3/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^2*d) + ((a - 5*b)*ArcTanh[Sin[
c + d*x]])/(2*(a - b)^2*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^
2*d) + 1/(4*(a - b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))
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 213
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 1181
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 1185
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 3302
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])
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-b x^4\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{4 (a-b) (-1+x)^2}+\frac {1}{4 (a-b) (1+x)^2}+\frac {-a+5 b}{2 (a-b)^2 \left (-1+x^2\right )}+\frac {b \left (a+b+2 b x^2\right )}{(a-b)^2 \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {1}{4 (a-b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}-\frac {(a-5 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sin (c+d x)\right )}{2 (a-b)^2 d}+\frac {b \text {Subst}\left (\int \frac {a+b+2 b x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{(a-b)^2 d}\\ &=\frac {(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 (a-b)^2 d}+\frac {1}{4 (a-b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {b^{3/2} \text {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 )^2 d}-\frac {b^{3/2} \text {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 )^2 d}\\ &=\frac {b^{3/4} \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 )^2 d}+\frac {(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 (a-b)^2 d}+\frac {b^{3/4} \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 )^2 d}+\frac {1}{4 (a-b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.73, size = 255, normalized size = 1.46 \begin {gather*} -\frac {-\frac {2 (a-5 b) \tanh ^{-1}(\sin (c+d x))}{(a-b)^2}+\frac {b^{3/4} \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^2}-\frac {i b^{3/4} \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^2}+\frac {i b^{3/4} \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^2}-\frac {b^{3/4} \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^2}+\frac {1}{(a-b) (-1+\sin (c+d x))}+\frac {1}{(a-b) (1+\sin (c+d x))}}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]
[Out]
-1/4*((-2*(a - 5*b)*ArcTanh[Sin[c + d*x]])/(a - b)^2 + (b^(3/4)*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]])/(a^(3/4)*
(Sqrt[a] - Sqrt[b])^2) - (I*b^(3/4)*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^2) + (
I*b^(3/4)*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^2) - (b^(3/4)*Log[a^(1/4) + b^(1
/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^2) + 1/((a - b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c +
d*x])))/d
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Maple [A]
time = 1.26, size = 241, normalized size = 1.38
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -5 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a +5 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a -b \right )^{2}}-\frac {b \left (\frac {\left (-a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a -b \right )^{2}}}{d}\) |
\(241\) |
| default |
\(\frac {-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -5 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a +5 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a -b \right )^{2}}-\frac {b \left (\frac {\left (-a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a -b \right )^{2}}}{d}\) |
\(241\) |
| risch |
\(-\frac {i \left ({\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left (a -b \right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{2 \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{2 \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{2 d \left (a^{2}-2 a b +b^{2}\right )}+8 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{7} d^{4}-4194304 a^{6} b \,d^{4}+6291456 a^{5} b^{2} d^{4}-4194304 a^{4} b^{3} d^{4}+1048576 a^{3} b^{4} d^{4}\right ) \textit {\_Z}^{4}+\left (-8192 a^{3} b^{2} d^{2}-8192 d^{2} b^{3} a^{2}\right ) \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {131072 i d^{3} a^{7}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}+\frac {524288 i d^{3} a^{6} b}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}-\frac {786432 i d^{3} a^{5} b^{2}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}+\frac {524288 i d^{3} a^{4} b^{3}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}-\frac {131072 i d^{3} a^{3} b^{4}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {64 i a^{4} b d}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}+\frac {960 i d \,a^{3} b^{2}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}+\frac {960 i d \,a^{2} b^{3}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}+\frac {64 i a \,b^{4} d}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{2} b^{2}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}-\frac {6 a \,b^{3}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}-\frac {b^{4}}{a^{2} b^{2}+6 a \,b^{3}+b^{4}}\right )\right )\) |
\(641\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/(4*a-4*b)/(1+sin(d*x+c))+1/4*(a-5*b)/(a-b)^2*ln(1+sin(d*x+c))-1/(4*a-4*b)/(sin(d*x+c)-1)+1/4/(a-b)^2*(
-a+5*b)*ln(sin(d*x+c)-1)-b/(a-b)^2*(1/4*(-a-b)*(1/b*a)^(1/4)/a*(ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b
*a)^(1/4)))+2*arctan(sin(d*x+c)/(1/b*a)^(1/4)))+1/2/(1/b*a)^(1/4)*(2*arctan(sin(d*x+c)/(1/b*a)^(1/4))-ln((sin(
d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b*a)^(1/4))))))
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Maxima [A]
time = 0.52, size = 244, normalized size = 1.39 \begin {gather*} -\frac {\frac {b {\left (\frac {2 \, {\left (b {\left (2 \, \sqrt {a} - \sqrt {b}\right )} - a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b {\left (2 \, \sqrt {a} + \sqrt {b}\right )} + a \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (a - 5 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (a - 5 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, \sin \left (d x + c\right )}{{\left (a - b\right )} \sin \left (d x + c\right )^{2} - a + b}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
-1/4*(b*(2*(b*(2*sqrt(a) - sqrt(b)) - a*sqrt(b))*arctan(sqrt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*s
qrt(sqrt(a)*sqrt(b))*sqrt(b)) + (b*(2*sqrt(a) + sqrt(b)) + a*sqrt(b))*log((sqrt(b)*sin(d*x + c) - sqrt(sqrt(a)
*sqrt(b)))/(sqrt(b)*sin(d*x + c) + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a^2 - 2*a
*b + b^2) - (a - 5*b)*log(sin(d*x + c) + 1)/(a^2 - 2*a*b + b^2) + (a - 5*b)*log(sin(d*x + c) - 1)/(a^2 - 2*a*b
+ b^2) + 2*sin(d*x + c)/((a - b)*sin(d*x + c)^2 - a + b))/d
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2529 vs.
\(2 (139) = 278\).
time = 0.89, size = 2529, normalized size = 14.45 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
-1/4*((a^2 - 2*a*b + b^2)*d*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b
^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a
^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*
cos(d*x + c)^2*log(1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3
+ a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a
^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 +
a*b^4)*d)*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 +
12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7
+ a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a^2 - 2*a*b +
b^2)*d*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12
*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 +
a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(1
/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt(
(a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4
- 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-((a^
5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a
^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) -
4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a^2 - 2*a*b + b^2)*d*sqrt(((a^5 -
4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11
- 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*
a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(-1/2*(a^2*b^2 + 6*a*b
^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b
^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a
^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(((a^5 - 4*a^4*b + 6*a^3*
b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*
a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((
a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) + (a^2 - 2*a*b + b^2)*d*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^
2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^
9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^
5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(-1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x +
c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 1
2*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7
+ a^3*b^8)*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 +
a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b
^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*
a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a - 5*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) + (a - 5*b)*cos(d*x + c)^
2*log(-sin(d*x + c) + 1) - 2*(a - b)*sin(d*x + c))/((a^2 - 2*a*b + b^2)*d*cos(d*x + c)^2)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{3}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)**3/(a-b*sin(d*x+c)**4),x)
[Out]
Integral(sec(c + d*x)**3/(a - b*sin(c + d*x)**4), x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 475 vs.
\(2 (139) = 278\).
time = 0.78, size = 475, normalized size = 2.71 \begin {gather*} \frac {\frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {1}{4}} {\left (a b + b^{2}\right )} + 2 \, \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b - 2 \, \sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {1}{4}} {\left (a b + b^{2}\right )} + 2 \, \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b - 2 \, \sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} - \frac {{\left (2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (\sqrt {2} a b + \sqrt {2} b^{2}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}} + \frac {{\left (2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (\sqrt {2} a b + \sqrt {2} b^{2}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}} + \frac {2 \, {\left (a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {2 \, {\left (a - 5 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {4 \, \sin \left (d x + c\right )}{{\left (\sin \left (d x + c\right )^{2} - 1\right )} {\left (a - b\right )}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
1/8*(4*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x +
c))/(-a/b)^(1/4))/(sqrt(2)*a^3*b - 2*sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 4*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*
b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^3*b - 2*sqrt(
2)*a^2*b^2 + sqrt(2)*a*b^3) - (2*sqrt(2)*(-a*b^3)^(3/4) - (-a*b^3)^(1/4)*(sqrt(2)*a*b + sqrt(2)*b^2))*log(sin(
d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^3*b - 2*a^2*b^2 + a*b^3) + (2*sqrt(2)*(-a*b^3)
^(3/4) - (-a*b^3)^(1/4)*(sqrt(2)*a*b + sqrt(2)*b^2))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) +
sqrt(-a/b))/(a^3*b - 2*a^2*b^2 + a*b^3) + 2*(a - 5*b)*log(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) - 2*(a -
5*b)*log(abs(sin(d*x + c) - 1))/(a^2 - 2*a*b + b^2) - 4*sin(d*x + c)/((sin(d*x + c)^2 - 1)*(a - b)))/d
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Mupad [B]
time = 19.19, size = 2500, normalized size = 14.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cos(c + d*x)^3*(a - b*sin(c + d*x)^4)),x)
[Out]
(atan(((((((128*a*b^11 + 256*a^2*b^10 - 3456*a^3*b^9 + 8960*a^4*b^8 - 10880*a^5*b^7 + 6912*a^6*b^6 - 2176*a^7*
b^5 + 256*a^8*b^4)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (sin(c + d*x)*(-(a^2*(a^3*b^3)^(1/2) + b^
2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 +
6*a^5*b^2)))^(1/2)*(512*a^2*b^11 - 2560*a^3*b^10 + 4608*a^4*b^9 - 2560*a^5*b^8 - 2560*a^6*b^7 + 4608*a^7*b^6 -
2560*a^8*b^5 + 512*a^9*b^4))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b
^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^
2)))^(1/2) + (sin(c + d*x)*(48*a*b^10 - 16*b^11 + 1024*a^2*b^9 - 2208*a^3*b^8 + 1264*a^4*b^7 - 144*a^5*b^6 + 3
2*a^6*b^5))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*
b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) - (200*
a*b^9 + 480*a^2*b^8 - 784*a^3*b^7 + 96*a^4*b^6 + 8*a^5*b^5)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(
-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*
b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) + (sin(c + d*x)*(11*a*b^8 + 27*b^9 - 7*a^2*b^7 + a^3*b^6))/(a^4 -
4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 +
6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2)*1i - (((((128*a*b^11 + 25
6*a^2*b^10 - 3456*a^3*b^9 + 8960*a^4*b^8 - 10880*a^5*b^7 + 6912*a^6*b^6 - 2176*a^7*b^5 + 256*a^8*b^4)/(2*(a^4
- 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (sin(c + d*x)*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b
^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2)*(512*a^2
*b^11 - 2560*a^3*b^10 + 4608*a^4*b^9 - 2560*a^5*b^8 - 2560*a^6*b^7 + 4608*a^7*b^6 - 2560*a^8*b^5 + 512*a^9*b^4
))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a
^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) - (sin(c + d*x)*
(48*a*b^10 - 16*b^11 + 1024*a^2*b^9 - 2208*a^3*b^8 + 1264*a^4*b^7 - 144*a^5*b^6 + 32*a^6*b^5))/(a^4 - 4*a^3*b
- 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^
3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) - (200*a*b^9 + 480*a^2*b^8 - 784*a
^3*b^7 + 96*a^4*b^6 + 8*a^5*b^5)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(-(a^2*(a^3*b^3)^(1/2) + b^2
*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6
*a^5*b^2)))^(1/2) - (sin(c + d*x)*(11*a*b^8 + 27*b^9 - 7*a^2*b^7 + a^3*b^6))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 +
6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*
(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2)*1i)/((((((128*a*b^11 + 256*a^2*b^10 - 3456*a^3*b^9 +
8960*a^4*b^8 - 10880*a^5*b^7 + 6912*a^6*b^6 - 2176*a^7*b^5 + 256*a^8*b^4)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 +
6*a^2*b^2)) - (sin(c + d*x)*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3
*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2)*(512*a^2*b^11 - 2560*a^3*b^10 + 460
8*a^4*b^9 - 2560*a^5*b^8 - 2560*a^6*b^7 + 4608*a^7*b^6 - 2560*a^8*b^5 + 512*a^9*b^4))/(a^4 - 4*a^3*b - 4*a*b^3
+ b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1
/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) + (sin(c + d*x)*(48*a*b^10 - 16*b^11 + 1024
*a^2*b^9 - 2208*a^3*b^8 + 1264*a^4*b^7 - 144*a^5*b^6 + 32*a^6*b^5))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2
))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*
a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) - (200*a*b^9 + 480*a^2*b^8 - 784*a^3*b^7 + 96*a^4*b^6 + 8*a^5
*b^5)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^
3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2) + (sin(c
+ d*x)*(11*a*b^8 + 27*b^9 - 7*a^2*b^7 + a^3*b^6))/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))*(-(a^2*(a^3*b^3
)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*b + a^3*b^4 -
4*a^4*b^3 + 6*a^5*b^2)))^(1/2) + (((((128*a*b^11 + 256*a^2*b^10 - 3456*a^3*b^9 + 8960*a^4*b^8 - 10880*a^5*b^7
+ 6912*a^6*b^6 - 2176*a^7*b^5 + 256*a^8*b^4)/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (sin(c + d*x)*(
-(a^2*(a^3*b^3)^(1/2) + b^2*(a^3*b^3)^(1/2) - 4*a^2*b^3 - 4*a^3*b^2 + 6*a*b*(a^3*b^3)^(1/2))/(16*(a^7 - 4*a^6*
b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))^(1/2)*(512*a^2*b^11 - 2560*a^3*b^10 + 4608*a^4*b^9 - 2560*a^5*b^8 - 256
0*a^6*b^7 + 4608*a^7*b^6 - 2560*a^8*b^5 + 512*a...
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