3.4.92 \(\int \frac {\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [392]

Optimal. Leaf size=1093 \[ -\frac {2 (-1)^{2/3} a^{2/3} b^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} \left (a^2-b^2\right )^2 d}-\frac {2 b^2 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} \left (a^2-b^2\right )^2 d}+\frac {2 a^{2/3} b^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{\sqrt {a^{2/3}-b^{2/3}} \left (a^2-b^2\right )^2 d}+\frac {2 b^2 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-b^{2/3}} \left (a^2-b^2\right )^2 d}+\frac {2 b^{4/3} \left (a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} \left (a^2-b^2\right )^2 d}-\frac {2 \sqrt [3]{-1} a^{2/3} b^{8/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} \left (a^2-b^2\right )^2 d}+\frac {2 b^2 \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} \left (a^2-b^2\right )^2 d}-\frac {2 b^{4/3} \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} \left (a^2-b^2\right )^2 d}-\frac {2 b^{4/3} \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} \left (a^2-b^2\right )^2 d}+\frac {\cos (c+d x)}{12 (a+b) d (1-\sin (c+d x))^2}+\frac {\cos (c+d x)}{12 (a+b) d (1-\sin (c+d x))}+\frac {(a+4 b) \cos (c+d x)}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{12 (a-b) d (1+\sin (c+d x))^2}-\frac {(a-4 b) \cos (c+d x)}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {\cos (c+d x)}{12 (a-b) d (1+\sin (c+d x))} \]

[Out]

1/12*cos(d*x+c)/(a+b)/d/(1-sin(d*x+c))^2+1/12*cos(d*x+c)/(a+b)/d/(1-sin(d*x+c))+1/4*(a+4*b)*cos(d*x+c)/(a+b)^2
/d/(1-sin(d*x+c))-1/12*cos(d*x+c)/(a-b)/d/(1+sin(d*x+c))^2-1/4*(a-4*b)*cos(d*x+c)/(a-b)^2/d/(1+sin(d*x+c))-1/1
2*cos(d*x+c)/(a-b)/d/(1+sin(d*x+c))+2*a^(2/3)*b^(8/3)*arctan((b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-b^(
2/3))^(1/2))/(a^2-b^2)^2/d/(a^(2/3)-b^(2/3))^(1/2)+2/3*b^2*(2*a^2+b^2)*arctan((b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2
*c))/(a^(2/3)-b^(2/3))^(1/2))/a^(2/3)/(a^2-b^2)^2/d/(a^(2/3)-b^(2/3))^(1/2)+2/3*b^(4/3)*(a^2+2*b^2)*arctan((b^
(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-b^(2/3))^(1/2))/(a^2-b^2)^2/d/(a^(2/3)-b^(2/3))^(1/2)-2/3*b^(4/3)*(
a^2+2*b^2)*arctanh((b^(1/3)+(-1)^(2/3)*a^(1/3)*tan(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3)+b^(2/3))^(1/2))/(a^2-b^
2)^2/d/((-1)^(1/3)*a^(2/3)+b^(2/3))^(1/2)-2/3*b^(4/3)*(a^2+2*b^2)*arctanh((b^(1/3)-(-1)^(1/3)*a^(1/3)*tan(1/2*
d*x+1/2*c))/(-(-1)^(2/3)*a^(2/3)+b^(2/3))^(1/2))/(a^2-b^2)^2/d/(-(-1)^(2/3)*a^(2/3)+b^(2/3))^(1/2)-2*(-1)^(1/3
)*a^(2/3)*b^(8/3)*arctan(((-1)^(2/3)*b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2))/(
a^2-b^2)^2/d/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2)+2/3*b^2*(2*a^2+b^2)*arctan(((-1)^(2/3)*b^(1/3)+a^(1/3)*tan(1/2
*d*x+1/2*c))/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2))/a^(2/3)/(a^2-b^2)^2/d/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2)-2*(-
1)^(2/3)*a^(2/3)*b^(8/3)*arctan(((-1)^(1/3)*b^(1/3)-a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(
1/2))/(a^2-b^2)^2/d/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)-2/3*b^2*(2*a^2+b^2)*arctan(((-1)^(1/3)*b^(1/3)-a^(1/3)*
tan(1/2*d*x+1/2*c))/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(2/3)/(a^2-b^2)^2/d/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/
2)

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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]

[Out]

Defer[Int][Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3), x]

Rubi steps

\begin {align*} \int \frac {\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \frac {\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.22, size = 679, normalized size = 0.62 \begin {gather*} \frac {4 i b^2 \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+4 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-2 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+6 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-20 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-16 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+10 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+8 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-12 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-6 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+4 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-2 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+\sec ^3(c+d x) \left (4 a^2 b+32 b^3-3 b \left (5 a^2+13 b^2\right ) \cos (c+d x)+12 b \left (a^2+2 b^2\right ) \cos (2 (c+d x))-5 a^2 b \cos (3 (c+d x))-13 b^3 \cos (3 (c+d x))+12 a^3 \sin (c+d x)-30 a b^2 \sin (c+d x)+4 a^3 \sin (3 (c+d x))-22 a b^2 \sin (3 (c+d x))\right )}{24 (a-b)^2 (a+b)^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]

[Out]

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*a^2*ArcTan[Sin[c + d*x]/(
Cos[c + d*x] - #1)] + 4*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]
- (2*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (12*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 6*a*
b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 16*b^2*ArcTan[
Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (8*I)*b^2*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 6*a*b*Log[1 - 2*C
os[c + d*x]*#1 + #1^2]*#1^3 + 2*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 4*b^2*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)]*#1^4 - I*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (2*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + Sec[c + d*x]^3*(4*a^2*b + 32*b^3 - 3*b*(5*a^2 +
13*b^2)*Cos[c + d*x] + 12*b*(a^2 + 2*b^2)*Cos[2*(c + d*x)] - 5*a^2*b*Cos[3*(c + d*x)] - 13*b^3*Cos[3*(c + d*x
)] + 12*a^3*Sin[c + d*x] - 30*a*b^2*Sin[c + d*x] + 4*a^3*Sin[3*(c + d*x)] - 22*a*b^2*Sin[3*(c + d*x)]))/(24*(a
- b)^2*(a + b)^2*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.96, size = 291, normalized size = 0.27

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a \textit {\_R} b +2 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (2 a -2 b \right )}+\frac {1}{\left (2 a -2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a -5 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (2 a +2 b \right )}-\frac {1}{\left (2 a +2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a +5 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(291\)
default	\(\frac {\frac {b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a \textit {\_R} b +2 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (2 a -2 b \right )}+\frac {1}{\left (2 a -2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a -5 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (2 a +2 b \right )}-\frac {1}{\left (2 a +2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a +5 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(291\)
risch	\(\text {Expression too large to display}\)	\(2867\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*b^2/(a-b)^2/(a+b)^2*sum(((2*a^2+b^2)*_R^4-6*a*b*_R^3+2*(4*a^2+5*b^2)*_R^2-6*a*_R*b+2*a^2+b^2)/(_R^5*a
+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-2/3/(tan(1/
2*d*x+1/2*c)+1)^3/(2*a-2*b)+1/(2*a-2*b)/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(2*a-5*b)/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)-
2/3/(tan(1/2*d*x+1/2*c)-1)^3/(2*a+2*b)-1/(2*a+2*b)/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(2*a+5*b)/(a+b)^2/(tan(1/2*d*x
+1/2*c)-1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [C] Result contains complex when optimal does not.
time = 11.21, size = 85064, normalized size = 77.83 \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)^4/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/12*(sqrt(2/3)*sqrt(1/6)*(a^4 - 2*a^2*b^2 + b^4)*d*sqrt(-(810*a^4*b^4 + 2754*a^2*b^6 + 810*b^8 - (a^10 - 5*a
^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*((5*b^6/(a^10*d^4 - 4*a^8*b^2*d^4 + 6*a^6*b^4*d^4 - 4*a^4
*b^6*d^4 + a^2*b^8*d^4) + 9*(5*a^4*b^4 + 17*a^2*b^6 + 5*b^8)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10
*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*b^8/(a^14*d^6 - 5*a^12*b^2*d^6 + 10*a^10
*b^4*d^6 - 10*a^8*b^6*d^6 + 5*a^6*b^8*d^6 - a^4*b^10*d^6) - 5/162*(5*a^4*b^4 + 17*a^2*b^6 + 5*b^8)*b^6/((a^10*
d^4 - 4*a^8*b^2*d^4 + 6*a^6*b^4*d^4 - 4*a^4*b^6*d^4 + a^2*b^8*d^4)*(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2
- 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)) - 1/27*(5*a^4*b^4 + 17*a^2*b^6 + 5*b^8)^3/(a^10*d^2 - 5*a^8*b^2*
d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 - 700*a^6*b^4
- 700*a^4*b^6 - 30*a^2*b^8 + b^10)*b^8/((a^2 - b^2)^10*a^4*d^6))^(1/3) + 81*(-1/1458*b^8/(a^14*d^6 - 5*a^12*b^
2*d^6 + 10 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{4}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4/(a+b*sin(d*x+c)**3),x)

[Out]

Integral(sec(c + d*x)**4/(a + b*sin(c + d*x)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^4/(b*sin(d*x + c)^3 + a), x)

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Mupad [B]
time = 25.85, size = 2500, normalized size = 2.29 \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)^4*(a + b*sin(c + d*x)^3)),x)

[Out]

(14*b^3*cos(c/2 + (d*x)/2)^6)/(3*(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d*x)/2)^6 - a^4*d*sin(c/2 + (d
*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*b^2*d*sin(c/2 + (d*x)/2)^6 +
3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2 + 3*b^4*
d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2 - 6*a^2*b^2*d*
cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2)) - (4*a^3*c
os(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^3)/(3*(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d*x)/2)^6 - a^4*d*
sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*b^2*d*sin(c/2 + (
d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2
)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2 -
6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2)
) + (6*b^3*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4)/(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d*x)/2)^6
- a^4*d*sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*b^2*d*si
n(c/2 + (d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2
+ (d*x)/2)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x
)/2)^2 - 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2)^2) - (8*b^3*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2)/(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d
*x)/2)^6 - a^4*d*sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*
b^2*d*sin(c/2 + (d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*
sin(c/2 + (d*x)/2)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/
2 + (d*x)/2)^2 - 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(
c/2 + (d*x)/2)^2) + (4*a^2*b*cos(c/2 + (d*x)/2)^6)/(3*(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d*x)/2)^6
- a^4*d*sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*b^2*d*si
n(c/2 + (d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2
+ (d*x)/2)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x
)/2)^2 - 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2)^2)) + (2*a^3*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^5)/(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2 + (d*
x)/2)^6 - a^4*d*sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2*a^2*b
^2*d*sin(c/2 + (d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/2)^4*s
in(c/2 + (d*x)/2)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*sin(c/2
+ (d*x)/2)^2 - 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4*sin(c
/2 + (d*x)/2)^2) + (2*a^3*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2))/(a^4*d*cos(c/2 + (d*x)/2)^6 + b^4*d*cos(c/2
+ (d*x)/2)^6 - a^4*d*sin(c/2 + (d*x)/2)^6 - b^4*d*sin(c/2 + (d*x)/2)^6 - 2*a^2*b^2*d*cos(c/2 + (d*x)/2)^6 + 2
*a^2*b^2*d*sin(c/2 + (d*x)/2)^6 + 3*a^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*a^4*d*cos(c/2 + (d*x)/
2)^4*sin(c/2 + (d*x)/2)^2 + 3*b^4*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 - 3*b^4*d*cos(c/2 + (d*x)/2)^4*s
in(c/2 + (d*x)/2)^2 - 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^4 + 6*a^2*b^2*d*cos(c/2 + (d*x)/2)^4
*sin(c/2 + (d*x)/2)^2) + (a^4*cos(c/2 + (d*x)/2)^6*symsum(log(-(8192*(6*a^2*b^38*cos(c/2 + (d*x)/2) - 20*a*b^3
9*sin(c/2 + (d*x)/2) - 84*a^4*b^36*cos(c/2 + (d*x)/2) + 546*a^6*b^34*cos(c/2 + (d*x)/2) - 2184*a^8*b^32*cos(c/
2 + (d*x)/2) + 6006*a^10*b^30*cos(c/2 + (d*x)/2) - 12012*a^12*b^28*cos(c/2 + (d*x)/2) + 18018*a^14*b^26*cos(c/
2 + (d*x)/2) - 20592*a^16*b^24*cos(c/2 + (d*x)/2) + 18018*a^18*b^22*cos(c/2 + (d*x)/2) - 12012*a^20*b^20*cos(c
/2 + (d*x)/2) + 6006*a^22*b^18*cos(c/2 + (d*x)/2) - 2184*a^24*b^16*cos(c/2 + (d*x)/2) + 546*a^26*b^14*cos(c/2
+ (d*x)/2) - 84*a^28*b^12*cos(c/2 + (d*x)/2) + 6*a^30*b^10*cos(c/2 + (d*x)/2) + 280*a^3*b^37*sin(c/2 + (d*x)/2
) - 1820*a^5*b^35*sin(c/2 + (d*x)/2) + 7280*a^7*b^33*sin(c/2 + (d*x)/2) - 20020*a^9*b^31*sin(c/2 + (d*x)/2) +
40040*a^11*b^29*sin(c/2 + (d*x)/2) - 60060*a^13*b^27*sin(c/2 + (d*x)/2) + 68640*a^15*b^25*sin(c/2 + (d*x)/2) -
60060*a^17*b^23*sin(c/2 + (d*x)/2) + 40040*a^19*b^21*sin(c/2 + (d*x)/2) - 20020*a^21*b^19*sin(c/2 + (d*x)/2)
+ 7280*a^23*b^17*sin(c/2 + (d*x)/2) - 1820*a^25*b^15*sin(c/2 + (d*x)/2) + 280*a^27*b^13*sin(c/2 + (d*x)/2) - 2
0*a^29*b^11*sin(c/2 + (d*x)/2) - 588*root(7290*...

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