Optimal. Leaf size=322 \[ \frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2708, 2709,
3557, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \sqrt {\tan (a+b x)} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {\tan (a+b x)} \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {d \csc (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2708
Rule 2709
Rule 3557
Rubi steps
\begin {align*} \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}} \, dx}{4 c^2}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \int \frac {1}{\sqrt {\tan (a+b x)}} \, dx}{4 c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{4 b c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{4 b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{4 b c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (a+b x)}\right )}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 1.55, size = 152, normalized size = 0.47 \begin {gather*} \frac {d \left (4 \cos ^2(a+b x)-3 \sqrt {2} \text {ArcTan}\left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}+3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}\right ) \sqrt {c \sec (a+b x)}}{8 b c^3 \sqrt {d \csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 34.88, size = 668, normalized size = 2.07
method | result | size |
default | \(\frac {\left (3 i \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 i \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+6 \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-2 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}\right ) \sqrt {\frac {d}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \sqrt {2}}{8 b \left (-1+\cos \left (b x +a \right )\right ) \cos \left (b x +a \right )^{3} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}}}\) | \(668\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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