Optimal. Leaf size=380 \[ -\frac {\left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (3 a^2-5 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {5 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.74, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2881, 3135,
3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} -\frac {5 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\left (3 a^2-5 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2881
Rule 2884
Rule 2886
Rule 3081
Rule 3134
Rule 3135
Rule 3138
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx &=\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (-\frac {5 b}{2}+\frac {3}{2} b \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{a}\\ &=\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\left (-\frac {15}{4} b \left (a^2-b^2\right )+\frac {3}{2} a b^2 \cos (c+d x)+\frac {1}{4} b \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\left (-\frac {15}{8} b \left (a^2-b^2\right )^2+\frac {1}{4} a b^2 \left (9 a^2-5 b^2\right ) \cos (c+d x)-\frac {1}{8} b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {4 \int \frac {\left (\frac {15}{8} b^2 \left (a^2-b^2\right )^2-\frac {1}{8} a b \left (3 a^4-8 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^3 b \left (a^2-b^2\right )^2}-\frac {\left (3 a^4-26 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {(5 b) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a^3}+\frac {\left (3 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{6 a^2 \left (a^2-b^2\right )}-\frac {\left (\left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{6 a^3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=-\frac {\left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\left (5 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{2 a^3 \sqrt {a+b \cos (c+d x)}}+\frac {\left (\left (3 a^2-5 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{6 a^2 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {\left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (3 a^2-5 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {5 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (3 a^2-5 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {b \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 16.61, size = 638, normalized size = 1.68 \begin {gather*} -\frac {b \left (\frac {2 \left (-36 a^3 b+20 a b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (33 a^4-86 a^2 b^2+45 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (3 a^4-26 a^2 b^2+15 b^4\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}\right )}{12 a^3 (-a+b)^2 (a+b)^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (-\frac {2 b^3 \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {4 \left (5 a^2 b^3 \sin (c+d x)-3 b^5 \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {\tan (c+d x)}{a^3}\right )}{d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1323\) vs.
\(2(441)=882\).
time = 0.91, size = 1324, normalized size = 3.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(1324\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \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 {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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