Optimal. Leaf size=220 \[ -\frac {8 \sqrt {2 \pi } b^{5/2} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 \sqrt {2 \pi } b^{5/2} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \sinh (a+b x) \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}+\frac {16 b^2}{15 d^3 \sqrt {c+d x}} \]
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Rubi [A] time = 0.31, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3314, 32, 3313, 12, 3308, 2180, 2204, 2205} \[ -\frac {8 \sqrt {2 \pi } b^{5/2} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 \sqrt {2 \pi } b^{5/2} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \sinh (a+b x) \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}+\frac {16 b^2}{15 d^3 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 3313
Rule 3314
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x)}{(c+d x)^{7/2}} \, dx &=-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {\left (8 b^2\right ) \int \frac {1}{(c+d x)^{3/2}} \, dx}{15 d^2}+\frac {\left (16 b^2\right ) \int \frac {\cosh ^2(a+b x)}{(c+d x)^{3/2}} \, dx}{15 d^2}\\ &=\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}+\frac {\left (64 i b^3\right ) \int -\frac {i \sinh (2 a+2 b x)}{2 \sqrt {c+d x}} \, dx}{15 d^3}\\ &=\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}+\frac {\left (32 b^3\right ) \int \frac {\sinh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{15 d^3}\\ &=\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}+\frac {\left (16 b^3\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3}-\frac {\left (16 b^3\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3}\\ &=\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {\left (32 b^3\right ) \operatorname {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{15 d^4}+\frac {\left (32 b^3\right ) \operatorname {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{15 d^4}\\ &=\frac {16 b^2}{15 d^3 \sqrt {c+d x}}-\frac {2 \cosh ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac {32 b^2 \cosh ^2(a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {8 b^{5/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {8 b^{5/2} e^{2 a-\frac {2 b c}{d}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{15 d^2 (c+d x)^{3/2}}\\ \end {align*}
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Mathematica [B] time = 3.17, size = 825, normalized size = 3.75 \[ \frac {e^{-\frac {2 b (c+d x)}{d}} \left (16 \sqrt {2} d^2 e^{\frac {2 b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac {2 b c}{d}\right )+\sinh \left (2 a-\frac {2 b c}{d}\right )\right ) \left (-\frac {b (c+d x)}{d}\right )^{5/2}-6 d^2 e^{\frac {2 b (c+d x)}{d}}-16 b^2 c^2 e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-4 b c d e^{\frac {4 b (c+d x)}{d}} \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 c^2 \cosh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 e^{\frac {4 b (c+d x)}{d}} x^2 \cosh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 x^2 \cosh \left (2 a-\frac {2 b c}{d}\right )+4 b c d \cosh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 e^{\frac {4 b (c+d x)}{d}} x \cosh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d e^{\frac {4 b (c+d x)}{d}} x \cosh \left (2 a-\frac {2 b c}{d}\right )+4 b d^2 x \cosh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d x \cosh \left (2 a-\frac {2 b c}{d}\right )+16 \sqrt {2} d^2 e^{\frac {2 b (c+d x)}{d}} \left (\frac {b (c+d x)}{d}\right )^{5/2} \Gamma \left (\frac {1}{2},\frac {2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac {2 b c}{d}\right )-\sinh \left (2 a-\frac {2 b c}{d}\right )\right )-16 b^2 c^2 e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )-3 d^2 e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b c d e^{\frac {4 b (c+d x)}{d}} \sinh \left (2 a-\frac {2 b c}{d}\right )+16 b^2 c^2 \sinh \left (2 a-\frac {2 b c}{d}\right )+3 d^2 \sinh \left (2 a-\frac {2 b c}{d}\right )-16 b^2 d^2 e^{\frac {4 b (c+d x)}{d}} x^2 \sinh \left (2 a-\frac {2 b c}{d}\right )+16 b^2 d^2 x^2 \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b c d \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 e^{\frac {4 b (c+d x)}{d}} x \sinh \left (2 a-\frac {2 b c}{d}\right )-32 b^2 c d e^{\frac {4 b (c+d x)}{d}} x \sinh \left (2 a-\frac {2 b c}{d}\right )-4 b d^2 x \sinh \left (2 a-\frac {2 b c}{d}\right )+32 b^2 c d x \sinh \left (2 a-\frac {2 b c}{d}\right )\right )}{30 d^3 (c+d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 1350, normalized size = 6.14 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}\left (b x +a \right )}{\left (d x +c \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 116, normalized size = 0.53 \[ -\frac {\frac {5 \, \sqrt {2} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {5 \, \sqrt {2} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {1}{{\left (d x + c\right )}^{\frac {5}{2}}}}{5 \, d} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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